diogo
/
NumericalAnalysisModule
Archived
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Solutions for FCUP's Numerical Analysis Assignments

This commit is contained in:
Diogo Cordeiro 2019-08-28 16:44:59 +01:00
commit b7b704b68b
108 changed files with 35323 additions and 0 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

674
1 - Numerical Series Approximation/LICENSE Normal file
View File

@ -0,0 +1,674 @@
GNU GENERAL PUBLIC LICENSE
Version 3, 29 June 2007
Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
Preamble
The GNU General Public License is a free, copyleft license for
software and other kinds of works.
The licenses for most software and other practical works are designed
to take away your freedom to share and change the works. By contrast,
the GNU General Public License is intended to guarantee your freedom to
share and change all versions of a program--to make sure it remains free
software for all its users. We, the Free Software Foundation, use the
GNU General Public License for most of our software; it applies also to
any other work released this way by its authors. You can apply it to
your programs, too.
When we speak of free software, we are referring to freedom, not
price. Our General Public Licenses are designed to make sure that you
have the freedom to distribute copies of free software (and charge for
them if you wish), that you receive source code or can get it if you
want it, that you can change the software or use pieces of it in new
free programs, and that you know you can do these things.
To protect your rights, we need to prevent others from denying you
these rights or asking you to surrender the rights. Therefore, you have
certain responsibilities if you distribute copies of the software, or if
you modify it: responsibilities to respect the freedom of others.
For example, if you distribute copies of such a program, whether
gratis or for a fee, you must pass on to the recipients the same
freedoms that you received. You must make sure that they, too, receive
or can get the source code. And you must show them these terms so they
know their rights.
Developers that use the GNU GPL protect your rights with two steps:
(1) assert copyright on the software, and (2) offer you this License
giving you legal permission to copy, distribute and/or modify it.
For the developers' and authors' protection, the GPL clearly explains
that there is no warranty for this free software. For both users' and
authors' sake, the GPL requires that modified versions be marked as
changed, so that their problems will not be attributed erroneously to
authors of previous versions.
Some devices are designed to deny users access to install or run
modified versions of the software inside them, although the manufacturer
can do so. This is fundamentally incompatible with the aim of
protecting users' freedom to change the software. The systematic
pattern of such abuse occurs in the area of products for individuals to
use, which is precisely where it is most unacceptable. Therefore, we
have designed this version of the GPL to prohibit the practice for those
products. If such problems arise substantially in other domains, we
stand ready to extend this provision to those domains in future versions
of the GPL, as needed to protect the freedom of users.
Finally, every program is threatened constantly by software patents.
States should not allow patents to restrict development and use of
software on general-purpose computers, but in those that do, we wish to
avoid the special danger that patents applied to a free program could
make it effectively proprietary. To prevent this, the GPL assures that
patents cannot be used to render the program non-free.
The precise terms and conditions for copying, distribution and
modification follow.
TERMS AND CONDITIONS
0. Definitions.
"This License" refers to version 3 of the GNU General Public License.
"Copyright" also means copyright-like laws that apply to other kinds of
works, such as semiconductor masks.
"The Program" refers to any copyrightable work licensed under this
License. Each licensee is addressed as "you". "Licensees" and
"recipients" may be individuals or organizations.
To "modify" a work means to copy from or adapt all or part of the work
in a fashion requiring copyright permission, other than the making of an
exact copy. The resulting work is called a "modified version" of the
earlier work or a work "based on" the earlier work.
A "covered work" means either the unmodified Program or a work based
on the Program.
To "propagate" a work means to do anything with it that, without
permission, would make you directly or secondarily liable for
infringement under applicable copyright law, except executing it on a
computer or modifying a private copy. Propagation includes copying,
distribution (with or without modification), making available to the
public, and in some countries other activities as well.
To "convey" a work means any kind of propagation that enables other
parties to make or receive copies. Mere interaction with a user through
a computer network, with no transfer of a copy, is not conveying.
An interactive user interface displays "Appropriate Legal Notices"
to the extent that it includes a convenient and prominently visible
feature that (1) displays an appropriate copyright notice, and (2)
tells the user that there is no warranty for the work (except to the
extent that warranties are provided), that licensees may convey the
work under this License, and how to view a copy of this License. If
the interface presents a list of user commands or options, such as a
menu, a prominent item in the list meets this criterion.
1. Source Code.
The "source code" for a work means the preferred form of the work
for making modifications to it. "Object code" means any non-source
form of a work.
A "Standard Interface" means an interface that either is an official
standard defined by a recognized standards body, or, in the case of
interfaces specified for a particular programming language, one that
is widely used among developers working in that language.
The "System Libraries" of an executable work include anything, other
than the work as a whole, that (a) is included in the normal form of
packaging a Major Component, but which is not part of that Major
Component, and (b) serves only to enable use of the work with that
Major Component, or to implement a Standard Interface for which an
implementation is available to the public in source code form. A
"Major Component", in this context, means a major essential component
(kernel, window system, and so on) of the specific operating system
(if any) on which the executable work runs, or a compiler used to
produce the work, or an object code interpreter used to run it.
The "Corresponding Source" for a work in object code form means all
the source code needed to generate, install, and (for an executable
work) run the object code and to modify the work, including scripts to
control those activities. However, it does not include the work's
System Libraries, or general-purpose tools or generally available free
programs which are used unmodified in performing those activities but
which are not part of the work. For example, Corresponding Source
includes interface definition files associated with source files for
the work, and the source code for shared libraries and dynamically
linked subprograms that the work is specifically designed to require,
such as by intimate data communication or control flow between those
subprograms and other parts of the work.
The Corresponding Source need not include anything that users
can regenerate automatically from other parts of the Corresponding
Source.
The Corresponding Source for a work in source code form is that
same work.
2. Basic Permissions.
All rights granted under this License are granted for the term of
copyright on the Program, and are irrevocable provided the stated
conditions are met. This License explicitly affirms your unlimited
permission to run the unmodified Program. The output from running a
covered work is covered by this License only if the output, given its
content, constitutes a covered work. This License acknowledges your
rights of fair use or other equivalent, as provided by copyright law.
You may make, run and propagate covered works that you do not
convey, without conditions so long as your license otherwise remains
in force. You may convey covered works to others for the sole purpose
of having them make modifications exclusively for you, or provide you
with facilities for running those works, provided that you comply with
the terms of this License in conveying all material for which you do
not control copyright. Those thus making or running the covered works
for you must do so exclusively on your behalf, under your direction
and control, on terms that prohibit them from making any copies of
your copyrighted material outside their relationship with you.
Conveying under any other circumstances is permitted solely under
the conditions stated below. Sublicensing is not allowed; section 10
makes it unnecessary.
3. Protecting Users' Legal Rights From Anti-Circumvention Law.
No covered work shall be deemed part of an effective technological
measure under any applicable law fulfilling obligations under article
11 of the WIPO copyright treaty adopted on 20 December 1996, or
similar laws prohibiting or restricting circumvention of such
measures.
When you convey a covered work, you waive any legal power to forbid
circumvention of technological measures to the extent such circumvention
is effected by exercising rights under this License with respect to
the covered work, and you disclaim any intention to limit operation or
modification of the work as a means of enforcing, against the work's
users, your or third parties' legal rights to forbid circumvention of
technological measures.
4. Conveying Verbatim Copies.
You may convey verbatim copies of the Program's source code as you
receive it, in any medium, provided that you conspicuously and
appropriately publish on each copy an appropriate copyright notice;
keep intact all notices stating that this License and any
non-permissive terms added in accord with section 7 apply to the code;
keep intact all notices of the absence of any warranty; and give all
recipients a copy of this License along with the Program.
You may charge any price or no price for each copy that you convey,
and you may offer support or warranty protection for a fee.
5. Conveying Modified Source Versions.
You may convey a work based on the Program, or the modifications to
produce it from the Program, in the form of source code under the
terms of section 4, provided that you also meet all of these conditions:
a) The work must carry prominent notices stating that you modified
it, and giving a relevant date.
b) The work must carry prominent notices stating that it is
released under this License and any conditions added under section
7. This requirement modifies the requirement in section 4 to
"keep intact all notices".
c) You must license the entire work, as a whole, under this
License to anyone who comes into possession of a copy. This
License will therefore apply, along with any applicable section 7
additional terms, to the whole of the work, and all its parts,
regardless of how they are packaged. This License gives no
permission to license the work in any other way, but it does not
invalidate such permission if you have separately received it.
d) If the work has interactive user interfaces, each must display
Appropriate Legal Notices; however, if the Program has interactive
interfaces that do not display Appropriate Legal Notices, your
work need not make them do so.
A compilation of a covered work with other separate and independent
works, which are not by their nature extensions of the covered work,
and which are not combined with it such as to form a larger program,
in or on a volume of a storage or distribution medium, is called an
"aggregate" if the compilation and its resulting copyright are not
used to limit the access or legal rights of the compilation's users
beyond what the individual works permit. Inclusion of a covered work
in an aggregate does not cause this License to apply to the other
parts of the aggregate.
6. Conveying Non-Source Forms.
You may convey a covered work in object code form under the terms
of sections 4 and 5, provided that you also convey the
machine-readable Corresponding Source under the terms of this License,
in one of these ways:
a) Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by the
Corresponding Source fixed on a durable physical medium
customarily used for software interchange.
b) Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by a
written offer, valid for at least three years and valid for as
long as you offer spare parts or customer support for that product
model, to give anyone who possesses the object code either (1) a
copy of the Corresponding Source for all the software in the
product that is covered by this License, on a durable physical
medium customarily used for software interchange, for a price no
more than your reasonable cost of physically performing this
conveying of source, or (2) access to copy the
Corresponding Source from a network server at no charge.
c) Convey individual copies of the object code with a copy of the
written offer to provide the Corresponding Source. This
alternative is allowed only occasionally and noncommercially, and
only if you received the object code with such an offer, in accord
with subsection 6b.
d) Convey the object code by offering access from a designated
place (gratis or for a charge), and offer equivalent access to the
Corresponding Source in the same way through the same place at no
further charge. You need not require recipients to copy the
Corresponding Source along with the object code. If the place to
copy the object code is a network server, the Corresponding Source
may be on a different server (operated by you or a third party)
that supports equivalent copying facilities, provided you maintain
clear directions next to the object code saying where to find the
Corresponding Source. Regardless of what server hosts the
Corresponding Source, you remain obligated to ensure that it is
available for as long as needed to satisfy these requirements.
e) Convey the object code using peer-to-peer transmission, provided
you inform other peers where the object code and Corresponding
Source of the work are being offered to the general public at no
charge under subsection 6d.
A separable portion of the object code, whose source code is excluded
from the Corresponding Source as a System Library, need not be
included in conveying the object code work.
A "User Product" is either (1) a "consumer product", which means any
tangible personal property which is normally used for personal, family,
or household purposes, or (2) anything designed or sold for incorporation
into a dwelling. In determining whether a product is a consumer product,
doubtful cases shall be resolved in favor of coverage. For a particular
product received by a particular user, "normally used" refers to a
typical or common use of that class of product, regardless of the status
of the particular user or of the way in which the particular user
actually uses, or expects or is expected to use, the product. A product
is a consumer product regardless of whether the product has substantial
commercial, industrial or non-consumer uses, unless such uses represent
the only significant mode of use of the product.
"Installation Information" for a User Product means any methods,
procedures, authorization keys, or other information required to install
and execute modified versions of a covered work in that User Product from
a modified version of its Corresponding Source. The information must
suffice to ensure that the continued functioning of the modified object
code is in no case prevented or interfered with solely because
modification has been made.
If you convey an object code work under this section in, or with, or
specifically for use in, a User Product, and the conveying occurs as
part of a transaction in which the right of possession and use of the
User Product is transferred to the recipient in perpetuity or for a
fixed term (regardless of how the transaction is characterized), the
Corresponding Source conveyed under this section must be accompanied
by the Installation Information. But this requirement does not apply
if neither you nor any third party retains the ability to install
modified object code on the User Product (for example, the work has
been installed in ROM).
The requirement to provide Installation Information does not include a
requirement to continue to provide support service, warranty, or updates
for a work that has been modified or installed by the recipient, or for
the User Product in which it has been modified or installed. Access to a
network may be denied when the modification itself materially and
adversely affects the operation of the network or violates the rules and
protocols for communication across the network.
Corresponding Source conveyed, and Installation Information provided,
in accord with this section must be in a format that is publicly
documented (and with an implementation available to the public in
source code form), and must require no special password or key for
unpacking, reading or copying.
7. Additional Terms.
"Additional permissions" are terms that supplement the terms of this
License by making exceptions from one or more of its conditions.
Additional permissions that are applicable to the entire Program shall
be treated as though they were included in this License, to the extent
that they are valid under applicable law. If additional permissions
apply only to part of the Program, that part may be used separately
under those permissions, but the entire Program remains governed by
this License without regard to the additional permissions.
When you convey a copy of a covered work, you may at your option
remove any additional permissions from that copy, or from any part of
it. (Additional permissions may be written to require their own
removal in certain cases when you modify the work.) You may place
additional permissions on material, added by you to a covered work,
for which you have or can give appropriate copyright permission.
Notwithstanding any other provision of this License, for material you
add to a covered work, you may (if authorized by the copyright holders of
that material) supplement the terms of this License with terms:
a) Disclaiming warranty or limiting liability differently from the
terms of sections 15 and 16 of this License; or
b) Requiring preservation of specified reasonable legal notices or
author attributions in that material or in the Appropriate Legal
Notices displayed by works containing it; or
c) Prohibiting misrepresentation of the origin of that material, or
requiring that modified versions of such material be marked in
reasonable ways as different from the original version; or
d) Limiting the use for publicity purposes of names of licensors or
authors of the material; or
e) Declining to grant rights under trademark law for use of some
trade names, trademarks, or service marks; or
f) Requiring indemnification of licensors and authors of that
material by anyone who conveys the material (or modified versions of
it) with contractual assumptions of liability to the recipient, for
any liability that these contractual assumptions directly impose on
those licensors and authors.
All other non-permissive additional terms are considered "further
restrictions" within the meaning of section 10. If the Program as you
received it, or any part of it, contains a notice stating that it is
governed by this License along with a term that is a further
restriction, you may remove that term. If a license document contains
a further restriction but permits relicensing or conveying under this
License, you may add to a covered work material governed by the terms
of that license document, provided that the further restriction does
not survive such relicensing or conveying.
If you add terms to a covered work in accord with this section, you
must place, in the relevant source files, a statement of the
additional terms that apply to those files, or a notice indicating
where to find the applicable terms.
Additional terms, permissive or non-permissive, may be stated in the
form of a separately written license, or stated as exceptions;
the above requirements apply either way.
8. Termination.
You may not propagate or modify a covered work except as expressly
provided under this License. Any attempt otherwise to propagate or
modify it is void, and will automatically terminate your rights under
this License (including any patent licenses granted under the third
paragraph of section 11).
However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
provisionally, unless and until the copyright holder explicitly and
finally terminates your license, and (b) permanently, if the copyright
holder fails to notify you of the violation by some reasonable means
prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from that
copyright holder, and you cure the violation prior to 30 days after
your receipt of the notice.
Termination of your rights under this section does not terminate the
licenses of parties who have received copies or rights from you under
this License. If your rights have been terminated and not permanently
reinstated, you do not qualify to receive new licenses for the same
material under section 10.
9. Acceptance Not Required for Having Copies.
You are not required to accept this License in order to receive or
run a copy of the Program. Ancillary propagation of a covered work
occurring solely as a consequence of using peer-to-peer transmission
to receive a copy likewise does not require acceptance. However,
nothing other than this License grants you permission to propagate or
modify any covered work. These actions infringe copyright if you do
not accept this License. Therefore, by modifying or propagating a
covered work, you indicate your acceptance of this License to do so.
10. Automatic Licensing of Downstream Recipients.
Each time you convey a covered work, the recipient automatically
receives a license from the original licensors, to run, modify and
propagate that work, subject to this License. You are not responsible
for enforcing compliance by third parties with this License.
An "entity transaction" is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations. If propagation of a covered
work results from an entity transaction, each party to that
transaction who receives a copy of the work also receives whatever
licenses to the work the party's predecessor in interest had or could
give under the previous paragraph, plus a right to possession of the
Corresponding Source of the work from the predecessor in interest, if
the predecessor has it or can get it with reasonable efforts.
You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License. For example, you may
not impose a license fee, royalty, or other charge for exercise of
rights granted under this License, and you may not initiate litigation
(including a cross-claim or counterclaim in a lawsuit) alleging that
any patent claim is infringed by making, using, selling, offering for
sale, or importing the Program or any portion of it.
11. Patents.
A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based. The
work thus licensed is called the contributor's "contributor version".
A contributor's "essential patent claims" are all patent claims
owned or controlled by the contributor, whether already acquired or
hereafter acquired, that would be infringed by some manner, permitted
by this License, of making, using, or selling its contributor version,
but do not include claims that would be infringed only as a
consequence of further modification of the contributor version. For
purposes of this definition, "control" includes the right to grant
patent sublicenses in a manner consistent with the requirements of
this License.
Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims, to
make, use, sell, offer for sale, import and otherwise run, modify and
propagate the contents of its contributor version.
In the following three paragraphs, a "patent license" is any express
agreement or commitment, however denominated, not to enforce a patent
(such as an express permission to practice a patent or covenant not to
sue for patent infringement). To "grant" such a patent license to a
party means to make such an agreement or commitment not to enforce a
patent against the party.
If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone
to copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
available, or (2) arrange to deprive yourself of the benefit of the
patent license for this particular work, or (3) arrange, in a manner
consistent with the requirements of this License, to extend the patent
license to downstream recipients. "Knowingly relying" means you have
actual knowledge that, but for the patent license, your conveying the
covered work in a country, or your recipient's use of the covered work
in a country, would infringe one or more identifiable patents in that
country that you have reason to believe are valid.
If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license to some of the parties
receiving the covered work authorizing them to use, propagate, modify
or convey a specific copy of the covered work, then the patent license
you grant is automatically extended to all recipients of the covered
work and works based on it.
A patent license is "discriminatory" if it does not include within
the scope of its coverage, prohibits the exercise of, or is
conditioned on the non-exercise of one or more of the rights that are
specifically granted under this License. You may not convey a covered
work if you are a party to an arrangement with a third party that is
in the business of distributing software, under which you make payment
to the third party based on the extent of your activity of conveying
the work, and under which the third party grants, to any of the
parties who would receive the covered work from you, a discriminatory
patent license (a) in connection with copies of the covered work
conveyed by you (or copies made from those copies), or (b) primarily
for and in connection with specific products or compilations that
contain the covered work, unless you entered into that arrangement,
or that patent license was granted, prior to 28 March 2007.
Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may
otherwise be available to you under applicable patent law.
12. No Surrender of Others' Freedom.
If conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License. If you cannot convey a
covered work so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you may
not convey it at all. For example, if you agree to terms that obligate you
to collect a royalty for further conveying from those to whom you convey
the Program, the only way you could satisfy both those terms and this
License would be to refrain entirely from conveying the Program.
13. Use with the GNU Affero General Public License.
Notwithstanding any other provision of this License, you have
permission to link or combine any covered work with a work licensed
under version 3 of the GNU Affero General Public License into a single
combined work, and to convey the resulting work. The terms of this
License will continue to apply to the part which is the covered work,
but the special requirements of the GNU Affero General Public License,
section 13, concerning interaction through a network will apply to the
combination as such.
14. Revised Versions of this License.
The Free Software Foundation may publish revised and/or new versions of
the GNU General Public License from time to time. Such new versions will
be similar in spirit to the present version, but may differ in detail to
address new problems or concerns.
Each version is given a distinguishing version number. If the
Program specifies that a certain numbered version of the GNU General
Public License "or any later version" applies to it, you have the
option of following the terms and conditions either of that numbered
version or of any later version published by the Free Software
Foundation. If the Program does not specify a version number of the
GNU General Public License, you may choose any version ever published
by the Free Software Foundation.
If the Program specifies that a proxy can decide which future
versions of the GNU General Public License can be used, that proxy's
public statement of acceptance of a version permanently authorizes you
to choose that version for the Program.
Later license versions may give you additional or different
permissions. However, no additional obligations are imposed on any
author or copyright holder as a result of your choosing to follow a
later version.
15. Disclaimer of Warranty.
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
16. Limitation of Liability.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
SUCH DAMAGES.
17. Interpretation of Sections 15 and 16.
If the disclaimer of warranty and limitation of liability provided
above cannot be given local legal effect according to their terms,
reviewing courts shall apply local law that most closely approximates
an absolute waiver of all civil liability in connection with the
Program, unless a warranty or assumption of liability accompanies a
copy of the Program in return for a fee.
END OF TERMS AND CONDITIONS
How to Apply These Terms to Your New Programs
If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.
To do so, attach the following notices to the program. It is safest
to attach them to the start of each source file to most effectively
state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.
<one line to give the program's name and a brief idea of what it does.>
Copyright (C) <year> <name of author>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:
<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<https://www.gnu.org/licenses/>.
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<https://www.gnu.org/licenses/why-not-lgpl.html>.

256
1 - Numerical Series Approximation/README.md Normal file
View File

@ -0,0 +1,256 @@
---
title: "Análise Numérica - Trabalho Prático 1"
author:
- Diogo Cordeiro
- Hugo Sales
- Pedro Costa
geometry: margin=2cm
output: pdf_document
---
## Motivação
Perceber e analisar técnicas de aproximação de séries numéricas e estratégias
para controlo do erro de cálculos em computador, assim como implementar
algoritmos com recurso a métodos numéricos adequados e interpretar os resultados.
## Questão 1
O _eps_ está definido como a diferença entre 1.0 e o menor valor
representável superior a um, por exemplo, 2^-23^, em precisão simples, e 2^-52^
em precisão dupla (em ambos os casos $Base^{-(precisão-1)}$). O ISO 9899 C define:
"a diferença entre 1 e o menor valor superior a 1 que é
representável num dado tipo de ponto flutuante"
Verificamos a existência de uma definição alternativa: "o menor número que
somado com 1 resulta num número maior do que 1". Não usamos esta definição, uma
vez que devido ao arredondamento para o valor mais próximo, bastaria usar um
valor ligeiramente maior do que metade do nosso _eps_ para satisfazer
a condição. E também porque os standards ISO
optam por "passo entre valores adjacentes".
O código seguinte implementa o cálculo de
$$ (2 - \sum _{n=1}^{\infty}\:\frac{1}{2^n}) - 1 = \textit{eps} $$
onde
$$ \lim _{n\to \infty }\sum _{n=1}^{\infty}\:\frac{1}{2^n} = 1^{-} $$
Logo este cálculo converge para 1 por números superiores.
\pagebreak
double machine_eps() {
// Pela definição, este valor tem que ser superior a 1, e
// como a máquina usa um sistema de vírgula flutuante de base 2,
// este tem que ser uma potência de 2
double epsilon_candidate = 2.0,
epsilon = epsilon_candidate,
// Primeiro termo da série geométrica com proporção 1/2,
// que converge para 1
power = 2.0;
// Diferente de 1 pela definição
while (epsilon_candidate != 1.0) {
epsilon = epsilon_candidate;
// Aproximar o epsilon candidate do epsilon,
// reduzindo este com a acumulação do termo da sucessão
epsilon_candidate -= 1/power;
// Razão da sucessão
power *= 2;
}
return epsilon-1.0;
}
Com a execução deste código obtivemos 2.2204460492503131e-16, o mesmo valor
definido, para aritmética IEEE, na C library `float.h`.
#define DBL_EPSILON 2.2204460492503131e-16
## Questão 2
Utilizamos o seguinte código na linguagem Java para computar a série dada e
obtivemos os resultados apresentados na tabela que segue.
Para evitar o cálculo de fatoriais e de grande magnitude, o que diminuiria a
performace e aumentaria o erro e a possibilidade de overflow,
optamos usar a definição por recorrência e por simplificar algebricamente o
cálculo dos termos da série, obtendo uma expressão mais simples.
$$a_{k+1} = a_k \cdot \frac{k + 1}{4 \cdot k + 6}$$
/**
* Computar a série 2 com um erro absoluto inferior a um dado
* epsilon
*/
public static double compute_series_2(double epsilon) {
// Fator constante
double factor = 9.0/(double)(2.0*Math.sqrt(3));
// Tirado do critério D'Alembert para L = 0.5 < 1
double super_L = 1.0/(double)(1-0.25);
// Index do sumatório
int k = 0;
// Acumulação do sumatório
double acc = 0;
// O valor do termo actual da série
double a = 1.0f; // a with k = 0
// Enquanto o nosso erro absoluto é superior ao
// epsilon dado
while(epsilon < factor*a*super_L) {
// Acumula com o termo anterior
acc += a;
// Computa o termo seguinte e posteriormente
// incrementa k
a = compute_serie_2_term(k++) * a;
}
System.out.println(factor*acc + " " + (k - 1));
}
/**
* Computa o termo k da série 2 dado um anterior
*/
public static double compute_serie_2_term(int k) {
return (double)(k+1.0f)/(double)(4.0f*k+6.0f);
}
### Tabela de resultados
$-log(\epsilon)$ | $S_n$ | Iterações | Tempo (s)
-----------------|---------------------|-------------|----------
8 | 3.141592651 | 13 | 0.094
9 | 3.1415926529 | 14 | 0.099
10 | 3.14159265355 | 16 | 0.100
11 | 3.141592653587 | 18 | 0.100
12 | 3.1415926535892 | 19 | 0.095
13 | 3.14159265358976 | 21 | 0.101
14 | 3.141592653589785 | 22 | 0.097
15 | 3.1415926535897936 | 24 | 0.097
A série aparenta aproximar o valor de $\pi$.
Reparamos que geralmente a ordem de grandeza do erro indica o número de casas
decimais exatas obtidas, contudo o valor obtido para $-log(\epsilon)$ igual a
9 e 14 produz um resultado com menos uma casa decimal exata. Ainda assim
o valor apresentado representa $\pi$ com erro absoluto inferior a
$5 \cdot 10^{-10}$ e $5 \cdot 10^{-15}$ respetivamente.
## Questão 3
Inicialmente implementamos o cálculo do valor aproximado desta série em Java,
mas deparamo-nos com um longo tempo de execução devido ao elevado número de
iterações necessárias para aproximar a série com o $\epsilon$ pretendido, pelo
que decidimos testar uma implementação na linguagem C++, na qual obtivemos maior
performace, o que permitiu o cãlculo para um valor menor de $\epsilon$ em tempo
útil.
double compute_serie_3_term(unsigned long n) {
return -((double)(2*n+1))/(double)(2*n+3);
}
void compute_serie_3(double err) {
// Termo atual
unsigned long k = 0;
// Valor do termo atual
double ak = 1;
// Acumulador do valor da série
double acc = 1;
// Parar quando o erro obtido for inferior ao pretendido
while (err < 4*std::abs(ak)) {
// Calcular o termo seguinte e incrementar k
ak = compute_serie_3_term(k++) * ak;
// Somar o termo ao total
acc += ak;
}
std::cout << k << " " << 4*acc << '\n';
}
### Tabela de Resultados em Java
$-log(\epsilon)$ | $S_n$ | Iterações | Tempo (s)
-----------------|---------------|-------------|-----------
8 | 3.141592659 | 200000000 | 2.237
9 | 3.1415926541 | 2000000000 | 21.522
10 | 3.14159265364 | 20000000012 | 215.442
### Tabela de Resultados em C++
$-log(\epsilon)$ | $S_n$ | Iterações | Tempo (s)
------------------|--------------------|----------------|----------
8 | 3.141592659 | 200000000 | 0.787
9 | 3.1415926541 | 2000000000 | 7.862
10 | 3.14159265364 | 20000000012 | 79.130
11 | 3.141592653593 | 200000002870 | 791.625
12 | 3.1415926535878 | 2000000614027 | 7871.842
Observando as tabelas notamos um padrão, e, motivados por establecer uma relação
entre $\epsilon$ e tanto o números de iterações e o tempo de execução, decidimos
traçar um gráfico com estes valores.
![Tempo em função de $\epsilon$](graficos/questao3_tempos.png){ width=12cm }
![Iterações em função de $\epsilon$](graficos/questao3_iteracoes.png){ width=12cm }
\pagebreak
Destes dois gráficos concluímos que o tempo cresce exponencialmente em função do
valor de $-log(\epsilon)$, sendo que ambos crescem com um fator de 10,
correspondendo aproximadamente a um número da forma $5*10^{-(\epsilon + 1)}$, mas
não exatamente a este valor, sendo que quando $\epsilon$ diminuiu, o valor
de $n$ afasta-se do valor $5*10^{-(\epsilon + 1)}$.
Descobrimos que se utilizarmos mais precisão nos cálculos (nomeadamente através
do uso de `long double` em C++, ou seja, 128 bits), o padrão verifica-se para
todos os casos testados.
Concluímos disto que este desvio se deve ao erro de
arredondamento ($|S_n - \hat{S}_n|$), que neste caso se deve à elevada magnitude
de $n$.
## Questão 4
Usamos como referência
$$ \pi = 3.14159265358979324$$
sendo este valor arredondado a 18 algarismos significativos, para que o erro de
arredondamento aqui utilizado não altere os resultados do cálculo do erro
absoluto efetivo.
Erro efetivo obtido na questão 2:
$-log(\epsilon)$ | $\Delta x$
-----------------|---------------------
8 | $2.6 \cdot 10^{-09}$
9 | $6.9 \cdot 10^{-10}$
10 | $4.0 \cdot 10^{-11}$
11 | $2.8 \cdot 10^{-12}$
12 | $6.0 \cdot 10^{-13}$
13 | $3.4 \cdot 10^{-14}$
14 | $8.0 \cdot 10^{-15}$
15 | $4.5 \cdot 10^{-16}$
Erro efetivo obtido na questão 3:
$-log(\epsilon)$ | $\delta x$
-----------------|---------------------
8 | $5.5 \cdot 10^{-09}$
9 | $5.2 \cdot 10^{-10}$
10 | $5.1 \cdot 10^{-11}$
11 | $3.3 \cdot 10^{-12}$
12 | $2.0 \cdot 10^{-12}$
Notamos que para $\epsilon = 10^{-12}$ o erro efetivamente cometido é maior que
o erro pretendido. Justificamos este facto com a elevada magnitude de $n$, o que
leva a termos de pequena magnitude, o que induz erros de cancelamento.
Concluímos que a série utilizada na questão dois possibilita o cálculo da
aproximação do valor de $\pi$ usando um número de termos muito menor, para uma
mesma precisão. Sendo que a série da questão dois dá resultados
com erro inferior a $10^{-15}$ com um número de iterações na ordem das dezenas,
enquanto que a série da questão três requer iterações na ordem das centenas de
milhões para obter um erro inferior a $10^{-8}$.

BIN
1 - Numerical Series Approximation/assignment.pdf Normal file
View File

Binary file not shown.

18
1 - Numerical Series Approximation/graficos/erro.cpp Normal file
View File

@ -0,0 +1,18 @@
#include <iostream>
#include <cmath>
int main() {
double vals[] = { 3.1415926029605323,
3.1415926506432696,
3.1415926528764806,
3.1415926535477534,
3.1415926535795631,
3.1415926535891852,
3.1415926535897571,
3.1415926535897851};
std::cout.precision(17);
for(double v : vals) {
std::cout << std::abs(3.14159265358979324 - v);
}
}

173
1 - Numerical Series Approximation/graficos/erro2.cpp Normal file
View File

@ -0,0 +1,173 @@
#include <iostream>
#include <cmath>
int main() {
// double vals[] = { 3.1415926029605323,
// 3.1415926506432696,
// 3.1415926528764806,
// 3.1415926535477534,
// 3.1415926535795631,
// 3.1415926535891852,
// 3.1415926535897571,
// 3.1415926535897851};
double vals[] = {
// 3.1415925645296694,
// 3.1415926484100587,
// 3.1415926523361879,
// 3.1415926535159437,
// 3.1415926535718257,
// 3.1415926535887251,
// 3.1415926535897296,
// 3.1415926535897780
// 1.2091995566689335,
// 1.2091995750220277
// ,
// 1.209199575881591
// ,
// 1.2091995761399641
// ,
// 1.2091995761522076
// ,
// 1.2091995761559111
// ,
// 1.2091995761561314,
// 1.209199576156142
// 3.1415926413913953,
// 3.1415926506432696,
// 3.1415926534167733,
// 3.1415926535795631,
// 3.1415926535873004,
// 3.1415926535896448,
// 3.1415926535897571,
// 3.1415926535897918
// 3.1415926436246062,
// 3.1415926511835623,
// 3.141592653448583,
// 3.1415926535814478,
// 3.1415926535877601,
// 3.1415926535896723,
// 3.1415926535897643,
// 3.1415926535897927
// 3.1415926441636568
// ,
// 3.1415926513142782
// ,
// 3.1415926534563079
// ,
// 3.141592653581907
// ,
// 3.141592653587872
// ,
// 3.1415926535896794
// ,
// 3.141592653589766
// ,
// 3.1415926535897927
// 3.1415926413913944
// ,
// 3.1415926506432688
// ,
// 3.1415926534167729
// ,
// 3.1415926535795626
// ,
// 3.1415926535873
// ,
// 3.1415926535896443
// ,
// 3.1415926535897567
// ,
// 3.1415926535897909
// 3.1415926413913948
// ,
// 3.1415926506432692
// ,
// 3.1415926534167732
// ,
// 3.1415926535795628
// ,
// 3.1415926535873003
// ,
// 3.1415926535896447
// ,
// 3.141592653589757
// ,
// 3.1415926535897911
// 3.1415926413913948
// ,
// 3.1415926506432692
// ,
// 3.1415926534167733
// ,
// 3.1415926535795628
// ,
// 3.1415926535873003
// ,
// 3.1415926535896447
// ,
// 3.141592653589757
// ,
// 3.1415926535897911
// 3.1415926413913948
// ,
// 3.1415926506432692
// ,
// 3.1415926534167733
// ,
// 3.1415926535795628
// ,
// 3.1415926535873003
// ,
// 3.1415926535896447
// ,
// 3.141592653589757
// ,
// 3.1415926535897911
// 3.1415926585894502,
// 3.1415926540880963,
// 3.1415926536384982,
// 3.1415926535932850,
// 3.1415926535877814
// 3.1415926506432696,
// 3.1415926528764806,
// 3.1415926535477534,
// 3.1415926535873004,
// 3.1415926535891850,
// 3.1415926535897570,
// 3.1415926535897850,
// 3.1415926535897936
3.141592651 ,
3.1415926529 ,
3.14159265355 ,
3.141592653587 ,
3.1415926535892 ,
3.14159265358976 ,
3.141592653589785,
3.1415926535897936
};
std::cout.precision(17);
for(double v : vals) {
std::cout << std::abs(3.14159265358979324 - v) << '\n';
}
}

30
1 - Numerical Series Approximation/graficos/erro3.cpp Normal file
View File

@ -0,0 +1,30 @@
#include <iostream>
#include <cmath>
int main() {
double vals[] = {
// 3.1415926335902893,
// 3.1415926515893244,
// 3.1415926533883067,
// 3.1415926535683196,
// 3.1415926535862631
// 3.1415926585894502,
// 3.1415926540880963,
// 3.1415926536384982,
// 3.1415926535932850,
// 3.1415926535877814
3.141592659 ,
3.1415926541 ,
3.14159265364 ,
3.141592653593,
3.1415926535878
};
std::cout.precision(17);
for(double v : vals) {
std::cout << std::abs(3.14159265358979324 - v) << '\n';
}
}

23
1 - Numerical Series Approximation/graficos/generate_graphics.gnuplot Executable file
View File

@ -0,0 +1,23 @@
#!/usr/bin/gnuplot
set output "questao3_tempos.png";
set terminal png size 1200,900 enhanced font "Helvetica,20"
set autoscale;
set xr [7:13];
set logscale y;
set xlabel "-log(epsilon)"
set ylabel "Tempo (s)"
set linetype 1 lc rgb 'black'
set linetype 2 lc rgb 'black
plot "../questao3_tempo_java.txt" with linespoints dashtype 0 lw 4 t "Java", "../questao3_tempo_Cpp.txt" with linespoints dashtype 2 lw 4 t "C++";
set output "questao3_iteracoes.png";
set terminal png size 1200,900 enhanced font "Helvetica,20"
set autoscale;
set xr [7:13];
set logscale y;
set xlabel "-log(epsilon)"
set ylabel "Iterações"
set linetype 1 lc rgb 'black'
set linetype 2 lc rgb 'black
plot "../questao3_iteracoes_java.txt" with linespoints dashtype 0 lw 4 t "Java", "../questao3_iteracoes_Cpp.txt" with linespoints dashtype 2 lw 4 t "C++";

BIN
1 - Numerical Series Approximation/graficos/questao3_iteracoes.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 11 KiB

5
1 - Numerical Series Approximation/graficos/questao3_iteracoes_Cpp.txt Normal file
View File

@ -0,0 +1,5 @@
8 200000000
9 2000000000
10 20000000012
11 200000002870
12 2000000614027

3
1 - Numerical Series Approximation/graficos/questao3_iteracoes_java.txt Normal file
View File

@ -0,0 +1,3 @@
8 200000000
9 2000000000
10 20000000012

5
1 - Numerical Series Approximation/graficos/questao3_tempo_Cpp.txt Normal file
View File

@ -0,0 +1,5 @@
8 0.787
9 7.862
10 79.130
11 791.625
12 7871.842

3
1 - Numerical Series Approximation/graficos/questao3_tempo_java.txt Normal file
View File

@ -0,0 +1,3 @@
8 2.237
9 21.522
10 215.442

BIN
1 - Numerical Series Approximation/graficos/questao3_tempos.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 10 KiB

10
1 - Numerical Series Approximation/graficos/questao4_erro3.txt Normal file
View File

@ -0,0 +1,10 @@
8 200000000 3.1415926585894502 0.787
9 2000000000 3.1415926540880963 7.862
10 20000000012 3.1415926536384982 79.130
11 200000002870 3.141592653593285 791.625
12 2000000614027 3.1415926535877814 7908.842
14
[0.787]

45
1 - Numerical Series Approximation/graficos/scratch Normal file
View File

@ -0,0 +1,45 @@
8 | 3.1415926029605323 | 12 | 0m0.002s
9 | 3.1415926506432696 | 14 | 0m0.002s
10 | 3.1415926528764806 | 15 | 0m0.002s
11 | 3.1415926535477534 | 17 | 0m0.002s
12 | 3.1415926535795631 | 18 | 0m0.002s
13 | 3.1415926535891852 | 20 | 0m0.002s
14 | 3.1415926535897571 | 22 | 0m0.002s
15 | 3.1415926535897851 | 23 | 0m0.001s
8 | 3.1415926335902893 | 50000000 |0m0.200s
9 | 3.1415926515893244 | 500000000 |0m1.979s
10 | 3.1415926533883067 | 5000000000 |0m19.783s
11 | 3.1415926535683196 | 50000000086 |3m18.003s
12 | ^C | 1m10.621s | 0m0.000s
8 | 50000000 | 3.1415926335902893 | 0.553s
9 | 500000000 | 3.1415926515893244 | 4.608s
10 | 5000000000 | 3.1415926533883067 | 45.069s
11 | 50000000086 | | 450.325s
3.1415926535683196
3.14159265358979324 (arredondado a 18 algorismos significativos, para que o erro de arredondamento aqui utilizado não altere os resultados do cálculo do erro absoluto efetivo)
8 | 3.1415926029605323 | 12 | 0m0.002s
9 | 3.1415926506432696 | 14 | 0m0.002s
10 | 3.1415926528764806 | 15 | 0m0.002s
11 | 3.1415926535477534 | 17 | 0m0.002s
12 | 3.1415926535795631 | 18 | 0m0.002s
13 | 3.1415926535891852 | 20 | 0m0.002s
14 | 3.1415926535897571 | 22 | 0m0.002s
15 | 3.1415926535897851 | 23 | 0m0.002s
8 | 3.1415926122124067 | 12 | 0m0.002s
9 | 3.1415926511835623 | 14 | 0m0.002s
10 | 3.1415926530074607 | 15 | 0m0.001s
11 | 3.1415926535554908 | 17 | 0m0.002s
12 | 3.1415926535814478 | 18 | 0m0.001s
13 | 3.1415926535892975 | 20 | 0m0.002s
14 | 3.1415926535897643 | 22 | 0m0.002s
15 | 3.1415926535897869 | 23 | 0m0.002s

BIN
1 - Numerical Series Approximation/print.pdf Normal file
View File

Binary file not shown.

27
1 - Numerical Series Approximation/source/Question1.cpp Normal file
View File

@ -0,0 +1,27 @@
#include <iostream>
double machine_eps() {
double epsilon_candidate = 2.0,
epsilon = epsilon_candidate,
power = 2.0;
while (epsilon_candidate != 1.0)
{
epsilon = epsilon_candidate;
epsilon_candidate -= 1/power;
power *= 2;
}
return epsilon-1.0;
}
int main() {
std::cout.precision(17);
std::cout << machine_eps() << " - macheps" << '\n';
/*for(double step = 0.0f; step < 100'000'000; step += 1.0)
{
std::cout << step << " " << machine_eps(step) << '\n';
}*/
}

22
1 - Numerical Series Approximation/source/Question1wrong.cpp Normal file
View File

@ -0,0 +1,22 @@
#include <iostream>
double
machine_eps(double start)
{
double epsilon = 1.0f;
while ((start + 0.5f * epsilon) != start)
{
epsilon = 0.5f * epsilon;
}
return epsilon;
}
int
main()
{
std::cout.precision(17);
std::cout << machine_eps(1) << " - macheps" << '\n';
}

56
1 - Numerical Series Approximation/source/Question2.cpp Normal file
View File

@ -0,0 +1,56 @@
#include <iostream>
#include <cmath>
#include <cstdlib>
/**
* Compute k term of Serie 2 given the previous one
*/
double _compute_serie_2_term(unsigned long k, double ak) {
return double(k+1)/(double(4*k+6)) * ak;
}
// double c(int k) {
// return std::pow(std::tgamma(k + 1), 2.0)/std::tgamma(2*k + 2);
// }
/**
* Compute serie 2 with absole error below a given epsilon
*/
void compute_serie_2(double epsilon) {
// Constant factor
double factor = 9.0/(2.0*std::sqrt(3));
// Taken from D'Alembert criterion for given that L = 0.5 < 1
double super_L = 1.0/(1.0-0.25);
// Summation index
unsigned long k = 0;
// Acumulation for the Summation
double acc = 0;
// Previous and current value on the series
// double a_prev = 1.0f, // a with k = 0
double a = 1.0; // _compute_serie_2_term(k, a_prev); // compute next
// While our absolute error is higher than the given epsilon
while(epsilon < factor*a*super_L) {
// Accumulate with previous term
acc += a;
// Update previous term to current term
// a_prev = a;
// Compute current term
// a = c(k);
a = _compute_serie_2_term(k++, a);
}
std::cout << k << " " << std::abs(M_PI - factor*acc) << '\n';
std::cout << factor * acc << " | " << k;
}
int main(int, char **argv) {
std::cout.precision(17);
double power_of_ten = std::pow(10.0, -std::atoi(argv[1]));
compute_serie_2(power_of_ten);
}

31
1 - Numerical Series Approximation/source/Question3.cpp Normal file
View File

@ -0,0 +1,31 @@
#include <iostream>
#include <cmath>
#include <numeric>
#include <ios>
#include <cstdlib>
double compute_serie_3_term(unsigned long n) {
return -((double)(2*n+1))/(double)(2*n+3);
}
double compute_serie_3(double err) {
unsigned long k = 0;
double ak = 1;
double acc = 1;
while (err < 4*std::abs(ak)) {
ak *= compute_serie_3_term(k++);
acc += ak;
}
std::cout << k << " " << 4*acc << " " << std::abs(3.14159265358979324$$ - 4*acc) << '\n';
return 4*acc;
}
int main(int, char **argv) {
const double err = std::pow(10.0, -std::atof(argv[1]));
std::cout.precision(17);
compute_serie_3(err);
}

16
1 - Numerical Series Approximation/source/java for speed comparison/Question1.java Normal file
View File

@ -0,0 +1,16 @@
public class Question1 {
public static void main(String args[]) {
System.out.println(machine_eps());
}
public static float machine_eps() {
float epsilon = 1.0f;
while ((1.0f + 0.5f * epsilon) != 1.0f){
epsilon = 0.5f * epsilon;
}
return epsilon;
}
}

49
1 - Numerical Series Approximation/source/java for speed comparison/Question2.java Normal file
View File

@ -0,0 +1,49 @@
public class Question2 {
public static void main(String[] args) {
double power_of_ten = Math.pow(10,-1 * Integer.parseInt(args[0]));
compute_serie_2(power_of_ten);
}
/**
* Compute serie 2 with absole error below a given epsilon
*/
public static void
compute_serie_2(double epsilon)
{
// Constant factor
double factor = 9.0f/(double)(2.0f*Math.sqrt(3));
// Taken from D'Alembert criterion for given that L = 0.5 < 1
double super_L = 1.0f/(double)(1-0.25);
// Summation index
int k = 0;
// Acumulation for the Summation
double acc = 0;
// Previous and current value on the series
double a = 1.0f; // a with k = 0
// While our absolute error is higher than the given epsilon
while(epsilon < factor*a*super_L)
{
// Accumulate with previous term
acc += a;
// Compute current term
a = _compute_serie_2_term(k++) * a;
}
// System.out.println(epsilon);
// System.out.println("Value: " + factor*acc);
// System.out.println("Error: " + (3.1415926535897932 - factor*acc));
// System.out.println("K: " +(k-1) + "\n--------------");
System.out.println(factor*acc + " " + (k - 1));
}
/**
* Compute k term of Serie 2 given the previous one
*/
public static double
_compute_serie_2_term(int k)
{
return (double)(k+1.0f)/(double)(4.0f*k+6.0f);
}
}

28
1 - Numerical Series Approximation/source/java for speed comparison/Question3.java Normal file
View File

@ -0,0 +1,28 @@
public class Question3 {
public static void main(String[] args) {
double err = Math.pow(10, -Integer.parseInt(args[0].trim()));
compute_serie_2(err);
}
public static double compute_serie_3_term(long n)
{
return -((double)(2*n+1))/(double)(2*n+3);
}
public static void compute_serie_2(double err)
{
long k = 0;
double ak = 1;
double acc = 1;
while (err < 4 * Math.abs(ak)) {
ak = compute_serie_3_term(k++) * ak;
acc += ak;
}
System.out.println(k + " " + 4 * acc);
}
}

674
2 - Newton and Iterative methods/LICENSE Normal file
View File

@ -0,0 +1,674 @@
GNU GENERAL PUBLIC LICENSE
Version 3, 29 June 2007
Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
Preamble
The GNU General Public License is a free, copyleft license for
software and other kinds of works.
The licenses for most software and other practical works are designed
to take away your freedom to share and change the works. By contrast,
the GNU General Public License is intended to guarantee your freedom to
share and change all versions of a program--to make sure it remains free
software for all its users. We, the Free Software Foundation, use the
GNU General Public License for most of our software; it applies also to
any other work released this way by its authors. You can apply it to
your programs, too.
When we speak of free software, we are referring to freedom, not
price. Our General Public Licenses are designed to make sure that you
have the freedom to distribute copies of free software (and charge for
them if you wish), that you receive source code or can get it if you
want it, that you can change the software or use pieces of it in new
free programs, and that you know you can do these things.
To protect your rights, we need to prevent others from denying you
these rights or asking you to surrender the rights. Therefore, you have
certain responsibilities if you distribute copies of the software, or if
you modify it: responsibilities to respect the freedom of others.
For example, if you distribute copies of such a program, whether
gratis or for a fee, you must pass on to the recipients the same
freedoms that you received. You must make sure that they, too, receive
or can get the source code. And you must show them these terms so they
know their rights.
Developers that use the GNU GPL protect your rights with two steps:
(1) assert copyright on the software, and (2) offer you this License
giving you legal permission to copy, distribute and/or modify it.
For the developers' and authors' protection, the GPL clearly explains
that there is no warranty for this free software. For both users' and
authors' sake, the GPL requires that modified versions be marked as
changed, so that their problems will not be attributed erroneously to
authors of previous versions.
Some devices are designed to deny users access to install or run
modified versions of the software inside them, although the manufacturer
can do so. This is fundamentally incompatible with the aim of
protecting users' freedom to change the software. The systematic
pattern of such abuse occurs in the area of products for individuals to
use, which is precisely where it is most unacceptable. Therefore, we
have designed this version of the GPL to prohibit the practice for those
products. If such problems arise substantially in other domains, we
stand ready to extend this provision to those domains in future versions
of the GPL, as needed to protect the freedom of users.
Finally, every program is threatened constantly by software patents.
States should not allow patents to restrict development and use of
software on general-purpose computers, but in those that do, we wish to
avoid the special danger that patents applied to a free program could
make it effectively proprietary. To prevent this, the GPL assures that
patents cannot be used to render the program non-free.
The precise terms and conditions for copying, distribution and
modification follow.
TERMS AND CONDITIONS
0. Definitions.
"This License" refers to version 3 of the GNU General Public License.
"Copyright" also means copyright-like laws that apply to other kinds of
works, such as semiconductor masks.
"The Program" refers to any copyrightable work licensed under this
License. Each licensee is addressed as "you". "Licensees" and
"recipients" may be individuals or organizations.
To "modify" a work means to copy from or adapt all or part of the work
in a fashion requiring copyright permission, other than the making of an
exact copy. The resulting work is called a "modified version" of the
earlier work or a work "based on" the earlier work.
A "covered work" means either the unmodified Program or a work based
on the Program.
To "propagate" a work means to do anything with it that, without
permission, would make you directly or secondarily liable for
infringement under applicable copyright law, except executing it on a
computer or modifying a private copy. Propagation includes copying,
distribution (with or without modification), making available to the
public, and in some countries other activities as well.
To "convey" a work means any kind of propagation that enables other
parties to make or receive copies. Mere interaction with a user through
a computer network, with no transfer of a copy, is not conveying.
An interactive user interface displays "Appropriate Legal Notices"
to the extent that it includes a convenient and prominently visible
feature that (1) displays an appropriate copyright notice, and (2)
tells the user that there is no warranty for the work (except to the
extent that warranties are provided), that licensees may convey the
work under this License, and how to view a copy of this License. If
the interface presents a list of user commands or options, such as a
menu, a prominent item in the list meets this criterion.
1. Source Code.
The "source code" for a work means the preferred form of the work
for making modifications to it. "Object code" means any non-source
form of a work.
A "Standard Interface" means an interface that either is an official
standard defined by a recognized standards body, or, in the case of
interfaces specified for a particular programming language, one that
is widely used among developers working in that language.
The "System Libraries" of an executable work include anything, other
than the work as a whole, that (a) is included in the normal form of
packaging a Major Component, but which is not part of that Major
Component, and (b) serves only to enable use of the work with that
Major Component, or to implement a Standard Interface for which an
implementation is available to the public in source code form. A
"Major Component", in this context, means a major essential component
(kernel, window system, and so on) of the specific operating system
(if any) on which the executable work runs, or a compiler used to
produce the work, or an object code interpreter used to run it.
The "Corresponding Source" for a work in object code form means all
the source code needed to generate, install, and (for an executable
work) run the object code and to modify the work, including scripts to
control those activities. However, it does not include the work's
System Libraries, or general-purpose tools or generally available free
programs which are used unmodified in performing those activities but
which are not part of the work. For example, Corresponding Source
includes interface definition files associated with source files for
the work, and the source code for shared libraries and dynamically
linked subprograms that the work is specifically designed to require,
such as by intimate data communication or control flow between those
subprograms and other parts of the work.
The Corresponding Source need not include anything that users
can regenerate automatically from other parts of the Corresponding
Source.
The Corresponding Source for a work in source code form is that
same work.
2. Basic Permissions.
All rights granted under this License are granted for the term of
copyright on the Program, and are irrevocable provided the stated
conditions are met. This License explicitly affirms your unlimited
permission to run the unmodified Program. The output from running a
covered work is covered by this License only if the output, given its
content, constitutes a covered work. This License acknowledges your
rights of fair use or other equivalent, as provided by copyright law.
You may make, run and propagate covered works that you do not
convey, without conditions so long as your license otherwise remains
in force. You may convey covered works to others for the sole purpose
of having them make modifications exclusively for you, or provide you
with facilities for running those works, provided that you comply with
the terms of this License in conveying all material for which you do
not control copyright. Those thus making or running the covered works
for you must do so exclusively on your behalf, under your direction
and control, on terms that prohibit them from making any copies of
your copyrighted material outside their relationship with you.
Conveying under any other circumstances is permitted solely under
the conditions stated below. Sublicensing is not allowed; section 10
makes it unnecessary.
3. Protecting Users' Legal Rights From Anti-Circumvention Law.
No covered work shall be deemed part of an effective technological
measure under any applicable law fulfilling obligations under article
11 of the WIPO copyright treaty adopted on 20 December 1996, or
similar laws prohibiting or restricting circumvention of such
measures.
When you convey a covered work, you waive any legal power to forbid
circumvention of technological measures to the extent such circumvention
is effected by exercising rights under this License with respect to
the covered work, and you disclaim any intention to limit operation or
modification of the work as a means of enforcing, against the work's
users, your or third parties' legal rights to forbid circumvention of
technological measures.
4. Conveying Verbatim Copies.
You may convey verbatim copies of the Program's source code as you
receive it, in any medium, provided that you conspicuously and
appropriately publish on each copy an appropriate copyright notice;
keep intact all notices stating that this License and any
non-permissive terms added in accord with section 7 apply to the code;
keep intact all notices of the absence of any warranty; and give all
recipients a copy of this License along with the Program.
You may charge any price or no price for each copy that you convey,
and you may offer support or warranty protection for a fee.
5. Conveying Modified Source Versions.
You may convey a work based on the Program, or the modifications to
produce it from the Program, in the form of source code under the
terms of section 4, provided that you also meet all of these conditions:
a) The work must carry prominent notices stating that you modified
it, and giving a relevant date.
b) The work must carry prominent notices stating that it is
released under this License and any conditions added under section
7. This requirement modifies the requirement in section 4 to
"keep intact all notices".
c) You must license the entire work, as a whole, under this
License to anyone who comes into possession of a copy. This
License will therefore apply, along with any applicable section 7
additional terms, to the whole of the work, and all its parts,
regardless of how they are packaged. This License gives no
permission to license the work in any other way, but it does not
invalidate such permission if you have separately received it.
d) If the work has interactive user interfaces, each must display
Appropriate Legal Notices; however, if the Program has interactive
interfaces that do not display Appropriate Legal Notices, your
work need not make them do so.
A compilation of a covered work with other separate and independent
works, which are not by their nature extensions of the covered work,
and which are not combined with it such as to form a larger program,
in or on a volume of a storage or distribution medium, is called an
"aggregate" if the compilation and its resulting copyright are not
used to limit the access or legal rights of the compilation's users
beyond what the individual works permit. Inclusion of a covered work
in an aggregate does not cause this License to apply to the other
parts of the aggregate.
6. Conveying Non-Source Forms.
You may convey a covered work in object code form under the terms
of sections 4 and 5, provided that you also convey the
machine-readable Corresponding Source under the terms of this License,
in one of these ways:
a) Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by the
Corresponding Source fixed on a durable physical medium
customarily used for software interchange.
b) Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by a
written offer, valid for at least three years and valid for as
long as you offer spare parts or customer support for that product
model, to give anyone who possesses the object code either (1) a
copy of the Corresponding Source for all the software in the
product that is covered by this License, on a durable physical
medium customarily used for software interchange, for a price no
more than your reasonable cost of physically performing this
conveying of source, or (2) access to copy the
Corresponding Source from a network server at no charge.
c) Convey individual copies of the object code with a copy of the
written offer to provide the Corresponding Source. This
alternative is allowed only occasionally and noncommercially, and
only if you received the object code with such an offer, in accord
with subsection 6b.
d) Convey the object code by offering access from a designated
place (gratis or for a charge), and offer equivalent access to the
Corresponding Source in the same way through the same place at no
further charge. You need not require recipients to copy the
Corresponding Source along with the object code. If the place to
copy the object code is a network server, the Corresponding Source
may be on a different server (operated by you or a third party)
that supports equivalent copying facilities, provided you maintain
clear directions next to the object code saying where to find the
Corresponding Source. Regardless of what server hosts the
Corresponding Source, you remain obligated to ensure that it is
available for as long as needed to satisfy these requirements.
e) Convey the object code using peer-to-peer transmission, provided
you inform other peers where the object code and Corresponding
Source of the work are being offered to the general public at no
charge under subsection 6d.
A separable portion of the object code, whose source code is excluded
from the Corresponding Source as a System Library, need not be
included in conveying the object code work.
A "User Product" is either (1) a "consumer product", which means any
tangible personal property which is normally used for personal, family,
or household purposes, or (2) anything designed or sold for incorporation
into a dwelling. In determining whether a product is a consumer product,
doubtful cases shall be resolved in favor of coverage. For a particular
product received by a particular user, "normally used" refers to a
typical or common use of that class of product, regardless of the status
of the particular user or of the way in which the particular user
actually uses, or expects or is expected to use, the product. A product
is a consumer product regardless of whether the product has substantial
commercial, industrial or non-consumer uses, unless such uses represent
the only significant mode of use of the product.
"Installation Information" for a User Product means any methods,
procedures, authorization keys, or other information required to install
and execute modified versions of a covered work in that User Product from
a modified version of its Corresponding Source. The information must
suffice to ensure that the continued functioning of the modified object
code is in no case prevented or interfered with solely because
modification has been made.
If you convey an object code work under this section in, or with, or
specifically for use in, a User Product, and the conveying occurs as
part of a transaction in which the right of possession and use of the
User Product is transferred to the recipient in perpetuity or for a
fixed term (regardless of how the transaction is characterized), the
Corresponding Source conveyed under this section must be accompanied
by the Installation Information. But this requirement does not apply
if neither you nor any third party retains the ability to install
modified object code on the User Product (for example, the work has
been installed in ROM).
The requirement to provide Installation Information does not include a
requirement to continue to provide support service, warranty, or updates
for a work that has been modified or installed by the recipient, or for
the User Product in which it has been modified or installed. Access to a
network may be denied when the modification itself materially and
adversely affects the operation of the network or violates the rules and
protocols for communication across the network.
Corresponding Source conveyed, and Installation Information provided,
in accord with this section must be in a format that is publicly
documented (and with an implementation available to the public in
source code form), and must require no special password or key for
unpacking, reading or copying.
7. Additional Terms.
"Additional permissions" are terms that supplement the terms of this
License by making exceptions from one or more of its conditions.
Additional permissions that are applicable to the entire Program shall
be treated as though they were included in this License, to the extent
that they are valid under applicable law. If additional permissions
apply only to part of the Program, that part may be used separately
under those permissions, but the entire Program remains governed by
this License without regard to the additional permissions.
When you convey a copy of a covered work, you may at your option
remove any additional permissions from that copy, or from any part of
it. (Additional permissions may be written to require their own
removal in certain cases when you modify the work.) You may place
additional permissions on material, added by you to a covered work,
for which you have or can give appropriate copyright permission.
Notwithstanding any other provision of this License, for material you
add to a covered work, you may (if authorized by the copyright holders of
that material) supplement the terms of this License with terms:
a) Disclaiming warranty or limiting liability differently from the
terms of sections 15 and 16 of this License; or
b) Requiring preservation of specified reasonable legal notices or
author attributions in that material or in the Appropriate Legal
Notices displayed by works containing it; or
c) Prohibiting misrepresentation of the origin of that material, or
requiring that modified versions of such material be marked in
reasonable ways as different from the original version; or
d) Limiting the use for publicity purposes of names of licensors or
authors of the material; or
e) Declining to grant rights under trademark law for use of some
trade names, trademarks, or service marks; or
f) Requiring indemnification of licensors and authors of that
material by anyone who conveys the material (or modified versions of
it) with contractual assumptions of liability to the recipient, for
any liability that these contractual assumptions directly impose on
those licensors and authors.
All other non-permissive additional terms are considered "further
restrictions" within the meaning of section 10. If the Program as you
received it, or any part of it, contains a notice stating that it is
governed by this License along with a term that is a further
restriction, you may remove that term. If a license document contains
a further restriction but permits relicensing or conveying under this
License, you may add to a covered work material governed by the terms
of that license document, provided that the further restriction does
not survive such relicensing or conveying.
If you add terms to a covered work in accord with this section, you
must place, in the relevant source files, a statement of the
additional terms that apply to those files, or a notice indicating
where to find the applicable terms.
Additional terms, permissive or non-permissive, may be stated in the
form of a separately written license, or stated as exceptions;
the above requirements apply either way.
8. Termination.
You may not propagate or modify a covered work except as expressly
provided under this License. Any attempt otherwise to propagate or
modify it is void, and will automatically terminate your rights under
this License (including any patent licenses granted under the third
paragraph of section 11).
However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
provisionally, unless and until the copyright holder explicitly and
finally terminates your license, and (b) permanently, if the copyright
holder fails to notify you of the violation by some reasonable means
prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from that
copyright holder, and you cure the violation prior to 30 days after
your receipt of the notice.
Termination of your rights under this section does not terminate the
licenses of parties who have received copies or rights from you under
this License. If your rights have been terminated and not permanently
reinstated, you do not qualify to receive new licenses for the same
material under section 10.
9. Acceptance Not Required for Having Copies.
You are not required to accept this License in order to receive or
run a copy of the Program. Ancillary propagation of a covered work
occurring solely as a consequence of using peer-to-peer transmission
to receive a copy likewise does not require acceptance. However,
nothing other than this License grants you permission to propagate or
modify any covered work. These actions infringe copyright if you do
not accept this License. Therefore, by modifying or propagating a
covered work, you indicate your acceptance of this License to do so.
10. Automatic Licensing of Downstream Recipients.
Each time you convey a covered work, the recipient automatically
receives a license from the original licensors, to run, modify and
propagate that work, subject to this License. You are not responsible
for enforcing compliance by third parties with this License.
An "entity transaction" is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations. If propagation of a covered
work results from an entity transaction, each party to that
transaction who receives a copy of the work also receives whatever
licenses to the work the party's predecessor in interest had or could
give under the previous paragraph, plus a right to possession of the
Corresponding Source of the work from the predecessor in interest, if
the predecessor has it or can get it with reasonable efforts.
You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License. For example, you may
not impose a license fee, royalty, or other charge for exercise of
rights granted under this License, and you may not initiate litigation
(including a cross-claim or counterclaim in a lawsuit) alleging that
any patent claim is infringed by making, using, selling, offering for
sale, or importing the Program or any portion of it.
11. Patents.
A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based. The
work thus licensed is called the contributor's "contributor version".
A contributor's "essential patent claims" are all patent claims
owned or controlled by the contributor, whether already acquired or
hereafter acquired, that would be infringed by some manner, permitted
by this License, of making, using, or selling its contributor version,
but do not include claims that would be infringed only as a
consequence of further modification of the contributor version. For
purposes of this definition, "control" includes the right to grant
patent sublicenses in a manner consistent with the requirements of
this License.
Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims, to
make, use, sell, offer for sale, import and otherwise run, modify and
propagate the contents of its contributor version.
In the following three paragraphs, a "patent license" is any express
agreement or commitment, however denominated, not to enforce a patent
(such as an express permission to practice a patent or covenant not to
sue for patent infringement). To "grant" such a patent license to a
party means to make such an agreement or commitment not to enforce a
patent against the party.
If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone
to copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
available, or (2) arrange to deprive yourself of the benefit of the
patent license for this particular work, or (3) arrange, in a manner
consistent with the requirements of this License, to extend the patent
license to downstream recipients. "Knowingly relying" means you have
actual knowledge that, but for the patent license, your conveying the
covered work in a country, or your recipient's use of the covered work
in a country, would infringe one or more identifiable patents in that
country that you have reason to believe are valid.
If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license to some of the parties
receiving the covered work authorizing them to use, propagate, modify
or convey a specific copy of the covered work, then the patent license
you grant is automatically extended to all recipients of the covered
work and works based on it.
A patent license is "discriminatory" if it does not include within
the scope of its coverage, prohibits the exercise of, or is
conditioned on the non-exercise of one or more of the rights that are
specifically granted under this License. You may not convey a covered
work if you are a party to an arrangement with a third party that is
in the business of distributing software, under which you make payment
to the third party based on the extent of your activity of conveying
the work, and under which the third party grants, to any of the
parties who would receive the covered work from you, a discriminatory
patent license (a) in connection with copies of the covered work
conveyed by you (or copies made from those copies), or (b) primarily
for and in connection with specific products or compilations that
contain the covered work, unless you entered into that arrangement,
or that patent license was granted, prior to 28 March 2007.
Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may
otherwise be available to you under applicable patent law.
12. No Surrender of Others' Freedom.
If conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License. If you cannot convey a
covered work so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you may
not convey it at all. For example, if you agree to terms that obligate you
to collect a royalty for further conveying from those to whom you convey
the Program, the only way you could satisfy both those terms and this
License would be to refrain entirely from conveying the Program.
13. Use with the GNU Affero General Public License.
Notwithstanding any other provision of this License, you have
permission to link or combine any covered work with a work licensed
under version 3 of the GNU Affero General Public License into a single
combined work, and to convey the resulting work. The terms of this
License will continue to apply to the part which is the covered work,
but the special requirements of the GNU Affero General Public License,
section 13, concerning interaction through a network will apply to the
combination as such.
14. Revised Versions of this License.
The Free Software Foundation may publish revised and/or new versions of
the GNU General Public License from time to time. Such new versions will
be similar in spirit to the present version, but may differ in detail to
address new problems or concerns.
Each version is given a distinguishing version number. If the
Program specifies that a certain numbered version of the GNU General
Public License "or any later version" applies to it, you have the
option of following the terms and conditions either of that numbered
version or of any later version published by the Free Software
Foundation. If the Program does not specify a version number of the
GNU General Public License, you may choose any version ever published
by the Free Software Foundation.
If the Program specifies that a proxy can decide which future
versions of the GNU General Public License can be used, that proxy's
public statement of acceptance of a version permanently authorizes you
to choose that version for the Program.
Later license versions may give you additional or different
permissions. However, no additional obligations are imposed on any
author or copyright holder as a result of your choosing to follow a
later version.
15. Disclaimer of Warranty.
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
16. Limitation of Liability.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
SUCH DAMAGES.
17. Interpretation of Sections 15 and 16.
If the disclaimer of warranty and limitation of liability provided
above cannot be given local legal effect according to their terms,
reviewing courts shall apply local law that most closely approximates
an absolute waiver of all civil liability in connection with the
Program, unless a warranty or assumption of liability accompanies a
copy of the Program in return for a fee.
END OF TERMS AND CONDITIONS
How to Apply These Terms to Your New Programs
If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.
To do so, attach the following notices to the program. It is safest
to attach them to the start of each source file to most effectively
state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.
<one line to give the program's name and a brief idea of what it does.>
Copyright (C) <year> <name of author>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:
<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<https://www.gnu.org/licenses/>.
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<https://www.gnu.org/licenses/why-not-lgpl.html>.

158
2 - Newton and Iterative methods/README.md Normal file
View File

@ -0,0 +1,158 @@
---
title: "Análise Numérica - Trabalho Prático 2"
author:
- Diogo Cordeiro
- Hugo Sales
- Pedro Costa
geometry: margin=2cm
output: pdf_document
---
### Motivação
Pretende-se usar os métodos de Newton e iterativo simples para determinar um valor aproximado de um zero de
$$x^2 - cos(x)^2$$
### 1.a)
Optamos por implementar o algoritmo pedido em C++ devido à possiblidade da aplicação de
templates e lambdas. Deste modo, foi-nos possível implementar os dois métodos pedidos partindo
de um algoritmo genérico pois a diferença entre o método de Newton
e o método iterativo consiste apenas na fórmula de recorrência. Assim, o método de Newton
pode ser visto como uma forma do método iterativo simples.
template<typename step_func>
double find_root(double x0, double epsilon, step_func step, long iter_limit) {
double x1 = x0, err;
long iter = 1;
do {
x0 = x1;
x1 = step(x0);
err = std::abs(x1 - x0);
} while(err > epsilon && iter++ < iter_limit);
return x1;
}
template<typename F_t, typename dF_t>
double newton(double x0, double epsilon, F_t F, dF_t dF, long iter_limit) {
return find_root(x0, epsilon,
[&F, &dF](double x0){ return x0 - F(x0)/dF(x0); },
iter_limit);
}
int main() {
auto F = [](double x){ return std::pow(x, 2.0) - std::pow(std::cos(x), 2.0); };
auto dF = [](double x){ return 2.0 * x + std::sin(2.0 * x); };
double epsilon = 5.0 * std::pow(10.0, -12.0);
std::cout << newton(0.8, epsilon, F, dF, 100000) << '\n';
}
\pagebreak
### 1.b)
![Gráfico para $x \in [-1;1]$](./graph.png){ width=8cm }
Como $\forall{x}: cos^2(x) \in [0:1]$, temos que $x^2 \geq cos^2(x)$ para $|x| > 1$, sabemos que o comportamento de $f(x)$ é dominado pelo comportamento de $x^2$, a qual só tem duas raízes.
A menor das raízes encontra-se no intervalo $]-\infty;0]$ e a maior destas em $[0;\infty[$.
Através da análise do gráfico, verificamos que o intervalo $[0.7;0.8]$ contém uma raíz. Definimos então $a = 0.7$ e $b = 0.8$.
![Gráfico para $x \in [0.7;0.8]$](./graph_small.png){ width=6cm }
### 1.c)
Queremos mostrar que as condições de aplicabilidade do método de Newton são satisfeitas no intervalo. Assim,
$$F(x) = x^2 - cos(x)^2$$
$$F'(x) = 2 \cdot x + sin(2 \cdot x)$$
$$F''(x) = 2 + 2 \cdot cos(2 \cdot x)$$
Como todas estas funções são compostas partindo de somas de funções contínuas em $\mathbb{R}$, são também continuas no intervalo considerado, verificando-se assim o primeiro critério.
Dado que,
$$F(a) < 0, F(b) < 0 \Rightarrow F(a) \cdot F(b) < 0$$
Verifica-se também o segundo critério.
Temos que $F''(x) = 2 + 2 \cdot cos(2 \cdot x)$, logo $cos(2 \cdot x) \in [-1;1]$, por isso $2 \cdot cos(2 \cdot x) \in [-2; 2]$, e $2 \cdot cos(2 \cdot x) + 2 \in [0;4]$ o que implica que $F''(x) \geq 0$, para $x \in \mathbb{R}$ e por isso $F'(x)$ é não decrescente em $\mathbb{R}$, ou seja, $F'(b) \geq F'(a)$ e $F'(a) > 3$ e por isso $F'(x) \neq 0 \forall{x} : \in [a;b]$, o que verifica o terceiro critério.
Como foi dito anteriormente, $F''(x) \geq 0$ em $\mathbb{R}$ e, por isso, também para $\forall{x} : \in [a;b]$, verificando-se a quarta condição.
Para $x_0 = b$, temos que $F(x_0) > 0$ e $F''(x_0) > 0$, logo $F(x_0) \cdot F''(x_0) > 0$, verificando-se a quinta condição.
Então a sucessão gerada converge para a unica raíz no intervalo $[a;b]$.
### 1.d)
Aplicamos o programa apresentado em $1.a$ e obtivemos os seguintes resultados.
Iterações | Erro estimado | Valor
----------|---------------|---------------
1 | 5.9e-02 | 0.740528800196
2 | 1.4e-03 | 0.739086050826
3 | 9.2e-07 | 0.739085133216
4 | 3.7e-13 | 0.739085133215
Verficamos que o erro estimado é aproximadamente metade do erro estimado da iteração amterior,
o que justificamos com o facto de que o resultado teórico
nos dizer que o erro converge segundo uma secessão de segunda ordem.
### 1.e)
Aplicando a fórmula
$$M = \frac{1}{2} \cdot \frac{\max\limits_{a \leq x \leq b} | F''(x) |}{\min\limits_{a \leq x \leq b} | F'(x) |}$$
Como $F''(x)$ é decrescente em $[0;\pi/2]$, então $\max\limits_{a \leq x \leq b} | F''(x) | = |F''(a)|$ e como $F'(x)$ é não decrescente em $\mathbb{R}$, $\min\limits_{a \leq x \leq b} | F'(x) | = F'(a)$.
Assim, obtemos $M = \frac{F''(0.7)}{2F'(0.7)} \leq \frac{2.4}{2 \cdot 2.3} \leq 0.53$ e
$n \geq \frac{ln(\alpha)}{ln(2)}, \alpha = \frac{ln(5 \cdot 10^{-14}) + ln(0.53)}{ln(0.53) + ln(10^{-1})}$, logo $\alpha \leq \frac{-32}{2.9} \leq 12$ e $n \geq 4$, ou seja que com 4 iterações conseguimos um erro absoluto inferior a $5 \cdot 10^{-14}$.
### 2.a)
Dada a implementação genérica do algoritmo, o método iterativo simples pode ser
implementado sucintamente como:
template<typename F_t>
double fixed_point(double x0, double epsilon, F_t f, long iter_limit) {
return find_root(x0, epsilon, f, iter_limit);
}
e usado como
auto f = [](double x){ return std::cos(x); };
std::cout << fixed_point(0.8, epsilon, f, 100000) << '\n';
A expressão de $\texttt{f}$ foi obtida por manipulação algébrica do seguinte modo:
$$F(x) = 0 \iff x^2 - cos^2(x) = 0 \iff x^2 = cos^2(x) \iff x = \pm cos(x)$$
Logo no intervalo $[a;b]$, temos que $x = cos(x)$ ou seja $f(x) = cos(x)$.
### 2.b)
Aplicando o programa supra apresentado, obtivemos os seguintes erros:
![Gŕafico da comparação do erro estimado dos dois métodos](err_fixed_point.png){ width=12cm }
Partindo destes resultados, verificamos empiricameente a diferença na ordem de convergência dos dois métodos.
Obtivemos como valor final para este método o valor $0.739085133217$, que difere do valor calculado
como o método de Newton em $-2 \cdot 10^{-12}$, por isso concluimos numericamente que, para os valores iniciais usados, este método pode ser aplicado, sendo que aproxima o valor real.
Verificando as condições de aplicabilidade deste método, temos que $f(x)$ é continua em $[a;b]$, verificando-se a primeira condição,
que $f(0.7) \approx 0.76484219$ e $f(0.7) \approx 0.69670671$ o que implica que $f([a;b]) \notin [a;b]$, o implica que não se verifica a segunda condição
e que $\forall x \in [0.7;0.8] : |-sin(x)| < 1$, o que significa que se verifica a terceira condição.
Apesar de não se verificar a segunda condição, verificamos na mesma a convergência da sucessão, o que não contradiz os resultados teóricos, uma vez que estas condições são suficientes, mas não necessárias para esta convergência.

BIN
2 - Newton and Iterative methods/assignment.pdf Normal file
View File

Binary file not shown.

BIN
2 - Newton and Iterative methods/err_fixed_point.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 11 KiB

BIN
2 - Newton and Iterative methods/graph.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 16 KiB

BIN
2 - Newton and Iterative methods/graph_small.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 13 KiB

62
2 - Newton and Iterative methods/graphics/err_fixed_point.txt Normal file
View File

@ -0,0 +1,62 @@
1.0e-01 1
7.0e-02 2
4.7e-02 3
3.2e-02 4
2.1e-02 5
1.4e-02 6
9.7e-03 7
6.5e-03 8
4.4e-03 9
3.0e-03 10
2.0e-03 11
1.3e-03 12
9.0e-04 13
6.1e-04 14
4.1e-04 15
2.8e-04 16
1.9e-04 17
1.3e-04 18
8.5e-05 19
5.7e-05 20
3.8e-05 21
2.6e-05 22
1.7e-05 23
1.2e-05 24
7.9e-06 25
5.3e-06 26
3.6e-06 27
2.4e-06 28
1.6e-06 29
1.1e-06 30
7.4e-07 31
5.0e-07 32
3.3e-07 33
2.3e-07 34
1.5e-07 35
1.0e-07 36
6.9e-08 37
4.6e-08 38
3.1e-08 39
2.1e-08 40
1.4e-08 41
9.6e-09 42
6.4e-09 43
4.3e-09 44
2.9e-09 45
2.0e-09 46
1.3e-09 47
8.9e-10 48
6.0e-10 49
4.1e-10 50
2.7e-10 51
1.8e-10 52
1.2e-10 53
8.3e-11 54
5.6e-11 55
3.8e-11 56
2.6e-11 57
1.7e-11 58
1.2e-11 59
7.8e-12 60
5.3e-12 61
3.5e-12 62

3
2 - Newton and Iterative methods/graphics/err_newton.txt Normal file
View File

@ -0,0 +1,3 @@
1.4e-03 1
9.2e-07 2
3.7e-13 3

27
2 - Newton and Iterative methods/graphics/generate_graphics.gnuplot Executable file
View File

@ -0,0 +1,27 @@
#!/usr/bin/gnuplot
set output "err_fixed_point.png";
set terminal png size 1200,900 enhanced font "Helvetica,20"
set autoscale;
set logscale x;
set xlabel "Erro"
set ylabel "Iterações"
set xrange [] reverse;
set linetype 1 lc rgb 'black'
set linetype 2 lc rgb 'black
plot "err_fixed_point.txt" with points pointtype 1 pointsize 2 lw 4 t "Fixed Point", "err_newton.txt" with points pointtype 2 pointsize 2 lw 4 t "Newton";
set terminal png size 1200,900 enhanced font "Helvetica,35"
set output "graph.png";
unset logscale x;
set xr [-1:1];
set zeroaxis ls 1 lw 3;
set xlabel "x"
set ylabel "y"
f(x)= x**2 - cos(x)**2;
plot f(x) lw 3 t "x^2 - cos(x)^2";
set output "graph_small.png";
set xr [0.7:0.8];
plot f(x) lw 3 t "x^2 - cos(x)^2";

BIN
2 - Newton and Iterative methods/print.pdf Normal file
View File

Binary file not shown.

1
2 - Newton and Iterative methods/slides/css/all.css Normal file
View File

File diff suppressed because one or more lines are too long

40
2 - Newton and Iterative methods/slides/css/css.css Normal file
View File

@ -0,0 +1,40 @@
/* cyrillic-ext */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-cSZMZ-Y.woff2) format('woff2');
unicode-range: U+0460-052F, U+1C80-1C88, U+20B4, U+2DE0-2DFF, U+A640-A69F, U+FE2E-FE2F;
}
/* cyrillic */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-eCZMZ-Y.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-cyZMZ-Y.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-ciZMZ-Y.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-fCZM.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

32
2 - Newton and Iterative methods/slides/css/css2.css Normal file
View File

@ -0,0 +1,32 @@
/* cyrillic */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752HT8Ghe4.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752Fj8Ghe4.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752Fz8Ghe4.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752GT8G.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

48
2 - Newton and Iterative methods/slides/css/css3.css Normal file
View File

@ -0,0 +1,48 @@
/* cyrillic-ext */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc3CsTKlA.woff2) format('woff2');
unicode-range: U+0460-052F, U+1C80-1C88, U+20B4, U+2DE0-2DFF, U+A640-A69F, U+FE2E-FE2F;
}
/* cyrillic */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc-CsTKlA.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* greek-ext */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc2CsTKlA.woff2) format('woff2');
unicode-range: U+1F00-1FFF;
}
/* greek */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc5CsTKlA.woff2) format('woff2');
unicode-range: U+0370-03FF;
}
/* latin-ext */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc0CsTKlA.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc6CsQ.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

224
2 - Newton and Iterative methods/slides/css/css4.css Normal file
View File

@ -0,0 +1,224 @@
/* latin-ext */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 400;
src: local('Quattrocento'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpEg_xvsDZQL_LKIF7q4jP3zWj6T4g.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 400;
src: local('Quattrocento'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpEg_xvsDZQL_LKIF7q4jP3w2j6.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Bold'), local('Quattrocento-Bold'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpbg_xvsDZQL_LKIF7q4jP_eE3vfqnYgXc.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Bold'), local('Quattrocento-Bold'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpbg_xvsDZQL_LKIF7q4jP_eE3vcKnY.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 400;
src: local('Quattrocento Sans Italic'), local('QuattrocentoSans-Italic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9a4lja2NVIDdIAAoMR5MfuElaRB0zMh0P2GEHJ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 400;
src: local('Quattrocento Sans Italic'), local('QuattrocentoSans-Italic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9a4lja2NVIDdIAAoMR5MfuElaRB0zMh032GA.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 700;
src: local('Quattrocento Sans Bold Italic'), local('QuattrocentoSans-BoldItalic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9X4lja2NVIDdIAAoMR5MfuElaRB0zMj_bTDXDojYsJ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 700;
src: local('Quattrocento Sans Bold Italic'), local('QuattrocentoSans-BoldItalic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9X4lja2NVIDdIAAoMR5MfuElaRB0zMj_bTDX7ojQ.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 400;
src: local('Quattrocento Sans'), local('QuattrocentoSans'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9c4lja2NVIDdIAAoMR5MfuElaRB0zHt0_uHA.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 400;
src: local('Quattrocento Sans'), local('QuattrocentoSans'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9c4lja2NVIDdIAAoMR5MfuElaRB0zJt08.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Sans Bold'), local('QuattrocentoSans-Bold'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9Z4lja2NVIDdIAAoMR5MfuElaRB0RyklrfPXzwiQ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Sans Bold'), local('QuattrocentoSans-Bold'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9Z4lja2NVIDdIAAoMR5MfuElaRB0RyklrRPXw.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8on7mTMuk.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8onXmTMuk.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8onTmTMuk.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8onrmTA.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfCFXUIJ.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfuFXUIJ.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfqFXUIJ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfSFXQ.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M9knj-SA.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M2knj-SA.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M3knj-SA.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M5kng.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23FafadWQ.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23OafadWQ.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23PafadWQ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23BafY.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

BIN
2 - Newton and Iterative methods/slides/css/fonts/KaTeX_Main-Regular.ttf Normal file
View File

Binary file not shown.

BIN
2 - Newton and Iterative methods/slides/css/fonts/KaTeX_Main-Regular.woff Normal file
View File

Binary file not shown.

BIN
2 - Newton and Iterative methods/slides/css/fonts/KaTeX_Main-Regular.woff2 Normal file
View File

Binary file not shown.

1
2 - Newton and Iterative methods/slides/css/katex.min.css vendored Normal file
View File

File diff suppressed because one or more lines are too long

8779
2 - Newton and Iterative methods/slides/css/style.css Normal file
View File

File diff suppressed because it is too large Load Diff

BIN
2 - Newton and Iterative methods/slides/err_fixed_point.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 11 KiB

BIN
2 - Newton and Iterative methods/slides/graph.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 16 KiB

BIN
2 - Newton and Iterative methods/slides/graph_small.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 13 KiB

71
2 - Newton and Iterative methods/slides/index.html Executable file
View File

@ -0,0 +1,71 @@
<!DOCTYPE html>
<html>
<head>
<meta name="robots" content="noindex">
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<title>Numerical Analysis - Slides</title>
<link rel="stylesheet" href="css/all.css">
<link href="css/css.css" rel="stylesheet">
<link href="css/css2.css" rel="stylesheet">
<link href="css/css3.css" rel="stylesheet">
<link href="css/css4.css" rel="stylesheet">
<link rel="stylesheet" href="css/style.css">
<link rel="stylesheet" href="css/katex.min.css">
<link rel="icon" href="https://hackersatporto.com/images/favicon.png">
<meta name="description" content="Porto's student computing society. We are a community focused on learning about and building things with technology.">
<script src="js/remark-latest.min.js"></script>
<script src="js/jquery.min.js"></script>
<script src="js/mark.min.js"></script>
<script src="js/MathJax.js"></script>
<script src="js/katex.min.js"></script>
<script src="js/auto-render.min.js"></script>
<script src="js/abcjs_basic_5.6.3-min.js"></script>
<script src="js/story.js"></script>
</head>
<body id="content" class="ma0 bg-white section-slides page-kind-page is-page-true feature-math feature-qrcode feature-nohighlight feature-music">
<script>
var hljs = remark.highlighter.engine;
var slideshow = remark.create({
ratio: "16:9",
countIncrementalSlides: false,
sourceUrl: 'text.md',
slideNumberFormat: '%current%',
highlightLanguage: "xaprb",
placeholder: "dummy value to end the list without a comma"
});
$(".remark-slide-content.fit-h1 h1").each(function(i, e) {
var $e = $(e);
var $p = $e.closest('div.remark-slide-container');
var needsToggle = !$p.hasClass('remark-visible');
if ( needsToggle ) $p.toggleClass('remark-visible');
while (e.scrollHeight > e.clientHeight + 1 && $e.css('font-size') != '1px') {
$e.css('font-size', (parseInt($e.css('font-size')) - 1) + "px");
}
if ( needsToggle ) $p.toggleClass('remark-visible');
});
$( function() {
if ( $( "body.feature-qrcode:not(.feature-noqrcode)" ).length ) {
if ( $( ".qrcode" ).length ) {
new QRCode($(".qrcode")[0], {
"useSVG": true,
"text": window.location.href.replace(/#\d*$/, "")
});
}
}
});
// Setup MathJax
MathJax.Hub.Config({
tex2jax: {
skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
}
});
MathJax.Hub.Configured();
</script>
</body>
</html>

19
2 - Newton and Iterative methods/slides/js/MathJax.js Normal file
View File

File diff suppressed because one or more lines are too long

3
2 - Newton and Iterative methods/slides/js/abcjs_basic_5.6.3-min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

1
2 - Newton and Iterative methods/slides/js/auto-render.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

19
2 - Newton and Iterative methods/slides/js/extensions/MathMenu.js Normal file
View File

File diff suppressed because one or more lines are too long

19
2 - Newton and Iterative methods/slides/js/extensions/MathZoom.js Normal file
View File

File diff suppressed because one or more lines are too long

2
2 - Newton and Iterative methods/slides/js/jquery.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

1
2 - Newton and Iterative methods/slides/js/katex.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

7
2 - Newton and Iterative methods/slides/js/mark.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

18
2 - Newton and Iterative methods/slides/js/remark-latest.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

146
2 - Newton and Iterative methods/slides/js/story.js Normal file
View File

@ -0,0 +1,146 @@
// Render KaTeX math typesetting, but only if the <body> has the class
// feature-math
$( function() {
if ( $( "body.feature-math:not(.feature-nomath)" ).length ) {
renderMathInElement(document.body);
}
});
// Render abcjs sheet music, but only if the <body> has the class
// feature-music. Do this by removing the <pre><code class="language-abc">
// and replacing it with a <p id="music-X"> which will be used to hold the
// generated sheet music. Remark's code syntax highlighting transforms the
// <code> block into a bunch of
// <div class="remark-code-line">K: Emin</div>
// one per line, so we have to reassemble those to get back linebreaks.
$( function() {
if ( $( "body.feature-music:not(.feature-nomusic)" ).length ) {
$( "code.language-abc, code.abc" ).each(function(i, e){
var $this = $(this);
var abc = $this.text();
if ( $this.hasClass("remark-code") ) {
abc = "";
$this.children().each(function(i, e) {
abc += "\n" + $(this).text();
});
abc = abc.trim();
}
var p = $this.parent().before('<p id="music-' + (i+1) + '">');
$this.parent().hide();
ABCJS.renderAbc("music-" + (i+1), abc, {
paddingtop: 0,
paddingbottom: 0,
paddingright: 0,
paddingleft: 0,
responsive: "resize"
});
});
}
});
// Render tweet-styled blockquotes, if <body> has the class feature-tweetquote and the blockquote ends
// with a twitter permalink URL.
$( function() {
if ( $( "body.feature-tweetquote:not(.feature-notweetquote)" ).length ) {
$("blockquote p:first-child a[href*='twitter.com']").each(function(i, e){
$(this.parentElement.parentElement).addClass("tweet sans-serif mw6");
$(this).addClass("no-underline b");
$(this).prepend('<i class="fl mr2 fab fa-twitter fa-2x pa2 ba br3">');
var m = $(this).attr('href').match(/twitter.com\/([^\/]*)\//);
if (m.length > 1) {
$(this).append('<br><span class="normal gray">@' + m[1] + '</span>');
}
});
}
});
// Render perspective book images, if <body> has the class
// feature-3dbook-covers
$( function() {
$( "body.feature-3dbook-covers article a" ).has( "img[src~='3dbook']" ).each(function(i, e) {
$( this ).addClass("book");
$( this ).parent().addClass("books");
});
});
// Highlight code listings, if <body> has the class
// feature-highlight
$( function() {
if ( $( "body.feature-highlight:not(.feature-nohighlight)" ).length ) {
hljs.initHighlightingOnLoad();
}
});
/* Turn images into figures with captions. Several scenarios are handled, in
* order:
* If the img is followed by an <em>, then transform the <em> into a caption.
* If the img has a title, use that as the caption.
* If the img has an alt, use that.
* The following features have to be enabled:
* - feature-figcaption has to be enabled or the whole thing is disabled
* - feature-figcaption-hidden makes the captions invisible till you hover
* - feature-fignum adds figure numbering
* - feature-figlink adds automatic links of text like "Figure 4"
*/
$( function() {
$("body.feature-figcaption:not(.feature-nofigcaption) article img").each(function(i, e) {
var $this = $(this);
// Don't put captions on images that have URL fragment pseudo-classes.
if ( $this.attr('src').match(/#/) ) return;
var $txt = false;
if ( $this.next().is("em") ) {
$txt = $this.next().html();
$this.next().remove();
} else {
$txt = $this.attr('title') ? $this.attr('title')
: $this.attr('alt') ? $this.attr('alt')
: false;
}
if ( $txt ) {
var $fig = $this.wrap('<figure id="fig-' + (i+1) + '">')
.after('<figcaption class="f5 lh-copy i ph3">' + $txt + '</figcaption>')
.parent();
}
});
if ( $("body.feature-figlink").length ) {
$("article p, article li").each(function(i, e) {
var $old = $(this).html();
var $new = $old.replace(/Figure\s+(\d+)/g, '<a href="#fig-$1">Figure $1</a>');
if ( $old !== $new ) {
$(this).html($new);
}
});
}
});
/* Add captions to tables.
* If the table is followed by a <p><em>, then transform the <em> into a caption.
* The following features have to be enabled:
* - feature-tablecaption has to be enabled or the whole thing is disabled
* - feature-fignum adds table numbering
* - feature-figlink adds automatic links of text like "Table 4"
*/
$( function() {
$("body.feature-tablecaption article table").each(function(i, e) {
var $this = $(this);
var $txt = false;
if ( $this.next().is("p") ) {
if ( $this.next().children().first().is("em:only-child") ) {
$txt = $this.next().children().first().html();
$this.next().remove();
}
}
if ( $txt ) {
$this.prepend('<caption id="tbl-' + (i+1) + '">' + $txt + '</caption>');
}
});
if ( $("body.feature-figlink").length ) {
$("article p, article li").each(function(i, e) {
var $old = $(this).html();
var $new = $old.replace(/Table\s+(\d+)/g, '<a href="#tbl-$1">Table $1</a>');
if ( $old !== $new ) {
$(this).html($new);
}
});
}
});

216
2 - Newton and Iterative methods/slides/text.md Normal file
View File

@ -0,0 +1,216 @@
class: title, smokescreen, shelf, no-footer
# Análise Numérica - Trabalho Prático 1
### Diogo Cordeiro | Hugo Sales | Pedro Costa | Ricardo Pimenta
---
name: motivacao
class: roomy
# Motivação
Pretende-se usar os métodos de Newton e iterativo simples para determinar um valor aproximado de um zero de
$$x^2 - cos(x)^2$$
---
name: 1a
class: compact
# 1.a)
Optamos por implementar o algoritmo pedido em C++ devido à possiblidade da aplicação de
templates e lambdas. Deste modo, foi-nos possível implementar os dois métodos pedidos partindo
de um algoritmo genérico pois a diferença entre o método de Newton
e o método iterativo consiste apenas na fórmula de recorrência. Assim, o método de Newton
pode ser visto como uma forma do método iterativo simples.
int main() {
auto F = [](double x){ return std::pow(x, 2.0) - std::pow(std::cos(x), 2.0); };
auto dF = [](double x){ return 2.0 * x + std::sin(2.0 * x); };
double epsilon = 5.0 * std::pow(10.0, -12.0);
std::cout << newton(0.8, epsilon, F, dF, 100000) << `\n`;
}
---
name: 1a
class: compact
# 1.a)
template<typename step_func>
double find_root(double x0, double epsilon, step_func step, long iter_limit) {
double x1 = x0, err;
long iter = 1;
do {
x0 = x1;
x1 = step(x0);
err = std::abs(x1 - x0);
} while(err > epsilon && iter++ < iter_limit);
return x1;
}
template<typename F_t, typename dF_t>
double newton(double x0, double epsilon, F_t F, dF_t dF, long iter_limit) {
return find_root(x0, epsilon,
[&F, &dF](double x0){ return x0 - F(x0)/dF(x0); },
iter_limit);
}
---
name: 1b
class: compact, img-right
# 1.b)
![Gráfico para `\(x \in [-1;1]\)`](./graph.png)
Como `\(\forall{x}: cos^2(x) \in [0:1]\)`, temos que `\(x^2 \geq cos^2(x)\)` para `\(|x| > 1\)`, sabemos que o comportamento de `\(f(x)\)` é dominado pelo comportamento de `\(x^2\)`, a qual só tem duas raízes.
A menor das raízes encontra-se no intervalo `\(]-\infty;0]\)` e a maior destas em `\([0;\infty[\)`.
---
name: 1b-cont
class: compact, img-right
# 1.b)
![Gráfico para `\(x \in [0.7;0.8]\)`](./graph_small.png)
Através da análise do gráfico, verificamos que o intervalo `\([0.7;0.8]\)` contém uma raíz. Definimos então `\(a = 0.7\)` e `\(b = 0.8\)`.
---
name: 1c
class: compact
# 1.c)
Queremos mostrar que as condições de aplicabilidade do método de Newton são satisfeitas no intervalo. Assim,
$$F(x) = x^2 - cos(x)^2$$
$$F'(x) = 2 \cdot x + sin(2 \cdot x)$$
$$F''(x) = 2 + 2 \cdot cos(2 \cdot x)$$
Como todas estas funções são compostas partindo de somas de funções contínuas em `\(\mathbb{R}\)`, são também continuas no intervalo considerado, verificando-se assim o primeiro critério.
Dado que,
$$F(a) < 0, F(b) < 0 \Rightarrow F(a) \cdot F(b) < 0$$
---
name: 1c-cont
class: compact
# 1.c)
Verifica-se também o segundo critério.
Temos que `\(F''(x) = 2 + 2 \cdot cos(2 \cdot x)\)`, logo `\(cos(2 \cdot x) \in [-1;1]\)`, por isso `\(2 \cdot cos(2 \cdot x) \in [-2; 2]\)`, e `\(2 \cdot cos(2 \cdot x) + 2 \in [0;4]\)` o que implica que `\(F''(x) \geq 0\)`, para `\(x \in \mathbb{R}\)` e por isso `\(F'(x)\)` é não decrescente em `\(\mathbb{R}\)`, ou seja, `\(F'(b) \geq F'(a)\)` e `\(F'(a) > 3\)` e por isso `\(F'(x) \neq 0 \forall{x} : \in [a;b]\)`, o que verifica o terceiro critério.
Como foi dito anteriormente, `\(F''(x) \geq 0\)` em `\(\mathbb{R}\)` e, por isso, também para `\(\forall{x} : \in [a;b]\)`, verificando-se a quarta condição.
Para `\(x_0 = b\)`, temos que `\(F(x_0) > 0\)` e `\(F''(x_0) > 0\)`, logo `\(F(x_0) \cdot F''(x_0) > 0\)`, verificando-se a quinta condição.
Então a sucessão gerada converge para a unica raíz no intervalo `\([a;b]\)`.
---
name: 1d
class: compact
# 1.d)
Aplicamos o programa apresentado em `\(1.a\)` e obtivemos os seguintes resultados.
Iterações | Erro estimado | Valor
----------|---------------|---------------
1 | 5.9e-02 | 0.740528800196
2 | 1.4e-03 | 0.739086050826
3 | 9.2e-07 | 0.739085133216
4 | 3.7e-13 | 0.739085133215
Verficamos que o erro estimado é aproximadamente metade do erro estimado da iteração amterior,
o que justificamos com o facto de que o resultado teórico
nos dizer que o erro converge segundo uma secessão de segunda ordem.
---
name: 1e
class: compact
# 1.e)
Aplicando a fórmula
<center>\( M = \frac{1}{2} \cdot \frac{\max \limits_{a \leq x \leq b} | F''(x) | }{\min \limits_{a \leq x \leq b} | F'(x) | } \)</center>
Como `\(F''(x)\)` é decrescente em `\([0;\pi/2]\)`, então `\(\max\limits_{a \leq x \leq b} | F''(x) | = |F''(a)|\)` e como `\(F'(x)\)` é não decrescente em `\(\mathbb{R}\)`, `\(\min\limits_{a \leq x \leq b} | F'(x) | = F'(a)\)`.
Assim, obtemos `\(M = \frac{F''(0.7)}{2F'(0.7)} \leq \frac{2.4}{2 \cdot 2.3} \leq 0.53\)` e
`\(n \geq \frac{ln(\alpha)}{ln(2)}, \alpha = \frac{ln(5 \cdot 10^{-14}) + ln(0.53)}{ln(0.53) + ln(10^{-1})}\)`, logo `\(\alpha \leq \frac{-32}{2.9} \leq 12\)` e `\(n \geq 4\)`, ou seja que com 4 iterações conseguimos um erro absoluto inferior a `\(5 \cdot 10^{-14}\)`.
---
name: 2a
class: compact
# 2.a)
Dada a implementação genérica do algoritmo, o método iterativo simples pode ser
implementado sucintamente como:
template<typename F_t>
double fixed_point(double x0, double epsilon, F_t f, long iter_limit) {
return find_root(x0, epsilon, f, iter_limit);
}
---
name: 2a-cont
class: compact
# 2.a)
e usado como
auto f = [](double x){ return std::cos(x); };
std::cout << fixed_point(0.8, epsilon, f, 100000) << `\n`;
A expressão de `\(\texttt{f}\)` foi obtida por manipulação algébrica do seguinte modo:
$$F(x) = 0 \iff x^2 - cos^2(x) = 0 \iff x^2 = cos^2(x) \iff x = \pm cos(x)$$
Logo no intervalo `\([a;b]\)`, temos que `\(x = cos(x)\)` ou seja `\(f(x) = cos(x)\)`.
---
name: 2b
class: img-right
![Gŕafico da comparação do erro estimado dos dois métodos](err_fixed_point.png)
# 2.b)
Partindo destes resultados, verificamos empiricameente a diferença na ordem de convergência dos dois métodos.
Obtivemos como valor final para este método o valor `\(0.739085133217\)`, que difere do valor calculado
como o método de Newton em `\(-2 \cdot 10^{-12}\)`, por isso concluimos numericamente que, para os valores iniciais usados, este método pode ser aplicado, sendo que aproxima o valor real.
---
name: 2b
class: img-right
![Gŕafico da comparação do erro estimado dos dois métodos](err_fixed_point.png)
# 2.b)
Verificando as condições de aplicabilidade deste método, temos que `\(f(x)\)` é continua em `\([a;b]\)`, verificando-se a primeira condição,
que `\(f(0.7) \approx 0.76484219\)` e `\(f(0.7) \approx 0.69670671\)` o que implica que `\(f([a;b]) \notin [a;b]\)`, o implica que não se verifica a segunda condição
e que `\(\forall x \in [0.7;0.8] : |-sin(x)| < 1\)`, o que significa que se verifica a terceira condição.
---
name: 2b
class: img-right
![Gŕafico da comparação do erro estimado dos dois métodos](err_fixed_point.png)
# 2.b)
Apesar de não se verificar a segunda condição, verificamos na mesma a convergência da sucessão, o que não contradiz os resultados teóricos, uma vez que estas condições são suficientes, mas não necessárias para esta convergência.

45
2 - Newton and Iterative methods/source/roots.cpp Normal file
View File

@ -0,0 +1,45 @@
#include <iostream>
#include <cmath>
#include <iomanip>
template<typename step_func>
double find_root(double x0, double epsilon, step_func step, long iter_limit) {
double x1 = x0, err;
long iter = 1;
std::cout << "Erros: ";
do {
x0 = x1;
x1 = step(x0);
err = std::abs(x1 - x0);
std::cout << iter << " "
<< std::setprecision(1) << std::scientific << err << " "
<< std::defaultfloat << std::setprecision(12) << x1 << '\n';
} while(err > epsilon && iter++ < iter_limit);
std::cout << '\n';
// std::cout << std::defaultfloat << std::setprecision(12);
return x1;
}
template<typename F_t, typename dF_t>
double newton(double x0, double epsilon, F_t F, dF_t dF, long iter_limit) {
return find_root(x0, epsilon, [&F, &dF](double x0){ return x0 - F(x0)/dF(x0); }, iter_limit);
}
template<typename F_t>
double fixed_point(double x0, double epsilon, F_t F, long iter_limit) {
return find_root(x0, epsilon, F, iter_limit);
}
int main() {
auto F = [](double x){ return std::pow(x, 2.0) - std::pow(std::cos(x), 2.0); };
auto dF = [](double x){ return 2.0 * x + std::sin(2.0 * x); };
// std::cout.precision(17);
double epsilon = 5.0 * std::pow(10.0, -12.0);
std::cout << newton(0.8, epsilon, F, dF, 100000) << '\n';
auto F2 = [](double x){ return std::cos(x); };
std::cout << fixed_point(0.8, epsilon, F2, 100000) << '\n';
}

674
3 - Splines/LICENSE Normal file
View File

@ -0,0 +1,674 @@
GNU GENERAL PUBLIC LICENSE
Version 3, 29 June 2007
Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>
Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.
Preamble
The GNU General Public License is a free, copyleft license for
software and other kinds of works.
The licenses for most software and other practical works are designed
to take away your freedom to share and change the works. By contrast,
the GNU General Public License is intended to guarantee your freedom to
share and change all versions of a program--to make sure it remains free
software for all its users. We, the Free Software Foundation, use the
GNU General Public License for most of our software; it applies also to
any other work released this way by its authors. You can apply it to
your programs, too.
When we speak of free software, we are referring to freedom, not
price. Our General Public Licenses are designed to make sure that you
have the freedom to distribute copies of free software (and charge for
them if you wish), that you receive source code or can get it if you
want it, that you can change the software or use pieces of it in new
free programs, and that you know you can do these things.
To protect your rights, we need to prevent others from denying you
these rights or asking you to surrender the rights. Therefore, you have
certain responsibilities if you distribute copies of the software, or if
you modify it: responsibilities to respect the freedom of others.
For example, if you distribute copies of such a program, whether
gratis or for a fee, you must pass on to the recipients the same
freedoms that you received. You must make sure that they, too, receive
or can get the source code. And you must show them these terms so they
know their rights.
Developers that use the GNU GPL protect your rights with two steps:
(1) assert copyright on the software, and (2) offer you this License
giving you legal permission to copy, distribute and/or modify it.
For the developers' and authors' protection, the GPL clearly explains
that there is no warranty for this free software. For both users' and
authors' sake, the GPL requires that modified versions be marked as
changed, so that their problems will not be attributed erroneously to
authors of previous versions.
Some devices are designed to deny users access to install or run
modified versions of the software inside them, although the manufacturer
can do so. This is fundamentally incompatible with the aim of
protecting users' freedom to change the software. The systematic
pattern of such abuse occurs in the area of products for individuals to
use, which is precisely where it is most unacceptable. Therefore, we
have designed this version of the GPL to prohibit the practice for those
products. If such problems arise substantially in other domains, we
stand ready to extend this provision to those domains in future versions
of the GPL, as needed to protect the freedom of users.
Finally, every program is threatened constantly by software patents.
States should not allow patents to restrict development and use of
software on general-purpose computers, but in those that do, we wish to
avoid the special danger that patents applied to a free program could
make it effectively proprietary. To prevent this, the GPL assures that
patents cannot be used to render the program non-free.
The precise terms and conditions for copying, distribution and
modification follow.
TERMS AND CONDITIONS
0. Definitions.
"This License" refers to version 3 of the GNU General Public License.
"Copyright" also means copyright-like laws that apply to other kinds of
works, such as semiconductor masks.
"The Program" refers to any copyrightable work licensed under this
License. Each licensee is addressed as "you". "Licensees" and
"recipients" may be individuals or organizations.
To "modify" a work means to copy from or adapt all or part of the work
in a fashion requiring copyright permission, other than the making of an
exact copy. The resulting work is called a "modified version" of the
earlier work or a work "based on" the earlier work.
A "covered work" means either the unmodified Program or a work based
on the Program.
To "propagate" a work means to do anything with it that, without
permission, would make you directly or secondarily liable for
infringement under applicable copyright law, except executing it on a
computer or modifying a private copy. Propagation includes copying,
distribution (with or without modification), making available to the
public, and in some countries other activities as well.
To "convey" a work means any kind of propagation that enables other
parties to make or receive copies. Mere interaction with a user through
a computer network, with no transfer of a copy, is not conveying.
An interactive user interface displays "Appropriate Legal Notices"
to the extent that it includes a convenient and prominently visible
feature that (1) displays an appropriate copyright notice, and (2)
tells the user that there is no warranty for the work (except to the
extent that warranties are provided), that licensees may convey the
work under this License, and how to view a copy of this License. If
the interface presents a list of user commands or options, such as a
menu, a prominent item in the list meets this criterion.
1. Source Code.
The "source code" for a work means the preferred form of the work
for making modifications to it. "Object code" means any non-source
form of a work.
A "Standard Interface" means an interface that either is an official
standard defined by a recognized standards body, or, in the case of
interfaces specified for a particular programming language, one that
is widely used among developers working in that language.
The "System Libraries" of an executable work include anything, other
than the work as a whole, that (a) is included in the normal form of
packaging a Major Component, but which is not part of that Major
Component, and (b) serves only to enable use of the work with that
Major Component, or to implement a Standard Interface for which an
implementation is available to the public in source code form. A
"Major Component", in this context, means a major essential component
(kernel, window system, and so on) of the specific operating system
(if any) on which the executable work runs, or a compiler used to
produce the work, or an object code interpreter used to run it.
The "Corresponding Source" for a work in object code form means all
the source code needed to generate, install, and (for an executable
work) run the object code and to modify the work, including scripts to
control those activities. However, it does not include the work's
System Libraries, or general-purpose tools or generally available free
programs which are used unmodified in performing those activities but
which are not part of the work. For example, Corresponding Source
includes interface definition files associated with source files for
the work, and the source code for shared libraries and dynamically
linked subprograms that the work is specifically designed to require,
such as by intimate data communication or control flow between those
subprograms and other parts of the work.
The Corresponding Source need not include anything that users
can regenerate automatically from other parts of the Corresponding
Source.
The Corresponding Source for a work in source code form is that
same work.
2. Basic Permissions.
All rights granted under this License are granted for the term of
copyright on the Program, and are irrevocable provided the stated
conditions are met. This License explicitly affirms your unlimited
permission to run the unmodified Program. The output from running a
covered work is covered by this License only if the output, given its
content, constitutes a covered work. This License acknowledges your
rights of fair use or other equivalent, as provided by copyright law.
You may make, run and propagate covered works that you do not
convey, without conditions so long as your license otherwise remains
in force. You may convey covered works to others for the sole purpose
of having them make modifications exclusively for you, or provide you
with facilities for running those works, provided that you comply with
the terms of this License in conveying all material for which you do
not control copyright. Those thus making or running the covered works
for you must do so exclusively on your behalf, under your direction
and control, on terms that prohibit them from making any copies of
your copyrighted material outside their relationship with you.
Conveying under any other circumstances is permitted solely under
the conditions stated below. Sublicensing is not allowed; section 10
makes it unnecessary.
3. Protecting Users' Legal Rights From Anti-Circumvention Law.
No covered work shall be deemed part of an effective technological
measure under any applicable law fulfilling obligations under article
11 of the WIPO copyright treaty adopted on 20 December 1996, or
similar laws prohibiting or restricting circumvention of such
measures.
When you convey a covered work, you waive any legal power to forbid
circumvention of technological measures to the extent such circumvention
is effected by exercising rights under this License with respect to
the covered work, and you disclaim any intention to limit operation or
modification of the work as a means of enforcing, against the work's
users, your or third parties' legal rights to forbid circumvention of
technological measures.
4. Conveying Verbatim Copies.
You may convey verbatim copies of the Program's source code as you
receive it, in any medium, provided that you conspicuously and
appropriately publish on each copy an appropriate copyright notice;
keep intact all notices stating that this License and any
non-permissive terms added in accord with section 7 apply to the code;
keep intact all notices of the absence of any warranty; and give all
recipients a copy of this License along with the Program.
You may charge any price or no price for each copy that you convey,
and you may offer support or warranty protection for a fee.
5. Conveying Modified Source Versions.
You may convey a work based on the Program, or the modifications to
produce it from the Program, in the form of source code under the
terms of section 4, provided that you also meet all of these conditions:
a) The work must carry prominent notices stating that you modified
it, and giving a relevant date.
b) The work must carry prominent notices stating that it is
released under this License and any conditions added under section
7. This requirement modifies the requirement in section 4 to
"keep intact all notices".
c) You must license the entire work, as a whole, under this
License to anyone who comes into possession of a copy. This
License will therefore apply, along with any applicable section 7
additional terms, to the whole of the work, and all its parts,
regardless of how they are packaged. This License gives no
permission to license the work in any other way, but it does not
invalidate such permission if you have separately received it.
d) If the work has interactive user interfaces, each must display
Appropriate Legal Notices; however, if the Program has interactive
interfaces that do not display Appropriate Legal Notices, your
work need not make them do so.
A compilation of a covered work with other separate and independent
works, which are not by their nature extensions of the covered work,
and which are not combined with it such as to form a larger program,
in or on a volume of a storage or distribution medium, is called an
"aggregate" if the compilation and its resulting copyright are not
used to limit the access or legal rights of the compilation's users
beyond what the individual works permit. Inclusion of a covered work
in an aggregate does not cause this License to apply to the other
parts of the aggregate.
6. Conveying Non-Source Forms.
You may convey a covered work in object code form under the terms
of sections 4 and 5, provided that you also convey the
machine-readable Corresponding Source under the terms of this License,
in one of these ways:
a) Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by the
Corresponding Source fixed on a durable physical medium
customarily used for software interchange.
b) Convey the object code in, or embodied in, a physical product
(including a physical distribution medium), accompanied by a
written offer, valid for at least three years and valid for as
long as you offer spare parts or customer support for that product
model, to give anyone who possesses the object code either (1) a
copy of the Corresponding Source for all the software in the
product that is covered by this License, on a durable physical
medium customarily used for software interchange, for a price no
more than your reasonable cost of physically performing this
conveying of source, or (2) access to copy the
Corresponding Source from a network server at no charge.
c) Convey individual copies of the object code with a copy of the
written offer to provide the Corresponding Source. This
alternative is allowed only occasionally and noncommercially, and
only if you received the object code with such an offer, in accord
with subsection 6b.
d) Convey the object code by offering access from a designated
place (gratis or for a charge), and offer equivalent access to the
Corresponding Source in the same way through the same place at no
further charge. You need not require recipients to copy the
Corresponding Source along with the object code. If the place to
copy the object code is a network server, the Corresponding Source
may be on a different server (operated by you or a third party)
that supports equivalent copying facilities, provided you maintain
clear directions next to the object code saying where to find the
Corresponding Source. Regardless of what server hosts the
Corresponding Source, you remain obligated to ensure that it is
available for as long as needed to satisfy these requirements.
e) Convey the object code using peer-to-peer transmission, provided
you inform other peers where the object code and Corresponding
Source of the work are being offered to the general public at no
charge under subsection 6d.
A separable portion of the object code, whose source code is excluded
from the Corresponding Source as a System Library, need not be
included in conveying the object code work.
A "User Product" is either (1) a "consumer product", which means any
tangible personal property which is normally used for personal, family,
or household purposes, or (2) anything designed or sold for incorporation
into a dwelling. In determining whether a product is a consumer product,
doubtful cases shall be resolved in favor of coverage. For a particular
product received by a particular user, "normally used" refers to a
typical or common use of that class of product, regardless of the status
of the particular user or of the way in which the particular user
actually uses, or expects or is expected to use, the product. A product
is a consumer product regardless of whether the product has substantial
commercial, industrial or non-consumer uses, unless such uses represent
the only significant mode of use of the product.
"Installation Information" for a User Product means any methods,
procedures, authorization keys, or other information required to install
and execute modified versions of a covered work in that User Product from
a modified version of its Corresponding Source. The information must
suffice to ensure that the continued functioning of the modified object
code is in no case prevented or interfered with solely because
modification has been made.
If you convey an object code work under this section in, or with, or
specifically for use in, a User Product, and the conveying occurs as
part of a transaction in which the right of possession and use of the
User Product is transferred to the recipient in perpetuity or for a
fixed term (regardless of how the transaction is characterized), the
Corresponding Source conveyed under this section must be accompanied
by the Installation Information. But this requirement does not apply
if neither you nor any third party retains the ability to install
modified object code on the User Product (for example, the work has
been installed in ROM).
The requirement to provide Installation Information does not include a
requirement to continue to provide support service, warranty, or updates
for a work that has been modified or installed by the recipient, or for
the User Product in which it has been modified or installed. Access to a
network may be denied when the modification itself materially and
adversely affects the operation of the network or violates the rules and
protocols for communication across the network.
Corresponding Source conveyed, and Installation Information provided,
in accord with this section must be in a format that is publicly
documented (and with an implementation available to the public in
source code form), and must require no special password or key for
unpacking, reading or copying.
7. Additional Terms.
"Additional permissions" are terms that supplement the terms of this
License by making exceptions from one or more of its conditions.
Additional permissions that are applicable to the entire Program shall
be treated as though they were included in this License, to the extent
that they are valid under applicable law. If additional permissions
apply only to part of the Program, that part may be used separately
under those permissions, but the entire Program remains governed by
this License without regard to the additional permissions.
When you convey a copy of a covered work, you may at your option
remove any additional permissions from that copy, or from any part of
it. (Additional permissions may be written to require their own
removal in certain cases when you modify the work.) You may place
additional permissions on material, added by you to a covered work,
for which you have or can give appropriate copyright permission.
Notwithstanding any other provision of this License, for material you
add to a covered work, you may (if authorized by the copyright holders of
that material) supplement the terms of this License with terms:
a) Disclaiming warranty or limiting liability differently from the
terms of sections 15 and 16 of this License; or
b) Requiring preservation of specified reasonable legal notices or
author attributions in that material or in the Appropriate Legal
Notices displayed by works containing it; or
c) Prohibiting misrepresentation of the origin of that material, or
requiring that modified versions of such material be marked in
reasonable ways as different from the original version; or
d) Limiting the use for publicity purposes of names of licensors or
authors of the material; or
e) Declining to grant rights under trademark law for use of some
trade names, trademarks, or service marks; or
f) Requiring indemnification of licensors and authors of that
material by anyone who conveys the material (or modified versions of
it) with contractual assumptions of liability to the recipient, for
any liability that these contractual assumptions directly impose on
those licensors and authors.
All other non-permissive additional terms are considered "further
restrictions" within the meaning of section 10. If the Program as you
received it, or any part of it, contains a notice stating that it is
governed by this License along with a term that is a further
restriction, you may remove that term. If a license document contains
a further restriction but permits relicensing or conveying under this
License, you may add to a covered work material governed by the terms
of that license document, provided that the further restriction does
not survive such relicensing or conveying.
If you add terms to a covered work in accord with this section, you
must place, in the relevant source files, a statement of the
additional terms that apply to those files, or a notice indicating
where to find the applicable terms.
Additional terms, permissive or non-permissive, may be stated in the
form of a separately written license, or stated as exceptions;
the above requirements apply either way.
8. Termination.
You may not propagate or modify a covered work except as expressly
provided under this License. Any attempt otherwise to propagate or
modify it is void, and will automatically terminate your rights under
this License (including any patent licenses granted under the third
paragraph of section 11).
However, if you cease all violation of this License, then your
license from a particular copyright holder is reinstated (a)
provisionally, unless and until the copyright holder explicitly and
finally terminates your license, and (b) permanently, if the copyright
holder fails to notify you of the violation by some reasonable means
prior to 60 days after the cessation.
Moreover, your license from a particular copyright holder is
reinstated permanently if the copyright holder notifies you of the
violation by some reasonable means, this is the first time you have
received notice of violation of this License (for any work) from that
copyright holder, and you cure the violation prior to 30 days after
your receipt of the notice.
Termination of your rights under this section does not terminate the
licenses of parties who have received copies or rights from you under
this License. If your rights have been terminated and not permanently
reinstated, you do not qualify to receive new licenses for the same
material under section 10.
9. Acceptance Not Required for Having Copies.
You are not required to accept this License in order to receive or
run a copy of the Program. Ancillary propagation of a covered work
occurring solely as a consequence of using peer-to-peer transmission
to receive a copy likewise does not require acceptance. However,
nothing other than this License grants you permission to propagate or
modify any covered work. These actions infringe copyright if you do
not accept this License. Therefore, by modifying or propagating a
covered work, you indicate your acceptance of this License to do so.
10. Automatic Licensing of Downstream Recipients.
Each time you convey a covered work, the recipient automatically
receives a license from the original licensors, to run, modify and
propagate that work, subject to this License. You are not responsible
for enforcing compliance by third parties with this License.
An "entity transaction" is a transaction transferring control of an
organization, or substantially all assets of one, or subdividing an
organization, or merging organizations. If propagation of a covered
work results from an entity transaction, each party to that
transaction who receives a copy of the work also receives whatever
licenses to the work the party's predecessor in interest had or could
give under the previous paragraph, plus a right to possession of the
Corresponding Source of the work from the predecessor in interest, if
the predecessor has it or can get it with reasonable efforts.
You may not impose any further restrictions on the exercise of the
rights granted or affirmed under this License. For example, you may
not impose a license fee, royalty, or other charge for exercise of
rights granted under this License, and you may not initiate litigation
(including a cross-claim or counterclaim in a lawsuit) alleging that
any patent claim is infringed by making, using, selling, offering for
sale, or importing the Program or any portion of it.
11. Patents.
A "contributor" is a copyright holder who authorizes use under this
License of the Program or a work on which the Program is based. The
work thus licensed is called the contributor's "contributor version".
A contributor's "essential patent claims" are all patent claims
owned or controlled by the contributor, whether already acquired or
hereafter acquired, that would be infringed by some manner, permitted
by this License, of making, using, or selling its contributor version,
but do not include claims that would be infringed only as a
consequence of further modification of the contributor version. For
purposes of this definition, "control" includes the right to grant
patent sublicenses in a manner consistent with the requirements of
this License.
Each contributor grants you a non-exclusive, worldwide, royalty-free
patent license under the contributor's essential patent claims, to
make, use, sell, offer for sale, import and otherwise run, modify and
propagate the contents of its contributor version.
In the following three paragraphs, a "patent license" is any express
agreement or commitment, however denominated, not to enforce a patent
(such as an express permission to practice a patent or covenant not to
sue for patent infringement). To "grant" such a patent license to a
party means to make such an agreement or commitment not to enforce a
patent against the party.
If you convey a covered work, knowingly relying on a patent license,
and the Corresponding Source of the work is not available for anyone
to copy, free of charge and under the terms of this License, through a
publicly available network server or other readily accessible means,
then you must either (1) cause the Corresponding Source to be so
available, or (2) arrange to deprive yourself of the benefit of the
patent license for this particular work, or (3) arrange, in a manner
consistent with the requirements of this License, to extend the patent
license to downstream recipients. "Knowingly relying" means you have
actual knowledge that, but for the patent license, your conveying the
covered work in a country, or your recipient's use of the covered work
in a country, would infringe one or more identifiable patents in that
country that you have reason to believe are valid.
If, pursuant to or in connection with a single transaction or
arrangement, you convey, or propagate by procuring conveyance of, a
covered work, and grant a patent license to some of the parties
receiving the covered work authorizing them to use, propagate, modify
or convey a specific copy of the covered work, then the patent license
you grant is automatically extended to all recipients of the covered
work and works based on it.
A patent license is "discriminatory" if it does not include within
the scope of its coverage, prohibits the exercise of, or is
conditioned on the non-exercise of one or more of the rights that are
specifically granted under this License. You may not convey a covered
work if you are a party to an arrangement with a third party that is
in the business of distributing software, under which you make payment
to the third party based on the extent of your activity of conveying
the work, and under which the third party grants, to any of the
parties who would receive the covered work from you, a discriminatory
patent license (a) in connection with copies of the covered work
conveyed by you (or copies made from those copies), or (b) primarily
for and in connection with specific products or compilations that
contain the covered work, unless you entered into that arrangement,
or that patent license was granted, prior to 28 March 2007.
Nothing in this License shall be construed as excluding or limiting
any implied license or other defenses to infringement that may
otherwise be available to you under applicable patent law.
12. No Surrender of Others' Freedom.
If conditions are imposed on you (whether by court order, agreement or
otherwise) that contradict the conditions of this License, they do not
excuse you from the conditions of this License. If you cannot convey a
covered work so as to satisfy simultaneously your obligations under this
License and any other pertinent obligations, then as a consequence you may
not convey it at all. For example, if you agree to terms that obligate you
to collect a royalty for further conveying from those to whom you convey
the Program, the only way you could satisfy both those terms and this
License would be to refrain entirely from conveying the Program.
13. Use with the GNU Affero General Public License.
Notwithstanding any other provision of this License, you have
permission to link or combine any covered work with a work licensed
under version 3 of the GNU Affero General Public License into a single
combined work, and to convey the resulting work. The terms of this
License will continue to apply to the part which is the covered work,
but the special requirements of the GNU Affero General Public License,
section 13, concerning interaction through a network will apply to the
combination as such.
14. Revised Versions of this License.
The Free Software Foundation may publish revised and/or new versions of
the GNU General Public License from time to time. Such new versions will
be similar in spirit to the present version, but may differ in detail to
address new problems or concerns.
Each version is given a distinguishing version number. If the
Program specifies that a certain numbered version of the GNU General
Public License "or any later version" applies to it, you have the
option of following the terms and conditions either of that numbered
version or of any later version published by the Free Software
Foundation. If the Program does not specify a version number of the
GNU General Public License, you may choose any version ever published
by the Free Software Foundation.
If the Program specifies that a proxy can decide which future
versions of the GNU General Public License can be used, that proxy's
public statement of acceptance of a version permanently authorizes you
to choose that version for the Program.
Later license versions may give you additional or different
permissions. However, no additional obligations are imposed on any
author or copyright holder as a result of your choosing to follow a
later version.
15. Disclaimer of Warranty.
THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
16. Limitation of Liability.
IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
SUCH DAMAGES.
17. Interpretation of Sections 15 and 16.
If the disclaimer of warranty and limitation of liability provided
above cannot be given local legal effect according to their terms,
reviewing courts shall apply local law that most closely approximates
an absolute waiver of all civil liability in connection with the
Program, unless a warranty or assumption of liability accompanies a
copy of the Program in return for a fee.
END OF TERMS AND CONDITIONS
How to Apply These Terms to Your New Programs
If you develop a new program, and you want it to be of the greatest
possible use to the public, the best way to achieve this is to make it
free software which everyone can redistribute and change under these terms.
To do so, attach the following notices to the program. It is safest
to attach them to the start of each source file to most effectively
state the exclusion of warranty; and each file should have at least
the "copyright" line and a pointer to where the full notice is found.
<one line to give the program's name and a brief idea of what it does.>
Copyright (C) <year> <name of author>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <https://www.gnu.org/licenses/>.
Also add information on how to contact you by electronic and paper mail.
If the program does terminal interaction, make it output a short
notice like this when it starts in an interactive mode:
<program> Copyright (C) <year> <name of author>
This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
This is free software, and you are welcome to redistribute it
under certain conditions; type `show c' for details.
The hypothetical commands `show w' and `show c' should show the appropriate
parts of the General Public License. Of course, your program's commands
might be different; for a GUI interface, you would use an "about box".
You should also get your employer (if you work as a programmer) or school,
if any, to sign a "copyright disclaimer" for the program, if necessary.
For more information on this, and how to apply and follow the GNU GPL, see
<https://www.gnu.org/licenses/>.
The GNU General Public License does not permit incorporating your program
into proprietary programs. If your program is a subroutine library, you
may consider it more useful to permit linking proprietary applications with
the library. If this is what you want to do, use the GNU Lesser General
Public License instead of this License. But first, please read
<https://www.gnu.org/licenses/why-not-lgpl.html>.

255
3 - Splines/README.md Normal file
View File

@ -0,0 +1,255 @@
---
title: "Análise Numérica - Trabalho Prático 3"
author:
- Diogo Cordeiro
- Hugo Sales
- Pedro Costa
geometry: margin=2cm
output: pdf_document
---
### Motivação
Pretende-se interpolar uma função através do método de Newton em diferenças divididas,
construir o spline cúbico natural e proceder a uma comparação e interpretação dos resultados obtidos.
### 1.a)
using points_t = std::pair<std::vector<double>, std::vector<double>>;
using matrix_t = std::vector<std::vector<double>>;
std::vector<double> newton_differences(points_t points) {
std::vector<double> factors{};
auto &[x, fx] = points;
int n = points.first.size();
for (int i = 1; i <= n; ++i) {
factors.push_back(fx[0]);
for (int j = 0; j < (n - i); ++j) {
fx[j] = (fx[j + 1] - fx[j]) / (x[j + i] - x[j]);
}
}
return factors;
}
double newton_polynomial(points_t points, std::vector<double> factors, double x) {
auto xs = points.first;
int n = points.second.size() - 1;
double val = 0;
for (int k = 0; k <= n; ++k) {
double acc = 1;
for(int i = 0; i < k; ++i) {
acc *= (x - xs[i]);
}
val += acc * factors[k];
}
return val;
}
\pagebreak
void exercise_a(points_t points) {
auto &[xs, fx] = points;
auto factors = newton_differences(points);
std::ofstream poly{"a_polynomial.txt"};
for(double x = 0; x < 4; x += 0.001)
poly << x << " " << newton_polynomial(points, factors, x) << '\n';
std::ofstream spline{"a_spline.txt"};
unsigned long n = xs.size();
matrix_t mat(n, std::vector<double>(n + 1));
calculate_natural_cubic_spline_matrix(points, mat);
for(double x = 0; x < 4; x += 0.001)
spline << x << " " << natural_cubic_spline(points, mat, x) << '\n';
}
### 1.b)
void calculate_natural_cubic_spline_matrix(points_t points, matrix_t &mat) {
auto &[xs, fx] = points;
int n = xs.size();
// Construção da matriz
for (int i = 1; i < n - 1; ++i) {
mat[i][i - 1] = (xs[i] - xs[i - 1])/6;
mat[i][i] = (xs[i + 1] - xs[i - 1])/3;
mat[i][i + 1] = (xs[i + 1] - xs[i])/6;
mat[i][n] = (fx[i + 1] - fx[i])/(xs[i + 1] - xs[i]) -
(fx[i] - fx[i - 1])/(xs[i] - xs[i - 1]);
}
mat[0][0] = 1;
mat[0][n - 1] = 0;
mat[n - 1][n - 1] = 1;
mat[n - 1][n] = 0;
// Passar para a forma triangular
for (int k = 0; k < n; ++k) {
for (int i = k + 1; i < n; ++i) {
if (mat[k][k] != 0) {
double mul = mat[i][k]/mat[k][k];
for (int j = k; j < n; ++j) {
mat[i][j] -= mul * mat[k][j];
}
mat[i][n] -= mul * mat[k][n];
}
}
}
\pagebreak
// Resolução da matriz
for (int i = n - 1; i > 0; --i) {
if (mat[i][i] != 0) {
double mul = mat[i - 1][i]/mat[i][i];
for (int j = 0; j < n + 1; ++j) {
mat[i - 1][j] -= mul * mat[i][j];
}
mat[i][n] /= mat[i][i];
mat[i][i] = 1;
}
}
}
double natural_cubic_spline(points_t points, matrix_t &mat, double x) {
auto &[xs, fx] = points;
int n = xs.size();
int i = 0;
for (int i_ = 0; i_ < n; ++i_) {
if (xs[i_] > x){
i = i_;
break;
}
}
double hi = xs[i] - xs[i - 1];
return mat[i - 1][n] * std::pow((xs[i] - x), 3)/(6 * hi) +
mat[i][n] * std::pow((x - xs[i - 1]), 3)/(6 * hi) +
(fx[i - 1] - mat[i - 1][n] * (hi * hi)/6)*(xs[i] - x)/hi +
(fx[i] - mat[i][n] * (hi * hi)/6)*(x - xs[i - 1])/hi;
}
void exercise_b() {
points_t points;
auto &[xs, fx] = points;
auto f = [](double x) { return 4 * std::pow(x, 2) + std::sin(9 * x); };
for (double x = -1; x <= 1; x += (1 - -1)/8.0) {
xs.push_back(x);
fx.push_back(f(x));
}
std::ofstream poly{"b_polynomial.txt"};
auto factors = newton_differences(points);
for(double x = -1; x < 1; x += 0.001)
poly << x << " " << newton_polynomial(points, factors, x) << '\n';
std::ofstream spline{"b_spline.txt"};
unsigned long n = xs.size();
matrix_t mat(n, std::vector<double>(n + 1));
calculate_natural_cubic_spline_matrix(points, mat);
for(double x = -1; x < 1; x += 0.001)
spline << x << " " << natural_cubic_spline(points, mat, x) << '\n';
}
### 2.a)
![Comparação Polinómio e Spline](a.png){ width=10cm }
Através dos graficos é possivel verificar que os valores das funções coincidem
no pontos de interpolação, como era esperado. Não é possivel determinal qual a
melhor aproximação sem conhecer a função original. O programa foi modificado para
imprimir a matriz inicial assim como o resultado final, que foi utilizado para calcular
o vetor resíduo e a norma deste. Este script encontra-se em baixo.
#!/usr/bin/env python3
import numpy as np
from scipy.linalg import solve
from numpy.linalg import norm
# Matriz construída pelo programa em C++
A = np.matrix([[1, 0, 0, 0, 0, 0],
[0.166667, 0.666667, 0.166667, 0, 0, 0],
[0, 0.166667, 0.5, 0.0833333, 0, 0],
[0, 0, 0.0833333, 0.333333, 0.0833333, 0],
[0, 0, 0, 0.0833333, 0.5, 0.166667],
[0, 0, 0, 0, 0, 1]])
b = np.array([0, 1.2, -1.2, 0.8, 0.4, 0])
x2 = np.array([0, 2.76846, -3.87386, 3.30622, 0.248963, 0])
x = solve(A, b)
print(norm(x - x2))
Obteu-se para a norma do vetor resíduo o valor $1.0e-05$ que
é considerado aceitável visto que os dados são apresentados
com dois algarismos significativos.
\pagebreak
### 2.b)
#### i)
$$
\begin{array}{c|ccccccccc}{x_{i}} & {-1} & {-0.75} & {-0.5} & {-0.25} & {0} &
{0.25} & {0.5} & {0.75} & {1} \\ \hline {f_{i}} & {3.58788} & {1.79996} &
{1.97753} & {-0.528073} & {0} & {1.02807} & {0.0224699} & {2.70004} & {4.41212}
\end{array}
$$
#### ii)
![Comparação de polinomio, spline e a função](b.png){ width=10cm }
De forma análoga ao exercício 2.a) foi calculada a norma do vetor resíduo
usando o seguinte script:
A = np.matrix(
[[1, 0, 0, 0, 0, 0, 0, 0, 0],
[0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0, 0, 0],
[0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0, 0],
[0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0],
[0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0],
[0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0],
[0, 0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0],
[0, 0, 0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667],
[0, 0, 0, 0, 0, 0, 0, 0, 1]])
b = np.array([0, 7.862, -10.7327, 12.1347, 2, -8.13471, 14.7327, -3.862, 0])
x2 = np.array([0, 73.9996, -107.31, 97.6564, 7.91753, -81.3265, 122.156, -53.7109, 0])
x = solve(A, b)
print(norm(x - x2))
Obteu-se para a norma do vetor resíduo o valor $9.0e-04$ que
é considerado aceitável visto que este valor tem uma magnitude pequena.
\pagebreak
#### iii)
x | f(x) | p(x) | abs(f(x)-p(x)) | s(x) | f(x)-s(x)
-----|------------|----------|--------------------|-------------------|-------------------
0.30 | 0.78737988 | 0.923318 | $$ 1.4*10^{-1} $$ | 0.826621 | $$ 4.0*10^{-2} $$
0.83 | 3.68278040 | 4.834190 | 1.2 | $$ 2.4*10^{-1} $$ | $$ 2.4*10^{-1} $$
Verificamos que o erro do spline cubico é inferior em ambos os casos. Além disso
em ambos os casos o erro é maior quando a abcissa é mais distante do centro do
intervalo.
#### iv)
É possivel observar que a interpolação pelo spline aproxima, em geral, melhor a
função dada do que o polinomio. Em particular verifica-se que o erro do polinomio
acentua-se à medida que as abcissas se afastam do centro do intervalo de
interpolação.

BIN
3 - Splines/a.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 14 KiB

BIN
3 - Splines/b.png Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 15 KiB

4001
3 - Splines/graphics/a_polynomial.txt Normal file
View File

File diff suppressed because it is too large Load Diff

4001
3 - Splines/graphics/a_spline.txt Normal file
View File

File diff suppressed because it is too large Load Diff

2000
3 - Splines/graphics/b_polynomial.txt Normal file
View File

File diff suppressed because it is too large Load Diff

2000
3 - Splines/graphics/b_spline.txt Normal file
View File

File diff suppressed because it is too large Load Diff

11
3 - Splines/graphics/graphs.gnuplot Executable file
View File

@ -0,0 +1,11 @@
#!/usr/bin/gnuplot
set output "a.png";
set terminal png size 1200,900 enhanced font "Helvetica, 20";
plot "a_polynomial.txt" with l lw 4 t "Polinomial", "a_spline.txt" with l lw 4 t "Spline Cúbico";
set output "b.png";
f(x) = 4*x**2 + sin(9*x)
plot f(x) with l lw 4 t "f(x)", "b_polynomial.txt" with l lw 4 t "Polinomial", "b_spline.txt" with l lw 4 t "Spline Cúbico";

BIN
3 - Splines/print.pdf Normal file
View File

Binary file not shown.

6
3 - Splines/source/input.txt Normal file
View File

@ -0,0 +1,6 @@
0 1.4
1 0.6
2 1.0
2.5 0.6
3 0.6
4 1.0

51
3 - Splines/source/matriz.txt Normal file
View File

@ -0,0 +1,51 @@
mat:
1,0,0,0,0,0,
0.166667,0.666667,0.166667,0,0,0,
0,0.166667,0.5,0.0833333,0,0,
0,0,0.0833333,0.333333,0.0833333,0,
0,0,0,0.0833333,0.5,0.166667,
0,0,0,0,0,1,
b:
0
1.2
-1.2
0.8
0.4
0
x:
0
2.76846
-3.87386
3.30622
0.248963
0
mat:
1,0,0,0,0,0,0,0,0,
0.0416667,0.166667,0.0416667,0,0,0,0,0,0,
0,0.0416667,0.166667,0.0416667,0,0,0,0,0,
0,0,0.0416667,0.166667,0.0416667,0,0,0,0,
0,0,0,0.0416667,0.166667,0.0416667,0,0,0,
0,0,0,0,0.0416667,0.166667,0.0416667,0,0,
0,0,0,0,0,0.0416667,0.166667,0.0416667,0,
0,0,0,0,0,0,0.0416667,0.166667,0.0416667,
0,0,0,0,0,0,0,0,1,
b:
0
7.862
-10.7327
12.1347
2
-8.13471
14.7327
-3.862
0
x:
0
73.9996
-107.31
97.6564
7.91753
-81.3265
122.156
-53.7109
0

0
3 - Splines/source/new.foo Normal file
View File

211
3 - Splines/source/newton.cpp Normal file
View File

@ -0,0 +1,211 @@
#include <algorithm>
#include <cmath>
#include <cstdlib>
#include <cstring>
#include <fstream>
#include <iomanip>
#include <ios>
#include <iostream>
#include <numeric>
#include <string>
#include <utility>
#include <vector>
using namespace std::literals;
using points_t = std::pair<std::vector<double>, std::vector<double>>;
using matrix_t = std::vector<std::vector<double>>;
std::vector<double> newton_differences(points_t points) {
std::vector<double> factors{};
auto &[x, fx] = points;
int n = points.first.size();
for (int i = 1; i <= n; ++i) {
factors.push_back(fx[0]);
for (int j = 0; j < (n - i); ++j) {
fx[j] = (fx[j + 1] - fx[j]) / (x[j + i] - x[j]);
}
}
return factors;
}
double newton_polynomial(points_t points, std::vector<double> factors, double x) {
auto xs = points.first;
int n = points.second.size() - 1;
double val = 0;
for (int k = 0; k <= n; ++k) {
double acc = 1;
for(int i = 0; i < k; ++i) {
acc *= (x - xs[i]);
}
val += acc * factors[k];
}
return val;
}
void calculate_natural_cubic_spline_matrix(points_t points, matrix_t &mat) {
auto &[xs, fx] = points;
int n = xs.size();
// Construção da matriz
for (int i = 1; i < n - 1; ++i) {
mat[i][i - 1] = (xs[i] - xs[i - 1])/6;
mat[i][i] = (xs[i + 1] - xs[i - 1])/3;
mat[i][i + 1] = (xs[i + 1] - xs[i])/6;
mat[i][n] = (fx[i + 1] - fx[i])/(xs[i + 1] - xs[i]) - (fx[i] - fx[i - 1])/(xs[i] - xs[i - 1]);
}
mat[0][0] = 1;
mat[0][n - 1] = 0;
mat[n - 1][n - 1] = 1;
mat[n - 1][n] = 0;
std::ofstream matriz("matriz.txt", std::ios_base::app);
matriz << "mat:\n";
for (auto i : mat){
for (auto j = i.begin(); j != i.end() - 1; ++ j)
matriz << *j << ",";
matriz << "\n";
}
matriz << "b:\n";
for (auto i : mat){
matriz << i[n] << '\n';
}
// Passar para a forma triangular
for (int k = 0; k < n; ++k) {
for (int i = k + 1; i < n; ++i) {
if (mat[k][k] != 0) {
double mul = mat[i][k]/mat[k][k];
for (int j = k; j < n; ++j) {
mat[i][j] -= mul * mat[k][j];
}
mat[i][n] -= mul * mat[k][n];
}
}
}
// Resolução da matriz
for (int i = n - 1; i > 0; --i) {
if (mat[i][i] != 0) {
double mul = mat[i - 1][i]/mat[i][i];
for (int j = 1; j < n + 1; ++j) {
mat[i - 1][j] -= mul * mat[i][j];
}
mat[i - 1][i] = 0;
mat[i][n] /= mat[i][i];
mat[i][i] = 1;
}
}
matriz << "x:\n";
for (auto i : mat){
matriz << i[n] << '\n';
}
}
double natural_cubic_spline(points_t points, matrix_t &mat, double x) {
auto &[xs, fx] = points;
int n = xs.size();
int i = 0;
for (int i_ = 0; i_ < n; ++i_) {
if (xs[i_] > x){
i = i_;
break;
}
}
double hi = xs[i] - xs[i - 1];
return mat[i - 1][n] * std::pow((xs[i] - x), 3)/(6 * hi) +
mat[i][n] * std::pow((x - xs[i - 1]), 3)/(6 * hi) +
(fx[i - 1] - mat[i - 1][n] * (hi * hi)/6)*(xs[i] - x)/hi +
(fx[i] - mat[i][n] * (hi * hi)/6)*(x - xs[i - 1])/hi;
}
void exercise_a(points_t points) {
auto &[xs, fx] = points;
auto factors = newton_differences(points);
std::ofstream poly{"a_polynomial.txt"};
for(double x = 0; x < 4; x += 0.001)
poly << x << " " << newton_polynomial(points, factors, x) << '\n';
std::ofstream spline{"a_spline.txt"};
unsigned long n = xs.size();
matrix_t mat(n, std::vector<double>(n + 1));
calculate_natural_cubic_spline_matrix(points, mat);
for(double x = 0; x < 4; x += 0.001)
spline << x << " " << natural_cubic_spline(points, mat, x) << '\n';
}
void exercise_b() {
points_t points;
auto &[xs, fx] = points;
auto f = [](double x) { return 4 * std::pow(x, 2) + std::sin(9 * x); };
for (double x = -1; x <= 1; x += (1 - -1)/8.0) {
xs.push_back(x);
fx.push_back(f(x));
}
std::ofstream poly{"b_polynomial.txt"};
auto factors = newton_differences(points);
for(double x = -1; x < 1; x += 0.001)
poly << x << " " << newton_polynomial(points, factors, x) << '\n';
std::ofstream spline{"b_spline.txt"};
unsigned long n = xs.size();
matrix_t mat(n, std::vector<double>(n + 1));
calculate_natural_cubic_spline_matrix(points, mat);
for(double x = -1; x < 1; x += 0.001)
spline << x << " " << natural_cubic_spline(points, mat, x) << '\n';
std::cout << "Erros poly: "
<< std::abs(std::round((f(0.3) - newton_polynomial(points, factors, 0.3)) * 100.0) / 100.0) << " "
<< std::abs(std::round((f(0.83) - newton_polynomial(points, factors, 0.83)) * 100.0) / 100.0) << '\n';
std::cout << "Erros spline: "
<< std::abs(std::round((f(0.3) - natural_cubic_spline(points, mat, 0.3))*100.0)/100.0) << " "
<< std::abs(std::round((f(0.83) - natural_cubic_spline(points, mat,0.83))*100.0)/100.0) << '\n';
std::cout << "p(0.3) = " << newton_polynomial(points, factors, 0.3) << ' '
<< "p(0.83) = " << newton_polynomial(points, factors, 0.83) << '\n';
std::cout << "p(0.3) = " << natural_cubic_spline(points, mat, 0.3) << ' '
<< "p(0.83) = " << natural_cubic_spline(points, mat, 0.83) << '\n';
}
int main(int argc, char **argv) {
std::istream *s;
std::ifstream file;
if (argc == 2) {
if (argv[1] == "-"s) {
s = &std::cin;
} else {
file.open(argv[1]);
s = &file;
}
} else{
std::cerr << "An argument required\n";
return 1;
}
points_t points;
std::pair<double, double> p;
while(true) {
*s >> p.first >> p.second;
if(s->eof())
break;
points.first.push_back(p.first);
points.second.push_back(p.second);
}
exercise_a(points);
exercise_b();
}

17
3 - Splines/source/old.foo Normal file
View File

@ -0,0 +1,17 @@
Mat:
1,0,0,0,0,0,0,
0,1,-7.14916e-18,2.73077e-18,-4.18062e-19,0,2.76846,
0,0,1,-7.50963e-18,1.14967e-18,0,-3.87386,
0,0,-3.57458e-18,1,-4.38965e-18,0,3.30622,
0,0,0,-7.50963e-18,1,0,0.248963,
0,0,0,0,0,1,0,
Mat:
1,0,0,0,0,0,0,0,0,0,
0,1,-1.73472e-18,4.64061e-19,2.85733e-19,5.30686e-20,-1.26382e-20,-4.49207e-21,0,73.9996,
0,0,1,-1.74023e-18,-1.0715e-18,-1.99007e-19,4.73931e-20,1.68453e-20,0,-107.31,
0,0,-1.73472e-18,1,4.00026e-18,7.4296e-19,-1.76934e-19,-6.2889e-20,0,97.6564,
0,0,0,-1.74023e-18,1,-2.77283e-18,6.60344e-19,2.34711e-19,0,7.91753,
0,0,0,0,4.00026e-18,1,-2.46444e-18,-8.75954e-19,0,-81.3265,
0,0,0,0,0,-2.77283e-18,1,3.26911e-18,0,122.156,
0,0,0,0,0,0,-2.46444e-18,1,0,-53.7109,
0,0,0,0,0,0,0,0,1,0,

39
3 - Splines/source/residue.py Executable file
View File

@ -0,0 +1,39 @@
#!/usr/bin/env python3
import numpy as np
from scipy.linalg import solve
from numpy.linalg import norm
# Matriz construída pelo programa em C++
A = np.matrix([[1, 0, 0, 0, 0, 0],
[0.166667, 0.666667, 0.166667, 0, 0, 0],
[0, 0.166667, 0.5, 0.0833333, 0, 0],
[0, 0, 0.0833333, 0.333333, 0.0833333, 0],
[0, 0, 0, 0.0833333, 0.5, 0.166667],
[0, 0, 0, 0, 0, 1]])
b = np.array([0, 1.2, -1.2, 0.8, 0.4, 0])
x2 = np.array([0, 2.76846, -3.87386, 3.30622, 0.248963, 0])
x = solve(A, b)
print(norm(x - x2))
A = np.matrix([[1, 0, 0, 0, 0, 0, 0, 0, 0],
[0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0, 0, 0],
[0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0, 0],
[0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0, 0],
[0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0, 0],
[0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0, 0],
[0, 0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667, 0],
[0, 0, 0, 0, 0, 0, 0.0416667, 0.166667, 0.0416667],
[0, 0, 0, 0, 0, 0, 0, 0, 1]])
b = np.array([0, 7.862, -10.7327, 12.1347, 2, -8.13471, 14.7327, -3.862, 0])
x2 = np.array([0, 73.9996, -107.31, 97.6564, 7.91753, -81.3265, 122.156, -53.7109, 0])
x = solve(A, b)
print(norm(x - x2))

17
3 - Splines/source/test.foo Normal file
View File

@ -0,0 +1,17 @@
Mat:
1,0,0,0,0,0,0,
0,1,0,0,0,0,2.76846,
0,0,1,0,0,0,-3.87386,
0,0,-3.57458e-18,1,0,0,3.30622,
0,0,0,-7.50963e-18,1,0,0.248963,
0,0,0,0,0,1,0,
Mat:
1,0,0,0,0,0,0,0,0,0,
0,1,0,0,0,0,0,0,0,73.9996,
0,0,1,0,0,0,0,0,0,-107.31,
0,0,-1.73472e-18,1,0,0,0,0,0,97.6564,
0,0,0,-1.74023e-18,1,0,0,0,0,7.91753,
0,0,0,0,4.00026e-18,1,0,0,0,-81.3265,
0,0,0,0,0,-2.77283e-18,1,0,0,122.156,
0,0,0,0,0,0,-2.46444e-18,1,0,-53.7109,
0,0,0,0,0,0,0,0,1,0,

150
4 - Simpson and Trapezoidal rules/README.md Executable file
View File

@ -0,0 +1,150 @@
---
title: "Análise Numérica - Trabalho Prático 3"
author:
- Diogo Cordeiro
- Hugo Sales
- Pedro Costa
geometry: margin=2cm
output: pdf_document
---
### Motivação
Pretende-se compreender o funcionamento conceptual bem como os desafios da implementação de dois métodos de integração
numéricos: `Regra de Simpson` e `Regra dos Trapézios.`
### 1
<!-- $$ \frac{d^4}{dx^4}_{x \in [0;2]}sin(sin(sin(sin(x)))) $$ -->
Através da análise do gráfico abaixo, verificamos que o majorante, em valor absoluto, da 4ª derivada
da função é menor que 12, este valor é usado para majorar a formula do erro para o calculo de n.
![Quarta derivada da função enunciada](majorante_erro_1.png){ width=8cm }
/* Função que implementa o método de Simpson */
double simpson (Function f, double a, double b, int n)
{
// Intervalo de passo
double h = (b - a)/n;
// Valor de f nos pontos de indice par
double evens = summation(f, 2, n - 2, 2, h, a);
// Valor de f nos pontos de indice impar
double odds = summation(f, 1, n - 1, 2, h, a);
// Aplicação da método de Simpson
value = (h/3)*(f(a) + f(b) + 2 * evens + 4 * odds);
}
\pagebreak
/* Função para calcular o sumatório de F entre os pontos de indice init
e stop saltando step pontos, usando h como intervalo de passo */
double summation (Function f, int init, int stop, int step, double h, double a)
{
// Acumulador
double total = 0;
for (int i = init; i <= stop; i += step) {
// Adicionamos o valor de f correspondente ao x de indice i
total += f(a + i * h);
}
return total;
}
/* Função para calcular o número de pontos necessários para o calculo
do integral com erro menor que error */
int calculateN (double A, double B, double error) {
int n = ceil( (B - A) / pow((15.0 * error) / 2.0, 1.0 / 4) );
return n + (n % 2);
}
void main ()
{
Function f(x) = sin(sin(sin(sin(x))));
// com 7 casas decimais correctas
print(simpson(f, 0, 2, calculateN(0, 2, pow(10, -7))));
// com 12 casas decimais correctas
print(simpson(f, 0, 2, calculateN(0, 2, pow(10, -12))));
}
#### Output
Erro | Resultado
---------------|-------------------
$$ 10^{-7} $$ | 1.0548418906594816
$$ 10^{-12} $$ | 1.0548418772492483
### 2)
// Valor exacto do integral calculado com o WolframAlpha, arredondado
// com 15 algarismos significativos, um a mais do que o erro majorado
// máximo para o caso de 2^20 pontos.
double I = 1.05484187724912;
/* Função para calcular o integral recorrendo ao método do Trapézio */
double trapezio (Function f, double a, double b) {
// Acumulador
double summation = 0;
// Intervalo de passo com n inicial = 2
double h = (b - a)/2;
// Valor constante
double fa_fb = f(a) + f(b);
// Para cada expoente de 1 a 20 (com passo 1)
for (int k = 1; k <= 20; ++k) {
// Guarda sumatorio dos pontos anteriores
double partial_sum = 0;
// Número de intervalos
// Com o left shift fazemos a potência de 2^k
int n = 1 << k;
// Evitamos recalcular pontos da função previamente computados
// guardando o sumatório destes em partial_sum e adicionamos a cada
// iteração os pontos novos, sendo estes de indice impar
for (int i = 1; i < n; i += 2) {
// Adicionamos o valor de f correspondente ao x de indice i
partial_sum += f(a + i * h);
}
summation += partial_sum;
// Aplicação da formula do Trapézio
double value = (h/2)*fa_fb + h * summation;
print(k + "\t| " + value + "\t| " + (I - value));
// Dividimos o intervalo por 2
h /= 2;
}
}
void main () {
Function f(x) = sin(sin(sin(sin(x))));
print(trapezio(f, 0, 2);
}
#### Output
k | $$ I_{n_{k}} $$ | $$ \mid I - I_{n_{k}} \mid $$
----|---------------------|--------------------------
1 | 0.9533749638740736 | $$ 1.1 \cdot 10^{-1} $$
2 | 1.0308378382617962 | $$ 2.5 \cdot 10^{-2} $$
3 | 1.0489039934457873 | $$ 6.0 \cdot 10^{-3} $$
4 | 1.053360809734676 | $$ 1.5 \cdot 10^{-3} $$
5 | 1.0544718169560368 | $$ 3.8 \cdot 10^{-4} $$
6 | 1.054749374997165 | $$ 9.3 \cdot 10^{-5} $$
7 | 1.0548187524860693 | $$ 2.4 \cdot 10^{-5} $$
8 | 1.0548360961083287 | $$ 5.8 \cdot 10^{-6} $$
9 | 1.054840431967042 | $$ 1.5 \cdot 10^{-6} $$
10 | 1.0548415159287925 | $$ 3.6 \cdot 10^{-7} $$
11 | 1.0548417869190467 | $$ 9.1 \cdot 10^{-8} $$
12 | 1.0548418546666 | $$ 2.3 \cdot 10^{-8} $$
13 | 1.054841871603487 | $$ 5.7 \cdot 10^{-9} $$
14 | 1.0548418758377078 | $$ 1.5 \cdot 10^{-9} $$
15 | 1.0548418768962615 | $$ 3.5 \cdot 10^{-10} $$
16 | 1.0548418771608998 | $$ 8.9 \cdot 10^{-11} $$
17 | 1.0548418772270582 | $$ 2.3 \cdot 10^{-11} $$
18 | 1.0548418772436017 | $$ 5.6 \cdot 10^{-12} $$
19 | 1.0548418772477444 | $$ 1.4 \cdot 10^{-12} $$
20 | 1.0548418772487873 | $$ 3.3 \cdot 10^{-13} $$

BIN
4 - Simpson and Trapezoidal rules/assignment.pdf Executable file
View File

Binary file not shown.

BIN
4 - Simpson and Trapezoidal rules/majorante_erro_1.png Executable file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 26 KiB

BIN
4 - Simpson and Trapezoidal rules/print.pdf Executable file
View File

Binary file not shown.

1
4 - Simpson and Trapezoidal rules/slides/css/all.css Normal file
View File

File diff suppressed because one or more lines are too long

40
4 - Simpson and Trapezoidal rules/slides/css/css.css Normal file
View File

@ -0,0 +1,40 @@
/* cyrillic-ext */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-cSZMZ-Y.woff2) format('woff2');
unicode-range: U+0460-052F, U+1C80-1C88, U+20B4, U+2DE0-2DFF, U+A640-A69F, U+FE2E-FE2F;
}
/* cyrillic */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-eCZMZ-Y.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-cyZMZ-Y.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-ciZMZ-Y.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Merriweather';
font-style: normal;
font-weight: 400;
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-fCZM.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

32
4 - Simpson and Trapezoidal rules/slides/css/css2.css Normal file
View File

@ -0,0 +1,32 @@
/* cyrillic */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752HT8Ghe4.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752Fj8Ghe4.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752Fz8Ghe4.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Oswald';
font-style: normal;
font-weight: 400;
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752GT8G.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

48
4 - Simpson and Trapezoidal rules/slides/css/css3.css Normal file
View File

@ -0,0 +1,48 @@
/* cyrillic-ext */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc3CsTKlA.woff2) format('woff2');
unicode-range: U+0460-052F, U+1C80-1C88, U+20B4, U+2DE0-2DFF, U+A640-A69F, U+FE2E-FE2F;
}
/* cyrillic */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc-CsTKlA.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* greek-ext */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc2CsTKlA.woff2) format('woff2');
unicode-range: U+1F00-1FFF;
}
/* greek */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc5CsTKlA.woff2) format('woff2');
unicode-range: U+0370-03FF;
}
/* latin-ext */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc0CsTKlA.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Ubuntu Mono';
font-style: normal;
font-weight: 400;
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc6CsQ.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

224
4 - Simpson and Trapezoidal rules/slides/css/css4.css Normal file
View File

@ -0,0 +1,224 @@
/* latin-ext */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 400;
src: local('Quattrocento'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpEg_xvsDZQL_LKIF7q4jP3zWj6T4g.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 400;
src: local('Quattrocento'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpEg_xvsDZQL_LKIF7q4jP3w2j6.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Bold'), local('Quattrocento-Bold'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpbg_xvsDZQL_LKIF7q4jP_eE3vfqnYgXc.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Bold'), local('Quattrocento-Bold'), url(https://fonts.gstatic.com/s/quattrocento/v10/OZpbg_xvsDZQL_LKIF7q4jP_eE3vcKnY.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 400;
src: local('Quattrocento Sans Italic'), local('QuattrocentoSans-Italic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9a4lja2NVIDdIAAoMR5MfuElaRB0zMh0P2GEHJ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 400;
src: local('Quattrocento Sans Italic'), local('QuattrocentoSans-Italic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9a4lja2NVIDdIAAoMR5MfuElaRB0zMh032GA.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 700;
src: local('Quattrocento Sans Bold Italic'), local('QuattrocentoSans-BoldItalic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9X4lja2NVIDdIAAoMR5MfuElaRB0zMj_bTDXDojYsJ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: italic;
font-weight: 700;
src: local('Quattrocento Sans Bold Italic'), local('QuattrocentoSans-BoldItalic'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9X4lja2NVIDdIAAoMR5MfuElaRB0zMj_bTDX7ojQ.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 400;
src: local('Quattrocento Sans'), local('QuattrocentoSans'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9c4lja2NVIDdIAAoMR5MfuElaRB0zHt0_uHA.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 400;
src: local('Quattrocento Sans'), local('QuattrocentoSans'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9c4lja2NVIDdIAAoMR5MfuElaRB0zJt08.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* latin-ext */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Sans Bold'), local('QuattrocentoSans-Bold'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9Z4lja2NVIDdIAAoMR5MfuElaRB0RyklrfPXzwiQ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Quattrocento Sans';
font-style: normal;
font-weight: 700;
src: local('Quattrocento Sans Bold'), local('QuattrocentoSans-Bold'), url(https://fonts.gstatic.com/s/quattrocentosans/v11/va9Z4lja2NVIDdIAAoMR5MfuElaRB0RyklrRPXw.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8on7mTMuk.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8onXmTMuk.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8onTmTMuk.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 400;
src: local('Spectral Italic'), local('Spectral-Italic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCt-xNNww_2s0amA9M8onrmTA.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfCFXUIJ.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfuFXUIJ.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfqFXUIJ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: italic;
font-weight: 700;
src: local('Spectral Bold Italic'), local('Spectral-BoldItalic'), url(https://fonts.gstatic.com/s/spectral/v5/rnCu-xNNww_2s0amA9M8qsHDWfSFXQ.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M9knj-SA.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M2knj-SA.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M3knj-SA.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 400;
src: local('Spectral Regular'), local('Spectral-Regular'), url(https://fonts.gstatic.com/s/spectral/v5/rnCr-xNNww_2s0amA9M5kng.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}
/* cyrillic */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23FafadWQ.woff2) format('woff2');
unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
}
/* vietnamese */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23OafadWQ.woff2) format('woff2');
unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
}
/* latin-ext */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23PafadWQ.woff2) format('woff2');
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
}
/* latin */
@font-face {
font-family: 'Spectral';
font-style: normal;
font-weight: 700;
src: local('Spectral Bold'), local('Spectral-Bold'), url(https://fonts.gstatic.com/s/spectral/v5/rnCs-xNNww_2s0amA9uCt23BafY.woff2) format('woff2');
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
}

BIN
4 - Simpson and Trapezoidal rules/slides/css/fonts/KaTeX_Main-Regular.ttf Normal file
View File

Binary file not shown.

BIN
4 - Simpson and Trapezoidal rules/slides/css/fonts/KaTeX_Main-Regular.woff Normal file
View File

Binary file not shown.

BIN
4 - Simpson and Trapezoidal rules/slides/css/fonts/KaTeX_Main-Regular.woff2 Normal file
View File

Binary file not shown.

1
4 - Simpson and Trapezoidal rules/slides/css/katex.min.css vendored Normal file
View File

File diff suppressed because one or more lines are too long

8779
4 - Simpson and Trapezoidal rules/slides/css/style.css Normal file
View File

File diff suppressed because it is too large Load Diff

BIN
4 - Simpson and Trapezoidal rules/slides/imgs/majorante_erro_1.png Executable file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 26 KiB

71
4 - Simpson and Trapezoidal rules/slides/index.html Executable file
View File

@ -0,0 +1,71 @@
<!DOCTYPE html>
<html>
<head>
<meta name="robots" content="noindex">
<meta http-equiv="content-type" content="text/html; charset=utf-8">
<title>Numerical Analysis - Slides</title>
<link rel="stylesheet" href="css/all.css">
<link href="css/css.css" rel="stylesheet">
<link href="css/css2.css" rel="stylesheet">
<link href="css/css3.css" rel="stylesheet">
<link href="css/css4.css" rel="stylesheet">
<link rel="stylesheet" href="css/style.css">
<link rel="stylesheet" href="css/katex.min.css">
<link rel="icon" href="https://hackersatporto.com/images/favicon.png">
<meta name="description" content="Porto's student computing society. We are a community focused on learning about and building things with technology.">
<script src="js/remark-latest.min.js"></script>
<script src="js/jquery.min.js"></script>
<script src="js/mark.min.js"></script>
<script src="js/MathJax.js"></script>
<script src="js/katex.min.js"></script>
<script src="js/auto-render.min.js"></script>
<script src="js/abcjs_basic_5.6.3-min.js"></script>
<script src="js/story.js"></script>
</head>
<body id="content" class="ma0 bg-white section-slides page-kind-page is-page-true feature-math feature-qrcode feature-nohighlight feature-music">
<script>
var hljs = remark.highlighter.engine;
var slideshow = remark.create({
ratio: "16:9",
countIncrementalSlides: false,
sourceUrl: 'text.md',
slideNumberFormat: '%current%',
highlightLanguage: "xaprb",
placeholder: "dummy value to end the list without a comma"
});
$(".remark-slide-content.fit-h1 h1").each(function(i, e) {
var $e = $(e);
var $p = $e.closest('div.remark-slide-container');
var needsToggle = !$p.hasClass('remark-visible');
if ( needsToggle ) $p.toggleClass('remark-visible');
while (e.scrollHeight > e.clientHeight + 1 && $e.css('font-size') != '1px') {
$e.css('font-size', (parseInt($e.css('font-size')) - 1) + "px");
}
if ( needsToggle ) $p.toggleClass('remark-visible');
});
$( function() {
if ( $( "body.feature-qrcode:not(.feature-noqrcode)" ).length ) {
if ( $( ".qrcode" ).length ) {
new QRCode($(".qrcode")[0], {
"useSVG": true,
"text": window.location.href.replace(/#\d*$/, "")
});
}
}
});
// Setup MathJax
MathJax.Hub.Config({
tex2jax: {
skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
}
});
MathJax.Hub.Configured();
</script>
</body>
</html>

19
4 - Simpson and Trapezoidal rules/slides/js/MathJax.js Normal file
View File

File diff suppressed because one or more lines are too long

3
4 - Simpson and Trapezoidal rules/slides/js/abcjs_basic_5.6.3-min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

1
4 - Simpson and Trapezoidal rules/slides/js/auto-render.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

19
4 - Simpson and Trapezoidal rules/slides/js/extensions/MathMenu.js Normal file
View File

File diff suppressed because one or more lines are too long

19
4 - Simpson and Trapezoidal rules/slides/js/extensions/MathZoom.js Normal file
View File

File diff suppressed because one or more lines are too long

2
4 - Simpson and Trapezoidal rules/slides/js/jquery.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

1
4 - Simpson and Trapezoidal rules/slides/js/katex.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

7
4 - Simpson and Trapezoidal rules/slides/js/mark.min.js vendored Normal file
View File

File diff suppressed because one or more lines are too long

Some files were not shown because too many files have changed in this diff Show More