HQR

The HQR function returns all eigenvalues of an upper Hessenberg array. Using the output produced by the ELMHES function, this function finds all eigenvalues of the original real, nonsymmetric array.

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

Examples

To compute the eigenvalues of a real, non-symmetric unbalanced array, first define the array A:

A = [[ 1.0, 2.0, 0.0, 0.0, 0.0], $
     [-2.0, 3.0, 0.0, 0.0, 0.0], $
     [ 3.0, 4.0, 50.0, 0.0, 0.0], $
     [-4.0, 5.0, -60.0, 7.0, 0.0], $
     [-5.0, 6.0, -70.0, 8.0, -9.0]]
; Compute the upper Hessenberg form of the array:
hes = ELMHES(A)
; Compute the eigenvalues:
evals = HQR(hes)
; Sort the eigenvalues into ascending order based on their 
; real components:
evals = evals(SORT(FLOAT(evals)))
;Print the result.
PRINT, evals

IDL prints:

  ( -9.00000, 0.00000)( 2.00000, -1.73205)

  (  2.00000, 1.73205)( 7.00000,  0.00000)

  ( 50.0000,  0.00000)

This is the exact solution vector to five-decimal accuracy.

Syntax

Result = HQR( A [, /COLUMN] [, /DOUBLE] )

Return Value

The result is an n-element complex vector.

Arguments

A

An n by n upper Hessenberg array. Typically, A would be an array resulting from an application of ELMHES.

Note: If HQR is complex then only the real part is used for the computation.

Keywords

COLUMN

Set this keyword if the input array A is in column-major format (composed of column vectors) rather than in row-major format (composed of row vectors).

DOUBLE

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

Version History

4.0	Introduced

References and Resources

HQR is based on the routine hqr described in section 11.6 of Numerical Recipes in C: The Art of Scientific Computing (Second Edition), published by Cambridge University Press, and is used by permission.

HQR