Computes the eigenexpansion of a real symmetric matrix A.

Synopsis

#include <imsl.h>

float *imsl_f_eig_sym (int n, float *a, …, 0)

The type double procedure is imsl_d_eig_sym.

Required Arguments

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

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

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_sym (int n, float *a,
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_EVECU_COL_DIM, int evecu_col_dim,
IMSL_RESULT_NUMBER, int *n_eval,
0)

Optional Arguments

IMSL_VECTORS, float **evec   (Output)
The address of a pointer to an array of size n × n containing the eigenvectors of the matrix. 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 orthogonal matrix of eigenvectors is returned in the space evecu.

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

IMSL_RANGE, float elow, float ehigh   (Input)
Return eigenvalues and optionally eigenvectors that lie in the interval with lower limit elow and upper limit ehigh.
Default: (elow, ehigh) = (−∞, +∞)

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

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

IMSL_RESULT_NUMBER, int *n_eval   (Output)
The number of output eigenvalues and eigenvectors in the range low, ehigh.

Description

The function imsl_f_eig_sym computes the eigenvalues of a symmetric real matrix by a two-phase process. The matrix is reduced to tridiagonal form by elementary orthogonal similarity transformations. Then, the eigenvalues are computed using a rational QR or bisection algorithm. Eigenvectors are calculated as required (Parlett 1980, pp. 169−173).

Examples

Example 1

#include <imsl.h>

main()
{
    int         n = 3;
    float       a[] =   {7.0,  -8.0,  -8.0,
                        -8.0, -16.0, -18.0,
                        -8.0, -18.0,  13.0};
    float       *eval;
                                /* Compute eigenvalues */
    eval = imsl_f_eig_sym(n, a, 0);
                                /* Print eigenvalues */
    imsl_f_write_matrix ("Eigenvalues", 1, 3, eval, 0);
}

Output

            Eigenvalues
         1           2           3
    -27.90       22.68        9.22

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[] =   {7.0,  -8.0,  -8.0,
                        -8.0, -16.0, -18.0,
                        -8.0, -18.0,  13.0};
    float       *eval;
    float       *evec;
                                /* Compute eigenvalues and eigenvectors */
    eval = imsl_f_eig_sym(n, a,
                          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
    -27.90       22.68        9.22

 

            Eigenvectors
            1           2           3
1      0.2945     -0.2722      0.9161
2      0.8521     -0.3591     -0.3806
3      0.4326      0.8927      0.1262

Warning Errors

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

IMSL_SLOW_CONVERGENCE_2                  Inverse iteration did not converge. Eigenvector is not correct for the specified eigenvalue.

IMSL_LOST_ORTHOGONALITY_2              The eigenvectors have lost orthogonality.

IMSL_NO_EIGENVALUES_RETURNED   The number of eigenvalues in the specified interval exceeds mxeval. The argument n_eval contains the number of eigenvalues in the interval. No eigenvalues will be returned.

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