Computes the eigenexpansion of a complex Hermitian matrix A.

Synopsis

#include <imsl.h>

float *imsl_c_eig_herm (int n, f_complex *a, …, 0)

The type double procedure is imsl_d_eig_herm.

Required Arguments

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

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

Return Value

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

IMSL_VECTORS_USER, f_complex evecu[]   (Output)
Compute eigenvectors of the matrix. An array of size n × n containing the unitary 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 X.
Default: evecu_col_dim = n

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

Description

The function imsl_c_eig_herm computes the eigenvalues of a complex Hermitian 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.

Examples

Example 1

#include <imsl.h>

main()
{
    int         n = 3;
    f_complex   a[] =   { {1,0}, {1,-7}, {0,-1},
                          {1,7}, {5,0}, {10,-3},
                          {0,1}, {10,3}, {-2,0} };
    float       *eval;
                                    /* Compute eigenvalues */
    eval = imsl_c_eig_herm(n, a, 0);
                                    /* Print eigenvalues */
    imsl_f_write_matrix ("Eigenvalues", 1, n, eval, 0);
}

Output

            Eigenvalues
         1           2           3
     15.38      -10.63       -0.75

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;
    f_complex   a[] =   { {1,0}, {1,-7},  {0,-1},
                          {1,7}, {5,0},  {10,-3},
                          {0,1}, {10,3}, {-2,0} };
    float       *eval;
    f_complex   *evec;
                                 /* Compute eigenvalues and eigenvectors */
    eval = imsl_c_eig_herm(n, a,
                           IMSL_VECTORS, &evec,
                           0);
                                 /* Print eigenvalues and eigenvectors */
    imsl_f_write_matrix ("Eigenvalues",  1, n, eval, 0);
    imsl_c_write_matrix ("Eigenvectors", n, n, evec, 0);
}

Output

            Eigenvalues
         1           2           3
     15.38      -10.63       -0.75
 
                                Eigenvectors
                        1                        2                        3
1  (   0.0631,   -0.4075)  (   -0.0598,   -0.3117)  (    0.8539,    0.0000)
2  (   0.7703,    0.0000)  (   -0.5939,    0.1841)  (   -0.0313,   -0.1380)
3  (   0.4668,    0.1366)  (    0.7160,    0.0000)  (    0.0808,   -0.4942)

Warning Errors

IMSL_LOST_ORTHOGONALITY                  The iteration for at least one eigenvector failed to converge. Some of the eigenvectors may be inaccurate.

IMSL_NEVAL_MXEVAL_MISMATCH     The determined number of eigenvalues in the interval (#, #) is #. However, the input value for the maximum number of eigenvalues in this interval is #.

Fatal Errors

IMSL_SLOW_CONVERGENCE_GEN              The iteration for the eigenvalues did not converge.

IMSL_HERMITIAN_DIAG_REAL                The matrix element A (#, #) = #. The diagonal of a Hermitian matrix must be real.

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