EIGENVEC

The EIGENVEC function computes the eigenvectors of an n-by-n real, non-symmetric array using Inverse Subspace Iteration. Use ELMHES and HQR to find the eigenvalues of an n-by-n real, nonsymmetric array.

This routine is written in the IDL language. Its source code can be found in the file eigenvec.pro in the lib subdirectory of the IDL distribution.

Note
If you are working with complex inputs, use the LA_EIGENVEC function instead.

Syntax

Result = EIGENVEC( A, Eval [, /DOUBLE] [, ITMAX=value] [, RESIDUAL=variable] )

Return Value

This function returns a complex array with a column dimension equal to n and a row dimension equal to the number of eigenvalues.

Arguments

A

An n-by-n nonsymmetric, single- or double-precision floating-point array.

EVAL

An n-element complex vector of eigenvalues.

Keywords

DOUBLE

Set this keyword to force the computation to be done in double-precision arithmetic.

ITMAX

The maximum number of iterations allowed in the computation of each eigenvector. The default value is 4.

RESIDUAL

Use this keyword to specify a named variable that will contain the residuals for each eigenvalue/eigenvector (λ/x) pair. The residual is based on the definition Ax – λx = 0 and is an array of the same size and type as that returned by the function. The rows of this array correspond to the residuals for each eigenvalue/eigenvector pair.

Examples

; Define an n-by-n real, nonsymmetric array:
A = [[1.0, -2.0, -4.0, 1.0], $
[0.0, -2.0, 3.0, 4.0], $
[2.0, -6.0, -1.0, 4.0], $
[3.0, -3.0, 1.0, -2.0]]

; Compute the eigenvalues of A using double-precision
; complex arithmetic and print the result:
eval = HQR(ELMHES(A), /DOUBLE)
PRINT, 'Eigenvalues: '
PRINT, eval

evec = EIGENVEC(A, eval, RESIDUAL = residual)

; Eigenvectors are not generally unique.
; Multiply each eigenvector by a complex scaling
; factor to force the initial term to be real.
; This normalization ensures a unique solution.
FOR i=0,3 DO evec[*,i] *= ABS(evec[0,i])/evec[0,i]

PRINT, 'Eigenvectors:'
PRINT, evec, FORMAT='(4("(",f8.5,",",f8.5,") "))'

PRINT, 'Residuals:'
FOR i=0,3 DO print, A ## evec[*,i] - eval[i]*evec[*,i], $
FORMAT ='(4("(",g9.2,",",g9.2,") "))'

IDL prints:

Eigenvalues:  
(0.26366252,-6.1925900)(0.26366252,6.1925900) 
(-4.9384492,0.00000000)(0.411124050.00000000) 
% Compiled module: EIGENVEC. 
Eigenvectors: 
( 0.42919, 0.00000) (-0.32241, 0.41235)  
( 0.29827, 0.54013) ( 0.23222, 0.32739)  
( 0.42919, 0.00000) (-0.32241,-0.41235)  
( 0.29827,-0.54013) ( 0.23222,-0.32739)  
( 0.54966, 0.00000) ( 0.18401, 0.00000)  
( 0.58125, 0.00000) (-0.57111, 0.00000)  
( 0.79297, 0.00000) ( 0.50289, 0.00000)  
(-0.04962, 0.00000) ( 0.34035, 0.00000)  
Residuals: 
( 3.1e-008, 2.9e-008) (-5.1e-008, 7.8e-009)  
(-1.5e-008, 5.9e-008) (-5.5e-009, 3.9e-008)  
( 3.1e-008,-2.9e-008) (-5.1e-008,-7.8e-009)  
(-1.5e-008,-5.9e-008) (-5.5e-009,-3.9e-008)  
(-1.1e-007,     0.00) ( 7.9e-009,     0.00)  
( 2.5e-008,     0.00) (-2.4e-008,     0.00)  
(-5.0e-008,     0.00) (-6.1e-008,     0.00)  
( 1.5e-010,     0.00) (-4.2e-008,     0.00)

You can check the accuracy of each eigenvalue/eigenvector (λ/x) pair by printing the residual array. All residual values should be floating-point zeros.

Note
Different machines may produce slightly different results.

Version History


4.0	Introduced