Vector Rosen34( Fun &F , size_t M , const Scalar &ti , const Scalar &tf , const Vector &xi , Vector &e ) { CPPAD_ASSERT_FIRST_CALL_NOT_PARALLEL; // check numeric type specifications CheckNumericType<Scalar>(); // check simple vector class specifications CheckSimpleVector<Scalar, Vector>(); // Parameters for Shampine's Rosenbrock method // are static to avoid recalculation on each call and // do not use Vector to avoid possible memory leak static Scalar a[3] = { Scalar(0), Scalar(1), Scalar(3) / Scalar(5) }; static Scalar b[2 * 2] = { Scalar(1), Scalar(0), Scalar(24) / Scalar(25), Scalar(3) / Scalar(25) }; static Scalar ct[4] = { Scalar(1) / Scalar(2), - Scalar(3) / Scalar(2), Scalar(121) / Scalar(50), Scalar(29) / Scalar(250) }; static Scalar cg[3 * 3] = { - Scalar(4), Scalar(0), Scalar(0), Scalar(186) / Scalar(25), Scalar(6) / Scalar(5), Scalar(0), - Scalar(56) / Scalar(125), - Scalar(27) / Scalar(125), - Scalar(1) / Scalar(5) }; static Scalar d3[3] = { Scalar(97) / Scalar(108), Scalar(11) / Scalar(72), Scalar(25) / Scalar(216) }; static Scalar d4[4] = { Scalar(19) / Scalar(18), Scalar(1) / Scalar(4), Scalar(25) / Scalar(216), Scalar(125) / Scalar(216) }; CPPAD_ASSERT_KNOWN( M >= 1, "Error in Rosen34: the number of steps is less than one" ); CPPAD_ASSERT_KNOWN( e.size() == xi.size(), "Error in Rosen34: size of e not equal to size of xi" ); size_t i, j, k, l, m; // indices size_t n = xi.size(); // number of components in X(t) Scalar ns = Scalar(double(M)); // number of steps as Scalar object Scalar h = (tf - ti) / ns; // step size Scalar zero = Scalar(0); // some constants Scalar one = Scalar(1); Scalar two = Scalar(2); // permutation vectors needed for LU factorization routine CppAD::vector<size_t> ip(n), jp(n); // vectors used to store values returned by F Vector E(n * n), Eg(n), f_t(n); Vector g(n * 3), x3(n), x4(n), xf(n), ftmp(n), xtmp(n), nan_vec(n); // initialize e = 0, nan_vec = nan for(i = 0; i < n; i++) { e[i] = zero; nan_vec[i] = nan(zero); } xf = xi; // initialize solution for(m = 0; m < M; m++) { // time at beginning of this interval Scalar t = ti * (Scalar(int(M - m)) / ns) + tf * (Scalar(int(m)) / ns); // value of x at beginning of this interval x3 = x4 = xf; // evaluate partial derivatives at beginning of this interval F.Ode_ind(t, xf, f_t); F.Ode_dep(t, xf, E); // E = f_x if( hasnan(f_t) || hasnan(E) ) { e = nan_vec; return nan_vec; } // E = I - f_x * h / 2 for(i = 0; i < n; i++) { for(j = 0; j < n; j++) E[i * n + j] = - E[i * n + j] * h / two; E[i * n + i] += one; } // LU factor the matrix E # ifndef NDEBUG int sign = LuFactor(ip, jp, E); # else LuFactor(ip, jp, E); # endif CPPAD_ASSERT_KNOWN( sign != 0, "Error in Rosen34: I - f_x * h / 2 not invertible" ); // loop over integration steps for(k = 0; k < 3; k++) { // set location for next function evaluation xtmp = xf; for(l = 0; l < k; l++) { // loop over previous function evaluations Scalar bkl = b[(k-1)*2 + l]; for(i = 0; i < n; i++) { // loop over elements of x xtmp[i] += bkl * g[i*3 + l] * h; } } // ftmp = F(t + a[k] * h, xtmp) F.Ode(t + a[k] * h, xtmp, ftmp); if( hasnan(ftmp) ) { e = nan_vec; return nan_vec; } // Form Eg for this integration step for(i = 0; i < n; i++) Eg[i] = ftmp[i] + ct[k] * f_t[i] * h; for(l = 0; l < k; l++) { for(i = 0; i < n; i++) Eg[i] += cg[(k-1)*3 + l] * g[i*3 + l]; } // Solve the equation E * g = Eg LuInvert(ip, jp, E, Eg); // save solution and advance x3, x4 for(i = 0; i < n; i++) { g[i*3 + k] = Eg[i]; x3[i] += h * d3[k] * Eg[i]; x4[i] += h * d4[k] * Eg[i]; } } // Form Eg for last update to x4 only for(i = 0; i < n; i++) Eg[i] = ftmp[i] + ct[3] * f_t[i] * h; for(l = 0; l < 3; l++) { for(i = 0; i < n; i++) Eg[i] += cg[2*3 + l] * g[i*3 + l]; } // Solve the equation E * g = Eg LuInvert(ip, jp, E, Eg); // advance x4 and accumulate error bound for(i = 0; i < n; i++) { x4[i] += h * d4[3] * Eg[i]; // cant use abs because cppad.hpp may not be included Scalar diff = x4[i] - x3[i]; if( diff < zero ) e[i] -= diff; else e[i] += diff; } // advance xf for this step using x4 xf = x4; } return xf; }
#define LOG(mess) { std::cout << space << mess << std::endl; } #define RUN(cmd) LOG(#cmd ";"); space-=2; cmd; space+=2; #define RETURN(cmd) LOG("return " #cmd ";"); return cmd; class Eg { public: Eg (const Eg & eg) { LOG("copy constructor"); } Eg () { LOG("default constructor"); } ~Eg () { LOG("destructor"); } const Eg & operator= (const Eg & eg) { LOG("copy"); } }; Eg fun1 () { RETURN( Eg() ) } Eg fun2 () { RUN( Eg x ) RUN( Eg y = x ) RETURN( x ) } Eg fun3 () { RUN( Eg x ) RUN( Eg y = x ) RETURN( y ) }