基本上我需要以上。我已经搜索了谷歌,但找不到实现这一点的方法。
我在这里找到了这个功能http://www.guwi17.de/ublas/examples/但它太慢了。我什至按照 MATLAB 的例程编写了自己的 Pade Approximation,但它只比链接中的快一点。
令我震惊的是 Mathematica 计算矩阵指数的速度有多快(它是否关心矩阵是否是 tridiag,我不知道)。
有人能帮忙吗?
编辑:这是我想出的,有什么意见吗?希望对以后的读者有用
我已经有一段时间没有使用 c++ 了,所以下面的代码可能有点杂乱/缓慢,所以如果你看到改进,请赐教。
//Program will compute the matrix exponential expm( i gA ). where i = sqrt(-1) and gA is a matrix
#include <iostream>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_blas.h>
int main() {
int n = 100;
unsigned i = 0;
unsigned j = 0;
gsl_complex img = gsl_complex_rect (0.,1.);
gsl_matrix * gA = gsl_matrix_alloc (n, n);
//Set gA:
for ( i = 0; i<n; i++) {
for ( j = 0; j<=i; j++) {
if( i == j ) {
if( i == 0 || i == n-1 ) {
gsl_matrix_set (gA, i, j, 1.);
} else {
gsl_matrix_set (gA, i, j, 2.);
}
} else if( j == i-1 ) {
gsl_matrix_set (gA, i, j, 1.);
gsl_matrix_set (gA, j, i, 1.);
} else {
gsl_matrix_set (gA, i, j, 0.);
gsl_matrix_set (gA, j, i, 0.);
}
}
}
//get SVD of gA
gsl_matrix * gA_t = gsl_matrix_alloc (n, n);
gsl_matrix_transpose_memcpy (gA_t, gA);
gsl_matrix * U = gsl_matrix_alloc (n, n);
gsl_matrix * V= gsl_matrix_alloc (n, n);
gsl_vector * S = gsl_vector_alloc (n);
gsl_vector * work = gsl_vector_alloc (n);
gsl_linalg_SV_decomp (gA_t, V, S, work);
gsl_vector_free(work);
gsl_matrix_memcpy (U, gA_t);
//Take exponential of S and form matrix
gsl_matrix_complex * Sp = gsl_matrix_complex_alloc (n, n);
gsl_matrix_complex_set_zero (Sp);
for (i = 0; i < n; i++) {
gsl_matrix_complex_set (Sp, i, i, gsl_complex_exp ( gsl_complex_mul_real ( img, gsl_vector_get(S, i) ) ) ); // Vector 'S' to matrix 'Sp'
}
gsl_matrix_complex * Uc = gsl_matrix_complex_alloc (n, n);
//convert U to a complex matrix for next step
for (i = 0; i < n; i++) {
for ( j = 0; j<n; j++) {
gsl_matrix_complex_set (Uc, i, j, gsl_complex_rect ( gsl_matrix_get(U, i, j), 0. ) );
}
}
//recombine U S Utranspose
gsl_matrix_complex * tA = gsl_matrix_complex_alloc (n, n);
gsl_matrix_complex * eA = gsl_matrix_complex_alloc (n, n);
gsl_blas_zgemm (CblasNoTrans, CblasNoTrans, GSL_COMPLEX_ONE, Uc, Sp, GSL_COMPLEX_ZERO, tA);
gsl_blas_zgemm (CblasNoTrans, CblasTrans, GSL_COMPLEX_ONE, tA, Uc, GSL_COMPLEX_ZERO, eA);
//Print result
std::cout << "eA:" << std::endl;
for (i = 0; i < n; i++)
for (j = 0; j < n; j++)
printf ("%g + %g i\n", GSL_REAL(gsl_matrix_complex_get (eA, i, j)), GSL_IMAG(gsl_matrix_complex_get (eA, i, j)));
std::cout << "\n" << std::endl;
//Free up
gsl_matrix_free(gA_t);
gsl_matrix_free(U);
gsl_matrix_free(gA);
gsl_matrix_free(V);
gsl_vector_free(S);
gsl_matrix_complex_free(Uc);
gsl_matrix_complex_free(tA);
gsl_matrix_complex_free(eA);
return 0;
}