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view liboctave/numeric/CollocWt.cc @ 21301:40de9f8f23a6
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Use '#include "config.h"' rather than <config.h>.
author | Rik <rik@octave.org> |
---|---|
date | Thu, 18 Feb 2016 13:34:50 -0800 |
parents | f7121e111991 |
children | 3d60ed163b70 |
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/* Copyright (C) 1993-2015 John W. Eaton This file is part of Octave. Octave is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. Octave is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with Octave; see the file COPYING. If not, see <http://www.gnu.org/licenses/>. */ #ifdef HAVE_CONFIG_H # include "config.h" #endif #include <iostream> #include <cfloat> #include "CollocWt.h" #include "f77-fcn.h" #include "lo-error.h" // The following routines jcobi, dif, and dfopr are based on the code // found in Villadsen, J. and M. L. Michelsen, Solution of Differential // Equation Models by Polynomial Approximation, Prentice-Hall (1978) // pages 418-420. // // Translated to C++ by jwe. // Compute the first three derivatives of the node polynomial. // // n0 (alpha,beta) n1 // p (x) = (x) * p (x) * (1 - x) // nt n // // at the interpolation points. Each of the parameters n0 and n1 // may be given the value 0 or 1. The total number of points // nt = n + n0 + n1 // // The values of root must be known before a call to dif is possible. // They may be computed using jcobi. static void dif (octave_idx_type nt, double *root, double *dif1, double *dif2, double *dif3) { // Evaluate derivatives of node polynomial using recursion formulas. for (octave_idx_type i = 0; i < nt; i++) { double x = root[i]; dif1[i] = 1.0; dif2[i] = 0.0; dif3[i] = 0.0; for (octave_idx_type j = 0; j < nt; j++) { if (j != i) { double y = x - root[j]; dif3[i] = y * dif3[i] + 3.0 * dif2[i]; dif2[i] = y * dif2[i] + 2.0 * dif1[i]; dif1[i] = y * dif1[i]; } } } } // Compute the zeros of the Jacobi polynomial. // // (alpha,beta) // p (x) // n // // Use dif to compute the derivatives of the node // polynomial // // n0 (alpha,beta) n1 // p (x) = (x) * p (x) * (1 - x) // nt n // // at the interpolation points. // // See Villadsen and Michelsen, pages 131-132 and 418. // // Input parameters: // // nd : the dimension of the vectors dif1, dif2, dif3, and root // // n : the degree of the jacobi polynomial, (i.e. the number // of interior interpolation points) // // n0 : determines whether x = 0 is included as an // interpolation point // // n0 = 0 ==> x = 0 is not included // n0 = 1 ==> x = 0 is included // // n1 : determines whether x = 1 is included as an // interpolation point // // n1 = 0 ==> x = 1 is not included // n1 = 1 ==> x = 1 is included // // alpha : the value of alpha in the description of the jacobi // polynomial // // beta : the value of beta in the description of the jacobi // polynomial // // For a more complete explanation of alpha an beta, see Villadsen // and Michelsen, pages 57 to 59. // // Output parameters: // // root : one dimensional vector containing on exit the // n + n0 + n1 zeros of the node polynomial used in the // interpolation routine // // dif1 : one dimensional vector containing the first derivative // of the node polynomial at the zeros // // dif2 : one dimensional vector containing the second derivative // of the node polynomial at the zeros // // dif3 : one dimensional vector containing the third derivative // of the node polynomial at the zeros static bool jcobi (octave_idx_type n, octave_idx_type n0, octave_idx_type n1, double alpha, double beta, double *dif1, double *dif2, double *dif3, double *root) { assert (n0 == 0 || n0 == 1); assert (n1 == 0 || n1 == 1); octave_idx_type nt = n + n0 + n1; assert (nt > 1); // -- first evaluation of coefficients in recursion formulas. // -- recursion coefficients are stored in dif1 and dif2. double ab = alpha + beta; double ad = beta - alpha; double ap = beta * alpha; dif1[0] = (ad / (ab + 2.0) + 1.0) / 2.0; dif2[0] = 0.0; if (n >= 2) { for (octave_idx_type i = 1; i < n; i++) { double z1 = i; double z = ab + 2 * z1; dif1[i] = (ab * ad / z / (z + 2.0) + 1.0) / 2.0; if (i == 1) dif2[i] = (ab + ap + z1) / z / z / (z + 1.0); else { z *= z; double y = z1 * (ab + z1); y *= (ap + y); dif2[i] = y / z / (z - 1.0); } } } // Root determination by Newton method with suppression of previously // determined roots. double x = 0.0; for (octave_idx_type i = 0; i < n; i++) { bool done = false; int k = 0; while (! done) { double xd = 0.0; double xn = 1.0; double xd1 = 0.0; double xn1 = 0.0; for (octave_idx_type j = 0; j < n; j++) { double xp = (dif1[j] - x) * xn - dif2[j] * xd; double xp1 = (dif1[j] - x) * xn1 - dif2[j] * xd1 - xn; xd = xn; xd1 = xn1; xn = xp; xn1 = xp1; } double zc = 1.0; double z = xn / xn1; if (i != 0) { for (octave_idx_type j = 1; j <= i; j++) zc -= z / (x - root[j-1]); } z /= zc; x -= z; // Famous last words: 100 iterations should be more than // enough in all cases. if (++k > 100 || xisnan (z)) return false; if (std::abs (z) <= 100 * std::numeric_limits<double>::epsilon ()) done = true; } root[i] = x; x += sqrt (std::numeric_limits<double>::epsilon ()); } // Add interpolation points at x = 0 and/or x = 1. if (n0 != 0) { for (octave_idx_type i = n; i > 0; i--) root[i] = root[i-1]; root[0] = 0.0; } if (n1 != 0) root[nt-1] = 1.0; dif (nt, root, dif1, dif2, dif3); return true; } // Compute derivative weights for orthogonal collocation. // // See Villadsen and Michelsen, pages 133-134, 419. // // Input parameters: // // nd : the dimension of the vectors dif1, dif2, dif3, and root // // n : the degree of the jacobi polynomial, (i.e. the number // of interior interpolation points) // // n0 : determines whether x = 0 is included as an // interpolation point // // n0 = 0 ==> x = 0 is not included // n0 = 1 ==> x = 0 is included // // n1 : determines whether x = 1 is included as an // interpolation point // // n1 = 0 ==> x = 1 is not included // n1 = 1 ==> x = 1 is included // // i : the index of the node for which the weights are to be // calculated // // id : indicator // // id = 1 ==> first derivative weights are computed // id = 2 ==> second derivative weights are computed // id = 3 ==> gaussian weights are computed (in this // case, the value of i is irrelevant) // // Output parameters: // // dif1 : one dimensional vector containing the first derivative // of the node polynomial at the zeros // // dif2 : one dimensional vector containing the second derivative // of the node polynomial at the zeros // // dif3 : one dimensional vector containing the third derivative // of the node polynomial at the zeros // // vect : one dimensional vector of computed weights static void dfopr (octave_idx_type n, octave_idx_type n0, octave_idx_type n1, octave_idx_type i, octave_idx_type id, double *dif1, double *dif2, double *dif3, double *root, double *vect) { assert (n0 == 0 || n0 == 1); assert (n1 == 0 || n1 == 1); octave_idx_type nt = n + n0 + n1; assert (nt > 1); assert (id == 1 || id == 2 || id == 3); if (id != 3) assert (i >= 0 && i < nt); // Evaluate discretization matrices and Gaussian quadrature weights. // Quadrature weights are normalized to sum to one. if (id != 3) { for (octave_idx_type j = 0; j < nt; j++) { if (j == i) { if (id == 1) vect[i] = dif2[i] / dif1[i] / 2.0; else vect[i] = dif3[i] / dif1[i] / 3.0; } else { double y = root[i] - root[j]; vect[j] = dif1[i] / dif1[j] / y; if (id == 2) vect[j] = vect[j] * (dif2[i] / dif1[i] - 2.0 / y); } } } else { double y = 0.0; for (octave_idx_type j = 0; j < nt; j++) { double x = root[j]; double ax = x * (1.0 - x); if (n0 == 0) ax = ax / x / x; if (n1 == 0) ax = ax / (1.0 - x) / (1.0 - x); vect[j] = ax / (dif1[j] * dif1[j]); y += vect[j]; } for (octave_idx_type j = 0; j < nt; j++) vect[j] = vect[j] / y; } } // Error handling. void CollocWt::error (const char* msg) { (*current_liboctave_error_handler) ("CollocWt: fatal error '%s'", msg); } CollocWt& CollocWt::set_left (double val) { if (val >= rb) error ("CollocWt: left bound greater than right bound"); lb = val; initialized = 0; return *this; } CollocWt& CollocWt::set_right (double val) { if (val <= lb) error ("CollocWt: right bound less than left bound"); rb = val; initialized = 0; return *this; } void CollocWt::init (void) { // Check for possible errors. double wid = rb - lb; if (wid <= 0.0) { error ("CollocWt: width less than or equal to zero"); return; } octave_idx_type nt = n + inc_left + inc_right; if (nt < 0) error ("CollocWt: total number of collocation points less than zero"); else if (nt == 0) return; Array<double> dif1 (dim_vector (nt, 1)); double *pdif1 = dif1.fortran_vec (); Array<double> dif2 (dim_vector (nt, 1)); double *pdif2 = dif2.fortran_vec (); Array<double> dif3 (dim_vector (nt, 1)); double *pdif3 = dif3.fortran_vec (); Array<double> vect (dim_vector (nt, 1)); double *pvect = vect.fortran_vec (); r.resize (nt, 1); q.resize (nt, 1); A.resize (nt, nt); B.resize (nt, nt); double *pr = r.fortran_vec (); // Compute roots. if (! jcobi (n, inc_left, inc_right, Alpha, Beta, pdif1, pdif2, pdif3, pr)) error ("jcobi: newton iteration failed"); octave_idx_type id; // First derivative weights. id = 1; for (octave_idx_type i = 0; i < nt; i++) { dfopr (n, inc_left, inc_right, i, id, pdif1, pdif2, pdif3, pr, pvect); for (octave_idx_type j = 0; j < nt; j++) A(i,j) = vect(j); } // Second derivative weights. id = 2; for (octave_idx_type i = 0; i < nt; i++) { dfopr (n, inc_left, inc_right, i, id, pdif1, pdif2, pdif3, pr, pvect); for (octave_idx_type j = 0; j < nt; j++) B(i,j) = vect(j); } // Gaussian quadrature weights. id = 3; double *pq = q.fortran_vec (); dfopr (n, inc_left, inc_right, id, id, pdif1, pdif2, pdif3, pr, pq); initialized = 1; } std::ostream& operator << (std::ostream& os, const CollocWt& a) { if (a.left_included ()) os << "left boundary is included\n"; else os << "left boundary is not included\n"; if (a.right_included ()) os << "right boundary is included\n"; else os << "right boundary is not included\n"; os << "\n"; os << a.Alpha << " " << a.Beta << "\n\n" << a.r << "\n\n" << a.q << "\n\n" << a.A << "\n" << a.B << "\n"; return os; }