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Re: Errors calculating the pseudoinverse of a mxn matrix (Part II)
- From: Frank Reininghaus <frank78ac at googlemail dot com>
- To: Leopold Palomo-Avellaneda <lepalom at wol dot es>
- Cc: gsl-discuss at sourceware dot org,leo at alaxarxa dot net,Josep Arnau <jsprnclrtrbrt at hotmail dot com>
- Date: Thu, 10 Apr 2008 21:16:11 +0200
- Subject: Re: Errors calculating the pseudoinverse of a mxn matrix (Part II)
- References: <200804081046.37469.lepalom@wol.es>
Hi,
On Tuesday 08 April 2008 10:46:37 Leopold Palomo-Avellaneda wrote:
> I have been lately working with the gsl library, using the SVD
> decomposition, but I am having real trouble, so i thought that i was
> probably doing something wrong and you could help me.
yes, you are doing something wrong: You calculate the pseudoinverse in a
completely different way in the C++ program and in the Matlab script and
expect to get the same results. In the C++ program, you do it using
gsl_linalg_LU_decomp () and gsl_linalg_LU_invert () which goes badly wrong
because the third diagonal entry of the matrix S is very close to zero. In
the Matlab script, you apparently thought of this problem and did it this
way:
% As S is diagonal, que can compute its inverse by dividing inverting its
% diagonal values
for i = 1:max
if (S(i,i) > 0.0000000001)
S(i,i) = 1/S(i,i);
else
S(i,i) = 0;
end
end
The "if (S(i,i) > 0.0000000001)" check avoids the huge matrix entries which
caused the problems in your C++ program. I've attached a fixed version of
your program which reproduces the results of the Matlab script (at least up
to the limited precision of the Matlab results).
Frank
P.S.: Apparently, my first reply was rejected by the list server because the
Google Mail web interface for some reason decided to send it as text/html.
#include <cstdlib>
#include <cstdio>
#include <string>
#include <iostream>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_permutation.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_blas.h>
int main() {
int n_fil = 3;
int n_col = 8;
double mat[3][8] = {{ 50,4.5, -23, 12, 1, 0, -1, 0}, // Example Rank-deficient matrix
{ 1, 2, 3, 4, 5, 1, 0, 0},
{ 2, 4, 6, 8, 10, 2, 0, 0}};
unsigned i = 0;
unsigned j = 0;
gsl_matrix * gA = gsl_matrix_alloc (n_fil, n_col);
for (i = 0; i < n_fil; i++)
for (j = 0; j < n_col; j++)
gsl_matrix_set (gA, i, j, mat[i][j]);
gsl_matrix * gA_t = gsl_matrix_alloc (n_col, n_fil);
gsl_matrix_transpose_memcpy (gA_t, gA); // Computing the transpose of gA
gsl_matrix * U = gsl_matrix_alloc (n_col, n_fil);
gsl_matrix * V= gsl_matrix_alloc (n_fil, n_fil);
gsl_vector * S = gsl_vector_alloc (n_fil);
// Computing the SVD of the transpose of A
// The matrix 'gA_t' will contain 'U' after the function is called
gsl_vector * work = gsl_vector_alloc (n_fil);
gsl_linalg_SV_decomp (gA_t, V, S, work);
gsl_vector_free(work);
gsl_matrix_memcpy (U, gA_t);
//Inverting S//
//----------------------------------------------------------
// Matrix 'S' is diagonal, so it is contained in a vector.
// We operate to convert the vector 'S' into the matrix 'Sp'.
//Then we invert 'Sp' to 'Spu'
//----------------------------------------------------------
gsl_matrix * Sp = gsl_matrix_alloc (n_fil, n_fil);
gsl_matrix_set_zero (Sp);
for (i = 0; i < n_fil; i++)
gsl_matrix_set (Sp, i, i, gsl_vector_get(S, i)); // Vector 'S' to matrix 'Sp'
gsl_permutation * p = gsl_permutation_alloc (n_fil);
int signum;
gsl_linalg_LU_decomp (Sp, p, &signum); // Computing the LU decomposition
// Compute the inverse like in the MATLAB script
gsl_matrix * SI = gsl_matrix_calloc (n_fil, n_fil);
for (i = 0; i < n_fil; i++) {
std::cout << "S [" << i << "] = " << gsl_vector_get (S, i) << std::endl;
if (gsl_vector_get (S, i) > 0.0000000001)
gsl_matrix_set (SI, i, i, 1.0 / gsl_vector_get (S, i));
}
gsl_matrix * VT = gsl_matrix_alloc (n_fil, n_fil);
gsl_matrix_transpose_memcpy (VT, V); // Tranpose of V
//THE PSEUDOINVERSE//
//----------------------------------------------------------
//Computation of the pseudoinverse of trans(A) as pinv(A) = U·inv(S).trans(V) with trans(A) = U.S.trans(V)
//----------------------------------------------------------
gsl_matrix * SIpVT = gsl_matrix_alloc (n_fil, n_fil);
gsl_blas_dgemm (CblasNoTrans, CblasNoTrans, // Calculating inv(S).trans(V)
1.0, SI, VT,
0.0, SIpVT);
gsl_matrix * pinv = gsl_matrix_alloc (n_col, n_fil); // Calculating U·inv(S).trans(V)
gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
1.0, U, SIpVT,
0.0, pinv);
//end THE PSEUDOINVERSE//
std::cout << "pinv:" << std::endl;
for (i = 0; i < n_col; i++)
for (j = 0; j < n_fil; j++)
printf ("m(%d,%d) = %g\n", i, j,
gsl_matrix_get (pinv, i, j));
std::cout << "\n" << std::endl;
gsl_matrix_free(VT);
gsl_matrix_free(SI);
gsl_matrix_free(SIpVT);
gsl_matrix_free(gA_t);
gsl_matrix_free(U);
gsl_matrix_free(gA);
gsl_matrix_free(V);
gsl_vector_free(S);
return 0;
}