Computes the generalized eigenexpansion of a system Ax = λBx. The matrices A and B are real and symmetric, and B is positive definite.

Synopsis

#include <imsl.h>

float *imsl_f_eig_symgen (int n, float *a, float *b, …, 0)

The type double procedure is imsl_d_eig_symgen.

Required Arguments

int n   (Input)
Number of rows and columns in the matrices.

float *a   (Input)
Array of size n × n containing the symmetric coefficient matrix A.

float *b   (Input)
Array of size n × n containing the positive definite symmetric coefficient matrix B.

Return Value

A pointer to the n eigenvalues of the symmetric matrix. To release this space, use free. If no value can be computed, then NULL is returned.

Synopsis with Optional Arguments

#include <imsl.h>

float *imsl_f_eig_symgen (int n, float *a, float *b,
IMSL_VECTORS, float **evec,
IMSL_VECTORS_USER, float evecu[],
IMSL_RETURN_USER, float evalu[],
IMSL_RANGE, float elow, float ehigh,
IMSL_A_COL_DIM, int a_col_dim,
IMSL_B_COL_DIM, int b_col_dim,
IMSL_EVECU_COL_DIM, int evecu_col_dim,
0)

Optional Arguments

IMSL_VECTORS, float **evec   (Output)
The address of a pointer to an array of size n × n containing eigenvectors of the problem. On return, the necessary space is allocated by the function. Typically, float *evec is declared, and &evec is used as an argument.

IMSL_VECTORS_USER, float evecu[]   (Output)
Compute eigenvectors of the matrix. An array of size n × n containing the matrix of generalized eigenvectors is returned in the space evecu.

IMSL_RETURN_USER, float evalu[]   (Output)
Store the n eigenvalues in the space evalu.

IMSL_A_COL_DIM, int a_col_dim   (Input)
The column dimension of A.
Default: a_col_dim = n

IMSL_B_COL_DIM, int b_col_dim   (Input)
The column dimension of B.
Default: b_col_dim = n

IMSL_EVECU_COL_DIM, int evecu_col_dim   (Input)
The column dimension of evecu.
Default: evecu_col_dim = n

Description

The function imsl_f_eig_symgen computes the eigenvalues of a symmetric, positive definite eigenvalue problem by a three-phase process (Martin and Wilkinson 1971). The matrix B is reduced to factored form using the Cholesky decomposition. These factors are used to form a congruence transformation that yields a symmetric real matrix whose eigenexpansion is obtained. The problem is then transformed back to the original coordinates. Eigenvectors are calculated and transformed as required.

Examples

Example 1

#include <imsl.h>

main()
{
    int         n = 3;
    float       a[] =   {1.1, 1.2, 1.4,
                         1.2, 1.3, 1.5,
                         1.4, 1.5, 1.6};
    float       b[] =   {2.0, 1.0, 0.0,
                         1.0, 2.0, 1.0,
                         0.0, 1.0, 2.0};
    float       *eval;
                                /* Solve for eigenvalues */
    eval = imsl_f_eig_symgen (n, a, b, 0);
                                /* Print eigenvalues */
    imsl_f_write_matrix ("Eigenvalues", 1, n, eval, 0);
}

Output

            Eigenvalues
         1           2           3
     1.386      -0.058      -0.003

Example 2

This example is a variation of the first example. Here, the eigenvectors are computed as well as the eigenvalues.

#include <imsl.h>

main()
{
    int         n = 3;
    float       a[] =   {1.1, 1.2, 1.4,
                         1.2, 1.3, 1.5,
                         1.4, 1.5, 1.6};
    float       b[] =   {2.0, 1.0, 0.0,
                         1.0, 2.0, 1.0,
                         0.0, 1.0, 2.0};
    float       *eval;
    float       *evec;
                               /* Solve for eigenvalues and eigenvectors */
    eval = imsl_f_eig_symgen (n, a, b,
                              IMSL_VECTORS, &evec,
                              0);
                               /* Print eigenvalues and eigenvectors */
    imsl_f_write_matrix ("Eigenvalues",  1, n, eval, 0);
    imsl_f_write_matrix ("Eigenvectors", n, n, evec, 0);
}

Output

            Eigenvalues
         1           2           3
     1.386      -0.058      -0.003
 
            Eigenvectors
            1           2           3
1      0.6431     -0.1147     -0.6817
2     -0.0224     -0.6872      0.7266
3      0.7655      0.7174     -0.0858

Warning Errors

IMSL_SLOW_CONVERGENCE_SYM             The iteration for an eigenvalue failed to converge in 100 iterations before deflating.

Fatal Errors

IMSL_SUBMATRIX_NOT_POS_DEFINITE  The leading # by # submatrix of the input matrix is not positive definite.

IMSL_MATRIX_B_NOT_POS_DEFINITE Matrix B is not positive definite.

Visual Numerics, Inc. Visual Numerics - Developers of IMSL and PV-WAVE http://www.vni.com/ PHONE: 713.784.3131 FAX:713.781.9260