Solutions for FCUP's Numerical Analysis Assignments
This commit is contained in:
Diogo Cordeiro
1
2 - Newton and Iterative methods/slides/css/all.css
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1
2 - Newton and Iterative methods/slides/css/all.css
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40
2 - Newton and Iterative methods/slides/css/css.css
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40
2 - Newton and Iterative methods/slides/css/css.css
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/* cyrillic-ext */
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||||
@font-face {
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font-family: 'Merriweather';
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font-style: normal;
|
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font-weight: 400;
|
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src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-cSZMZ-Y.woff2) format('woff2');
|
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unicode-range: U+0460-052F, U+1C80-1C88, U+20B4, U+2DE0-2DFF, U+A640-A69F, U+FE2E-FE2F;
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}
|
||||
/* cyrillic */
|
||||
@font-face {
|
||||
font-family: 'Merriweather';
|
||||
font-style: normal;
|
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font-weight: 400;
|
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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;
|
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}
|
||||
/* vietnamese */
|
||||
@font-face {
|
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font-family: 'Merriweather';
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font-style: normal;
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font-weight: 400;
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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;
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}
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/* latin-ext */
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@font-face {
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||||
font-family: 'Merriweather';
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font-style: normal;
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font-weight: 400;
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||||
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;
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}
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||||
/* latin */
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@font-face {
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font-family: 'Merriweather';
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font-style: normal;
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||||
font-weight: 400;
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||||
src: local('Merriweather Regular'), local('Merriweather-Regular'), url(https://fonts.gstatic.com/s/merriweather/v19/u-440qyriQwlOrhSvowK_l5-fCZM.woff2) format('woff2');
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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;
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}
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32
2 - Newton and Iterative methods/slides/css/css2.css
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32
2 - Newton and Iterative methods/slides/css/css2.css
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@@ -0,0 +1,32 @@
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/* cyrillic */
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@font-face {
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font-family: 'Oswald';
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font-style: normal;
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font-weight: 400;
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src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752HT8Ghe4.woff2) format('woff2');
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unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
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}
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/* vietnamese */
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@font-face {
|
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font-family: 'Oswald';
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font-style: normal;
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font-weight: 400;
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src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752Fj8Ghe4.woff2) format('woff2');
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unicode-range: U+0102-0103, U+0110-0111, U+1EA0-1EF9, U+20AB;
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}
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/* latin-ext */
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@font-face {
|
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font-family: 'Oswald';
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font-style: normal;
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font-weight: 400;
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src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752Fz8Ghe4.woff2) format('woff2');
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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;
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}
|
||||
/* latin */
|
||||
@font-face {
|
||||
font-family: 'Oswald';
|
||||
font-style: normal;
|
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font-weight: 400;
|
||||
src: local('Oswald Regular'), local('Oswald-Regular'), url(https://fonts.gstatic.com/s/oswald/v16/TK3iWkUHHAIjg752GT8G.woff2) format('woff2');
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||||
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;
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}
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48
2 - Newton and Iterative methods/slides/css/css3.css
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48
2 - Newton and Iterative methods/slides/css/css3.css
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@@ -0,0 +1,48 @@
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/* cyrillic-ext */
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@font-face {
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font-family: 'Ubuntu Mono';
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font-style: normal;
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font-weight: 400;
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src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc3CsTKlA.woff2) format('woff2');
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unicode-range: U+0460-052F, U+1C80-1C88, U+20B4, U+2DE0-2DFF, U+A640-A69F, U+FE2E-FE2F;
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}
|
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/* cyrillic */
|
||||
@font-face {
|
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font-family: 'Ubuntu Mono';
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font-style: normal;
|
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font-weight: 400;
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src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc-CsTKlA.woff2) format('woff2');
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unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
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}
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/* greek-ext */
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@font-face {
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font-family: 'Ubuntu Mono';
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font-style: normal;
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font-weight: 400;
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src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc2CsTKlA.woff2) format('woff2');
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unicode-range: U+1F00-1FFF;
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}
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/* greek */
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@font-face {
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font-family: 'Ubuntu Mono';
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font-style: normal;
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font-weight: 400;
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src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc5CsTKlA.woff2) format('woff2');
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||||
unicode-range: U+0370-03FF;
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||||
}
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/* latin-ext */
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@font-face {
|
||||
font-family: 'Ubuntu Mono';
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font-style: normal;
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||||
font-weight: 400;
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||||
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;
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||||
}
|
||||
/* 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;
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}
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||||
224
2 - Newton and Iterative methods/slides/css/css4.css
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224
2 - Newton and Iterative methods/slides/css/css4.css
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@@ -0,0 +1,224 @@
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/* latin-ext */
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@font-face {
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||||
font-family: 'Quattrocento';
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font-style: normal;
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font-weight: 400;
|
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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;
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||||
}
|
||||
/* latin */
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||||
@font-face {
|
||||
font-family: 'Quattrocento';
|
||||
font-style: normal;
|
||||
font-weight: 400;
|
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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');
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class: title, smokescreen, shelf, no-footer
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||||
# 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
|
||||
|
||||
|
||||
|
||||
# 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
|
||||
|
||||
|
||||
|
||||
# 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
|
||||
|
||||
|
||||
|
||||
# 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.
|
||||
Reference in New Issue
Block a user