geneig
Computes the generalized eigenexpansion of a system Ax = λBx, with A and B real.
Synopsis
#include <imsl.h>
void imsl_f_geneig (int n, float *a, float *b, f_complex *alpha, float *beta, ..., 0)
The double analogue is imsl_d_geneig.
Required Arguments
int n (Input)
Number of rows and columns in A and B.
Number of rows and columns in A and B.
float *a (Input)
Array of size n × n containing the coefficient matrix A.
Array of size n × n containing the coefficient matrix A.
float *b (Input)
Array of size n × n containing the coefficient matrix B.
Array of size n × n containing the coefficient matrix B.
f_complex *alpha (Output)
Vector of size n containing scalars αi. If βi ≠ 0, λi = αi/βi for i = 0, …, n − 1 are the eigenvalues of the system.
Vector of size n containing scalars αi. If βi ≠ 0, λi = αi/βi for i = 0, …, n − 1 are the eigenvalues of the system.
float *beta (Output)
Vector of size n .
Vector of size n .
Synopsis with Optional Arguments
#include <imsl.h>
void imsl_f_geneig (int n, float *a, float *b,
IMSL_VECTORS, f_complex **evec,
IMSL_VECTORS_USER, f_complex evecu[],
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, f_complex **evec (Output)
The address of a pointer to an array of size n × n containing eigenvectors of the problem. Each vector is normalized to have Euclidean length equal to the value one. 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 containing eigenvectors of the problem. Each vector is normalized to have Euclidean length equal to the value one. 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 matrix of generalized eigenvectors is returned in the space evecu. Each vector is normalized to have Euclidean length equal to the value one.
Compute eigenvectors of the matrix. An array of size n × n containing the matrix of generalized eigenvectors is returned in the space evecu. Each vector is normalized to have Euclidean length equal to the value one.
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_B_COL_DIM, int b_col_dim (Input)
The column dimension of B.
Default: b_col_dim = n .
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
The column dimension of evecu.
Default: evecu_col_dim = n
Description
The function imsl_f_geneig uses the QZ algorithm to compute the eigenvalues and eigenvectors of the generalized eigensystem Ax = λBx, where A and B are real matrices of order n. The eigenvalues for this problem can be infinite, so α and β are returned instead of λ. If β is nonzero, λ = α/β.
The first step of the QZ algorithm is to simultaneously reduce A to upper-Hessenberg form and B to upper-triangular form. Then, orthogonal transformations are used to reduce A to quasi-upper-triangular form while keeping B upper triangular. The generalized eigenvalues and eigenvectors for the reduced problem are then computed.
The function imsl_f_geneig is based on the QZ algorithm due to Moler and Stewart (1973), as implemented by the EISPACK routines QZHES, QZIT and QZVAL; see Garbow et al. (1977).
Examples
Example 1
In this example, the eigenvalue, λ, of system Ax = λBx is computed, where
#include <imsl.h>
#include <stdio.h>
int main()
{
int n = 3;
f_complex alpha[3];
float beta[3];
int i;
f_complex eval[3];
float a[] =
{1.0, 0.5, 0.0,
-10.0, 2.0, 0.0,
5.0, 1.0, 0.5};
float b[] =
{0.5, 0.0, 0.0,
3.0, 3.0, 0.0,
4.0, 0.5, 1.0};
/* Compute eigenvalues */
imsl_f_geneig (n, a, b, alpha, beta,
0);
for (i=0; i<n; i++)
if (beta[i] != 0.0)
eval[i] = imsl_c_div(alpha[i],
imsl_cf_convert(beta[i], 0.0));
else
printf ("Infinite eigenvalue\n");
/* Print eigenvalues */
imsl_c_write_matrix ("Eigenvalues", 1, n, eval,
0);
}
Output
Eigenvalues
1 2 3
( 0.833, 1.993) ( 0.833, -1.993) ( 0.500, 0.000)
Example 2
This example finds the eigenvalues and eigenvectors of the same eigensystem given in the last example.
#include <imsl.h>
#include <stdio.h>
int main()
{
int n = 3;
f_complex alpha[3];
float beta[3];
int i;
f_complex eval[3];
f_complex *evec;
float a[] =
{1.0, 0.5, 0.0,
-10.0, 2.0, 0.0,
5.0, 1.0, 0.5};
float b[] =
{0.5, 0.0, 0.0,
3.0, 3.0, 0.0,
4.0, 0.5, 1.0};
imsl_f_geneig (n, a, b, alpha, beta,
IMSL_VECTORS, &evec,
0);
for (i=0; i<n; i++)
if (beta[i] != 0.0)
eval[i] = imsl_c_div(alpha[i],
imsl_cf_convert(beta[i], 0.0));
else
printf ("Infinite eigenvalue\n");
/* Print eigenvalues */
imsl_c_write_matrix ("Eigenvalues", 1, n, eval,
0);
/* Print eigenvectors */
imsl_c_write_matrix ("Eigenvectors", n, n, evec,
0);
}
Output
Eigenvalues
1 2 3
( 0.833, 1.993) ( 0.833, -1.993) ( 0.500, -0.000)
Eigenvectors
1 2 3
1 ( -0.197, 0.150) ( -0.197, -0.150) ( -0.000, 0.000)
2 ( -0.069, -0.568) ( -0.069, 0.568) ( -0.000, 0.000)
3 ( 0.782, 0.000) ( 0.782, 0.000) ( 1.000, 0.000)
