Prev Next Index-> contents reference index search external Up-> CppAD utility Runge45 runge45_1.cpp CppAD-> Install Introduction AD ADFun preprocessor multi_thread utility ipopt_solve Example speed Appendix utility-> ErrorHandler NearEqual speed_test SpeedTest time_test test_boolofvoid NumericType CheckNumericType SimpleVector CheckSimpleVector nan pow_int Poly LuDetAndSolve RombergOne RombergMul Runge45 Rosen34 OdeErrControl OdeGear OdeGearControl CppAD_vector thread_alloc index_sort to_string set_union sparse_rc sparse_rcv Runge45-> runge45_1.cpp runge45_2.cpp runge45_1.cpp Headings

$\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }$
Runge45: Example and Test
Define $X : \B{R} \rightarrow \B{R}^n$ by $$X_i (t) = t^{i+1}$$ for $i = 1 , \ldots , n-1$. It follows that $$\begin{array}{rclr} X_i(0) & = & 0 & {\rm for \; all \;} i \\ X_i ' (t) & = & 1 & {\rm if \;} i = 0 \\ X_i '(t) & = & (i+1) t^i = (i+1) X_{i-1} (t) & {\rm if \;} i > 0 \end{array}$$ The example tests Runge45 using the relations above:  # include <cstddef> // for size_t # include <cppad/utility/near_equal.hpp> // for CppAD::NearEqual # include <cppad/utility/vector.hpp> // for CppAD::vector # include <cppad/utility/runge_45.hpp> // for CppAD::Runge45 // Runge45 requires fabs to be defined (not std::fabs) // <cppad/cppad.hpp> defines this for doubles, but runge_45.hpp does not. # include <math.h> // for fabs without std in front namespace { class Fun { public: // constructor Fun(bool use_x_) : use_x(use_x_) { } // set f = x'(t) void Ode( const double &t, const CppAD::vector<double> &x, CppAD::vector<double> &f) { size_t n = x.size(); double ti = 1.; f[0] = 1.; size_t i; for(i = 1; i < n; i++) { ti *= t; if( use_x ) f[i] = double(i+1) * x[i-1]; else f[i] = double(i+1) * ti; } } private: const bool use_x; }; } bool runge_45_1(void) { bool ok = true; // initial return value size_t i; // temporary indices using CppAD::NearEqual; double eps99 = 99.0 * std::numeric_limits<double>::epsilon(); size_t n = 5; // number components in X(t) and order of method size_t M = 2; // number of Runge45 steps in [ti, tf] double ti = 0.; // initial time double tf = 2.; // final time // xi = X(0) CppAD::vector<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 Runge45 approximation for X(tf) CppAD::vector<double> xf(n), e(n); xf = CppAD::Runge45(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.); // 5th order method is exact for i < 5 if( i < 5 ) ok &= NearEqual(xf[i], check, eps99, eps99); // 4th order method is exact for i < 4 if( i < 4 ) ok &= (e[i] <= eps99); // check value for next i check *= tf; } } return ok; } 
Input File: example/utility/runge45_1.cpp