Solutions for FCUP's Numerical Analysis Assignments

1 - Numerical Series Approximation/LICENSE

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+                    GNU GENERAL PUBLIC LICENSE
+                       Version 3, 29 June 2007
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+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>.

1 - Numerical Series Approximation/README.md

@@ -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 $10^{-8}$.

1 - Numerical Series Approximation/graficos/erro.cpp

@@ -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);
+	}
+}

1 - Numerical Series Approximation/graficos/erro2.cpp

@@ -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';
+	}
+}

1 - Numerical Series Approximation/graficos/erro3.cpp

@@ -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';
+	}
+}

1 - Numerical Series Approximation/graficos/generate_graphics.gnuplot

@@ -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++";
+

1 - Numerical Series Approximation/graficos/questao3_iteracoes_Cpp.txt

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

1 - Numerical Series Approximation/graficos/questao3_iteracoes_java.txt

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

1 - Numerical Series Approximation/graficos/questao3_tempo_Cpp.txt

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

1 - Numerical Series Approximation/graficos/questao3_tempo_java.txt

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

1 - Numerical Series Approximation/graficos/questao4_erro3.txt

@@ -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]

1 - Numerical Series Approximation/graficos/scratch

@@ -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

1 - Numerical Series Approximation/source/Question1.cpp

@@ -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';
+	}*/
+}

1 - Numerical Series Approximation/source/Question1wrong.cpp

@@ -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';
+}

1 - Numerical Series Approximation/source/Question2.cpp

@@ -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);
+}

1 - Numerical Series Approximation/source/Question3.cpp

@@ -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);
+}

1 - Numerical Series Approximation/source/java for speed comparison/Question1.java

@@ -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;
+	}
+}

1 - Numerical Series Approximation/source/java for speed comparison/Question2.java

@@ -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);
+        }
+}

1 - Numerical Series Approximation/source/java for speed comparison/Question3.java

@@ -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);
+    }
+
+}
+

2 - Newton and Iterative methods/LICENSE

@@ -0,0 +1,674 @@
+                    GNU GENERAL PUBLIC LICENSE
+                       Version 3, 29 June 2007
+
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2 - Newton and Iterative methods/README.md

@@ -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.

2 - Newton and Iterative methods/graphics/err_fixed_point.txt

@@ -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

2 - Newton and Iterative methods/graphics/err_newton.txt

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

2 - Newton and Iterative methods/graphics/generate_graphics.gnuplot

@@ -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";

2 - Newton and Iterative methods/slides/css/css.css

@@ -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;
+}

2 - Newton and Iterative methods/slides/css/css2.css

@@ -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;
+}

2 - Newton and Iterative methods/slides/css/css3.css

@@ -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;
+}

2 - Newton and Iterative methods/slides/css/css4.css

@@ -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');
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+  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');
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+/* vietnamese */
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+  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;
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+/* latin-ext */
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+  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');
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+}
+/* 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');
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+}

2 - Newton and Iterative methods/slides/index.html

@@ -0,0 +1,71 @@
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2 - Newton and Iterative methods/slides/js/story.js

@@ -0,0 +1,146 @@
+// Render KaTeX math typesetting, but only if the <body> has the class
+// feature-math
+$( function() {
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+      renderMathInElement(document.body);
+   }
+});
+
+// Render abcjs sheet music, but only if the <body> has the class
+// feature-music. Do this by removing the <pre><code class="language-abc">
+// and replacing it with a <p id="music-X"> which will be used to hold the
+// generated sheet music. Remark's code syntax highlighting transforms the
+// <code> block into a bunch of
+//		<div class="remark-code-line">K: Emin</div>
+// one per line, so we have to reassemble those to get back linebreaks.
+$( function() {
+   if ( $( "body.feature-music:not(.feature-nomusic)" ).length ) {
+      $( "code.language-abc, code.abc" ).each(function(i, e){
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+   if ( $("body.feature-figlink").length ) {
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+/* Add captions to tables.
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+ * - feature-fignum adds table numbering
+ * - feature-figlink adds automatic links of text like "Table 4"
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+});

2 - Newton and Iterative methods/slides/text.md

@@ -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.

2 - Newton and Iterative methods/source/roots.cpp

@@ -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';
+}

3 - Splines/LICENSE

@@ -0,0 +1,674 @@
+                    GNU GENERAL PUBLIC LICENSE
+                       Version 3, 29 June 2007
+
+ Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>
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+  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>.

3 - Splines/README.md

@@ -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.
+

3 - Splines/graphics/graphs.gnuplot

@@ -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";

3 - Splines/source/input.txt

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

3 - Splines/source/matriz.txt

@@ -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

3 - Splines/source/newton.cpp

@@ -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();
+
+}

3 - Splines/source/old.foo

@@ -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,

3 - Splines/source/residue.py

@@ -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))

3 - Splines/source/test.foo

@@ -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,

4 - Simpson and Trapezoidal rules/README.md

@@ -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  | $$ 3.3 \cdot 10^{-13} $$

4 - Simpson and Trapezoidal rules/slides/css/css.css

@@ -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;
+}

4 - Simpson and Trapezoidal rules/slides/css/css2.css

@@ -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;
+}

4 - Simpson and Trapezoidal rules/slides/css/css3.css

@@ -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;
+}

4 - Simpson and Trapezoidal rules/slides/css/css4.css

@@ -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;
+}

4 - Simpson and Trapezoidal rules/slides/index.html

@@ -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*$/, "")
+        			});
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+        	}
+        });
+
+        // Setup MathJax
+        MathJax.Hub.Config({
+            tex2jax: {
+                skipTags: ['script', 'noscript', 'style', 'textarea', 'pre']
+            }
+        });
+        MathJax.Hub.Configured();
+    </script>
+</body>
+</html>