<?xml version='1.0'?> <?xml-stylesheet type='text/xsl' href='pmathml.xsl'?> <html xmlns='http://www.w3.org/1999/xhtml'> <head> <title>Rosen34: Example and Test</title> <meta name="description" id="description" content="Rosen34: Example and Test"/> <meta name="keywords" id="keywords" content=" Rosen34 example test "/> <style type='text/css'> body { color : black } body { background-color : white } A:link { color : blue } A:visited { color : purple } A:active { color : purple } </style> <script type='text/javascript' language='JavaScript' src='_rosen34.cpp_xml.js'> </script> </head> <body> <table><tr> <td> <a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a> </td> <td><a href="rosen34.xml" target="_top">Prev</a> </td><td><a href="odeerrcontrol.xml" target="_top">Next</a> </td><td> <select onchange='choose_across0(this)'> <option>Index-></option> <option>contents</option> <option>reference</option> <option>index</option> <option>search</option> <option>external</option> </select> </td> <td> <select onchange='choose_up0(this)'> <option>Up-></option> <option>CppAD</option> <option>library</option> <option>Rosen34</option> <option>Rosen34.cpp</option> </select> </td> <td> <select onchange='choose_down3(this)'> <option>CppAD-></option> <option>Install</option> <option>Introduction</option> <option>AD</option> <option>ADFun</option> <option>library</option> <option>Example</option> <option>configure</option> <option>Appendix</option> </select> </td> <td> <select onchange='choose_down2(this)'> <option>library-></option> <option>ErrorHandler</option> <option>NearEqual</option> <option>speed_test</option> <option>SpeedTest</option> <option>NumericType</option> <option>CheckNumericType</option> <option>SimpleVector</option> <option>CheckSimpleVector</option> <option>nan</option> <option>pow_int</option> <option>Poly</option> <option>LuDetAndSolve</option> <option>RombergOne</option> <option>RombergMul</option> <option>Runge45</option> <option>Rosen34</option> <option>OdeErrControl</option> <option>OdeGear</option> <option>OdeGearControl</option> <option>BenderQuad</option> <option>LuRatio</option> <option>std_math_unary</option> <option>CppAD_vector</option> <option>TrackNewDel</option> </select> </td> <td> <select onchange='choose_down1(this)'> <option>Rosen34-></option> <option>Rosen34.cpp</option> </select> </td> <td>Rosen34.cpp</td> <td>Headings</td> </tr></table><br/> <center><b><big><big>Rosen34: Example and Test</big></big></b></center> Define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>X</mi> <mo stretchy="false">:</mo> <mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi> </mstyle></mrow> <mo stretchy="false">→</mo> <msup><mrow><mstyle mathvariant='bold'><mi mathvariant='bold'>R</mi> </mstyle></mrow> <mi mathvariant='italic'>n</mi> </msup> </mrow></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <msub><mi mathvariant='italic'>X</mi> <mi mathvariant='italic'>i</mi> </msub> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">=</mo> <msup><mi mathvariant='italic'>t</mi> <mrow><mi mathvariant='italic'>i</mi> <mo stretchy="false">+</mo> <mn>1</mn> </mrow> </msup> </mrow></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow> <mi mathvariant='italic'>i</mi> <mo stretchy="false">=</mo> <mn>1</mn> <mo stretchy="false">,</mo> <mo stretchy="false">…</mo> <mo stretchy="false">,</mo> <mi mathvariant='italic'>n</mi> <mn>-1</mn> </mrow></math> . It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <mtable rowalign="center" ><mtr><mtd columnalign="right" > <msub><mi mathvariant='italic'>X</mi> <mi mathvariant='italic'>i</mi> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mn>0</mn> </mtd><mtd columnalign="right" > <mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>for</mi> <mspace width='.3em'/> <mi mathvariant='normal'>all</mi> <mspace width='.3em'/> </mstyle></mrow> <mi mathvariant='italic'>i</mi> </mtd></mtr><mtr><mtd columnalign="right" > <msub><mi mathvariant='italic'>X</mi> <mi mathvariant='italic'>i</mi> </msub> <mo stretchy="false">'</mo> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mn>1</mn> </mtd><mtd columnalign="right" > <mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi> <mspace width='.3em'/> </mstyle></mrow> <mi mathvariant='italic'>i</mi> <mo stretchy="false">=</mo> <mn>0</mn> </mtd></mtr><mtr><mtd columnalign="right" > <msub><mi mathvariant='italic'>X</mi> <mi mathvariant='italic'>i</mi> </msub> <mo stretchy="false">'</mo> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> </mtd><mtd columnalign="center" > <mo stretchy="false">=</mo> </mtd><mtd columnalign="left" > <mo stretchy="false">(</mo> <mi mathvariant='italic'>i</mi> <mo stretchy="false">+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup><mi mathvariant='italic'>t</mi> <mi mathvariant='italic'>i</mi> </msup> <mo stretchy="false">=</mo> <mo stretchy="false">(</mo> <mi mathvariant='italic'>i</mi> <mo stretchy="false">+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub><mi mathvariant='italic'>X</mi> <mrow><mi mathvariant='italic'>i</mi> <mn>-1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant='italic'>t</mi> <mo stretchy="false">)</mo> </mtd><mtd columnalign="right" > <mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>if</mi> <mspace width='.3em'/> </mstyle></mrow> <mi mathvariant='italic'>i</mi> <mo stretchy="false">></mo> <mn>0</mn> </mtd></mtr></mtable> </mrow></math> The example tests Rosen34 using the relations above: <code><font color="blue"> <pre style='display:inline'> # include <cppad/cppad.hpp> // For automatic differentiation namespace { class Fun { public: // constructor Fun(bool use_x_) : use_x(use_x_) { } // compute f(t, x) both for double and <a href="ad.xml" target="_top">AD</a><double> template <typename Scalar> void Ode( const Scalar &t, const <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><Scalar> &x, <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><Scalar> &f) { size_t n = x.size(); Scalar ti(1); f[0] = Scalar(1); size_t i; for(i = 1; i < n; i++) { ti *= t; // convert int(size_t) to avoid warning // on _MSC_VER systems if( use_x ) f[i] = int(i+1) * x[i-1]; else f[i] = int(i+1) * ti; } } // compute partial of f(t, x) w.r.t. t using AD void Ode_ind( const double &t, const <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> &x, <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> &f_t) { using namespace CppAD; size_t n = x.size(); <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a>< <a href="ad.xml" target="_top">AD</a><double> > T(1); <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a>< <a href="ad.xml" target="_top">AD</a><double> > X(n); <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a>< <a href="ad.xml" target="_top">AD</a><double> > F(n); // set argument values T[0] = t; size_t i; for(i = 0; i < n; i++) X[i] = x[i]; // declare independent variables <a href="independent.xml" target="_top">Independent</a>(T); // compute f(t, x) this->Ode(T[0], X, F); // define AD function object <a href="funconstruct.xml" target="_top">ADFun</a><double> Fun(T, F); // compute partial of f w.r.t t <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> dt(1); dt[0] = 1.; f_t = Fun.<a href="forward.xml" target="_top">Forward</a>(1, dt); } // compute partial of f(t, x) w.r.t. x using AD void Ode_dep( const double &t, const <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> &x, <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> &f_x) { using namespace CppAD; size_t n = x.size(); <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a>< <a href="ad.xml" target="_top">AD</a><double> > T(1); <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a>< <a href="ad.xml" target="_top">AD</a><double> > X(n); <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a>< <a href="ad.xml" target="_top">AD</a><double> > F(n); // set argument values T[0] = t; size_t i, j; for(i = 0; i < n; i++) X[i] = x[i]; // declare independent variables <a href="independent.xml" target="_top">Independent</a>(X); // compute f(t, x) this->Ode(T[0], X, F); // define AD function object <a href="funconstruct.xml" target="_top">ADFun</a><double> Fun(X, F); // compute partial of f w.r.t x <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> dx(n); <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> df(n); for(j = 0; j < n; j++) dx[j] = 0.; for(j = 0; j < n; j++) { dx[j] = 1.; df = Fun.<a href="forward.xml" target="_top">Forward</a>(1, dx); for(i = 0; i < n; i++) f_x [i * n + j] = df[i]; dx[j] = 0.; } } private: const bool use_x; }; } bool Rosen34(void) { bool ok = true; // initial return value size_t i; // temporary indices size_t n = 4; // number components in X(t) and order of method size_t M = 2; // number of Rosen34 steps in [ti, tf] double ti = 0.; // initial time double tf = 2.; // final time // xi = X(0) <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> xi(n); for(i = 0; i <n; i++) xi[i] = 0.; size_t use_x; for( use_x = 0; use_x < 2; use_x++) { // function object depends on value of use_x Fun F(use_x > 0); // compute Rosen34 approximation for X(tf) <a href="test_vector.xml" target="_top">CPPAD_TEST_VECTOR</a><double> xf(n), e(n); xf = CppAD::Rosen34(F, M, ti, tf, xi, e); double check = tf; for(i = 0; i < n; i++) { // check that error is always positive ok &= (e[i] >= 0.); // 4th order method is exact for i < 4 if( i < 4 ) ok &= CppAD::<a href="nearequal.xml" target="_top">NearEqual</a>(xf[i], check, 1e-10, 1e-10); // 3rd order method is exact for i < 3 if( i < 3 ) ok &= (e[i] <= 1e-10); // check value for next i check *= tf; } } return ok; } </pre> </font></code> <hr/>Input File: example/rosen_34.cpp </body> </html>