Solutions for FCUP's Numerical Analysis Assignments
This commit is contained in:
Diogo Cordeiro
1
4 - Simpson and Trapezoidal rules/slides/css/all.css
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1
4 - Simpson and Trapezoidal rules/slides/css/all.css
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40
4 - Simpson and Trapezoidal rules/slides/css/css.css
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40
4 - Simpson and Trapezoidal rules/slides/css/css.css
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/* cyrillic-ext */
|
||||
@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';
|
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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-eCZMZ-Y.woff2) format('woff2');
|
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unicode-range: U+0400-045F, U+0490-0491, U+04B0-04B1, U+2116;
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}
|
||||
/* vietnamese */
|
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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-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
4 - Simpson and Trapezoidal rules/slides/css/css2.css
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32
4 - Simpson and Trapezoidal rules/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;
|
||||
}
|
||||
/* latin */
|
||||
@font-face {
|
||||
font-family: 'Oswald';
|
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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
4 - Simpson and Trapezoidal rules/slides/css/css3.css
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48
4 - Simpson and Trapezoidal rules/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 */
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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/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');
|
||||
unicode-range: U+0370-03FF;
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||||
}
|
||||
/* latin-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;
|
||||
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc0CsTKlA.woff2) format('woff2');
|
||||
unicode-range: U+0100-024F, U+0259, U+1E00-1EFF, U+2020, U+20A0-20AB, U+20AD-20CF, U+2113, U+2C60-2C7F, U+A720-A7FF;
|
||||
}
|
||||
/* latin */
|
||||
@font-face {
|
||||
font-family: 'Ubuntu Mono';
|
||||
font-style: normal;
|
||||
font-weight: 400;
|
||||
src: local('Ubuntu Mono'), local('UbuntuMono-Regular'), url(https://fonts.gstatic.com/s/ubuntumono/v8/KFOjCneDtsqEr0keqCMhbCc6CsQ.woff2) format('woff2');
|
||||
unicode-range: U+0000-00FF, U+0131, U+0152-0153, U+02BB-02BC, U+02C6, U+02DA, U+02DC, U+2000-206F, U+2074, U+20AC, U+2122, U+2191, U+2193, U+2212, U+2215, U+FEFF, U+FFFD;
|
||||
}
|
||||
224
4 - Simpson and Trapezoidal rules/slides/css/css4.css
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224
4 - Simpson and Trapezoidal rules/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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}
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||||
/* latin */
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||||
@font-face {
|
||||
font-family: 'Quattrocento';
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font-style: normal;
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||||
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;
|
||||
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|
||||
class: title, smokescreen, shelf, no-footer
|
||||
# Análise Numérica - Trabalho Prático 4
|
||||
### Diogo Cordeiro, Hugo Sales, Pedro Costa, Ricardo Pimenta
|
||||
|
||||
---
|
||||
class: roomy
|
||||
|
||||
# Motivação
|
||||
|
||||
Pretende-se compreender o funcionamento conceptual bem como os desafios da implementação de dois métodos numéricos: `Regra de Simpson` e `Regra dos Trapézios.`
|
||||
|
||||
---
|
||||
class: img-right
|
||||
|
||||
.center[]
|
||||
|
||||
# 1
|
||||
|
||||
Através da análise do gráfico, 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.
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
# Simpson
|
||||
|
||||
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);
|
||||
}
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
# Summation
|
||||
|
||||
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;
|
||||
}
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
# calculateN
|
||||
|
||||
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);
|
||||
}
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
# main
|
||||
|
||||
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))));
|
||||
}
|
||||
|
||||
---
|
||||
class: roomy
|
||||
|
||||
<style>
|
||||
table td {
|
||||
background: inherit;
|
||||
border-radius: .5em;
|
||||
color: #000;
|
||||
}
|
||||
</style>
|
||||
|
||||
#### Output
|
||||
|
||||
Erro | Resultado
|
||||
---------------|-------------------
|
||||
$$ 10^{-7} $$ | 1.0548418906594816
|
||||
$$ 10^{-12} $$ | 1.0548418772492483
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
### 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);
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
### 2)
|
||||
|
||||
// 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;
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
### 2)
|
||||
|
||||
// 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);
|
||||
}
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
<style>
|
||||
table td {
|
||||
background: inherit;
|
||||
border-radius: .5em;
|
||||
color: #000;
|
||||
}
|
||||
</style>
|
||||
|
||||
#### 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 | $$ 2.4 \cdot 10^{-5} $$
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
<style>
|
||||
table td {
|
||||
background: inherit;
|
||||
border-radius: .5em;
|
||||
color: #000;
|
||||
}
|
||||
</style>
|
||||
|
||||
#### Output
|
||||
|
||||
k | $$ I_{n_{k}} $$ | $$ \mid I - I_{n_{k}} \mid $$
|
||||
----|---------------------|--------------------------
|
||||
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 | $$ 1.5 \cdot 10^{-9} $$
|
||||
|
||||
---
|
||||
class: compact
|
||||
|
||||
<style>
|
||||
table td {
|
||||
background: inherit;
|
||||
border-radius: .5em;
|
||||
color: #000;
|
||||
}
|
||||
</style>
|
||||
|
||||
#### Output
|
||||
|
||||
k | $$ I_{n_{k}} $$ | $$ \mid I - I_{n_{k}} \mid $$
|
||||
----|---------------------|--------------------------
|
||||
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 | $$ 3.3 \cdot 10^{-13} $$
|
||||
Reference in New Issue
Block a user