eig_herm (complex)
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_z_eig_herm.
Required Arguments
int n (Input)
Number of rows and columns in the matrix.
Number of rows and columns in the matrix.
f_complex *a (Input)
Array of size n × n containing the matrix.
Array of size n × n containing the matrix.
Return Value
A pointer to the eigenvalues of the matrix in decreasing order of magnitude. To release this space, use imsl_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_EXTREME_VALUES, int small, int n_extreme,
IMSL_A_COL_DIM, int a_col_dim,
IMSL_EVECU_COL_DIM, int evecu_col_dim,
IMSL_RETURN_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_eval containing the orthonormal eigenvectors of the matrix. In the special case of n_eval = 0, a one-element array is returned. On return, the necessary space is allocated by the function. Typically, f_complex *evec is declared, and &evec is used as an argument.
The address of a pointer to an array of size n × n_eval containing the orthonormal eigenvectors of the matrix. In the special case of n_eval = 0, a one-element array is returned. 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. Array evecu, which contains the orthonormal eigenvectors, is user-defined and must be of size n × k, where k >= n_extreme if optional argument IMSL_EXTREME_VALUES is used, and k >= n otherwise.
Compute eigenvectors of the matrix. Array evecu, which contains the orthonormal eigenvectors, is user-defined and must be of size n × k, where k >= n_extreme if optional argument IMSL_EXTREME_VALUES is used, and k >= n otherwise.
IMSL_RETURN_USER, float evalu[] (Output)
Store the eigenvalues in decreasing order of magnitude in a user-defined array. Array evalu must be of size k, where k >= n_extreme if optional argument IMSL_EXTREME_VALUES is used, and k >= n otherwise.
Store the eigenvalues in decreasing order of magnitude in a user-defined array. Array evalu must be of size k, where k >= n_extreme if optional argument IMSL_EXTREME_VALUES is used, and k >= n otherwise.
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) = (−∞, +∞).
Return eigenvalues and optionally eigenvectors that lie in the interval with lower limit elow and upper limit ehigh.
Default: (elow, ehigh) = (−∞, +∞).
IMSL_EXTREME_VALUES, int small, int n_extreme (Input)
Return extreme eigenvalues and optionally eigenvectors of the matrix. If small = 0, the largest n_extreme eigenvalues are returned, if small = 1, the smallest n_extreme eigenvalues are returned.
Return extreme eigenvalues and optionally eigenvectors of the matrix. If small = 0, the largest n_extreme eigenvalues are returned, if small = 1, the smallest n_extreme eigenvalues are returned.
IMSL_A_COL_DIM, int a_col_dim (Input)
The column dimension of A.
Default: a_col_dim = n
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_extreme, if argument IMSL_EXTREME_VALUES is used, evecu_col_dim = n otherwise.
The column dimension of evecu.
Default: evecu_col_dim = n_extreme, if argument IMSL_EXTREME_VALUES is used, evecu_col_dim = n otherwise.
IMSL_RETURN_NUMBER, int *n_eval (Output)
The number of output eigenvalues and eigenvectors in the range (elow, ehigh) or, if optional argument IMSL_EXTREME_VALUES is used, the number of extreme eigenvalues computed (that is, n_extreme).
The number of output eigenvalues and eigenvectors in the range (elow, ehigh) or, if optional argument IMSL_EXTREME_VALUES is used, the number of extreme eigenvalues computed (that is, n_extreme).
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>
int 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>
int 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 #. |
IMSL_HERMITIAN_DIAG_REAL_1 | The matrix element “a[#][#]” =#. The diagonal of a Hermitian matrix must be real; its imaginary part is set to zero. |
IMSL_BEST_ESTIMATE_RETURNED | The iteration for an eigenvalue failed to converge. The best estimate will be returned. |
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. |
