The TRIQL procedure uses the QL algorithm with implicit shifts to determine the eigenvalues and eigenvectors of a real, symmetric, tridiagonal array. The routine TRIRED can be used to reduce a real, symmetric array to the tridiagonal form suitable for input to this procedure.
TRIQL is based on the routine tqli described in section 11.3 of Numerical Recipes in C: The Art of Scientific Computing (Second Edition), published by Cambridge University Press, and is used by permission.
Note: If you are working with complex inputs, use the LA_TRIQL procedure instead.
TRIQL, D, E, A [, /DOUBLE]
On input, this argument should be an n-element vector containing the diagonal elements of the array being analyzed. On output, D contains the eigenvalues.
An n-element vector containing the off-diagonal elements of the array. E0 is arbitrary. On output, this parameter is destroyed.
A named variable that returns the n eigenvectors. If the eigenvectors of a tridiagonal array are desired, A should be input as an identity array. If the eigenvectors of an array that has been reduced by TRIRED are desired, A is input as the array Q output by TRIRED.
Set this keyword to force the computation to be done in double-precision arithmetic.
To compute eigenvalues and eigenvectors of a real, symmetric, tridiagonal array, begin with an array A representing a symmetric array:
; Create the array A:
A = [[ 3.0, 1.0, -4.0], $
[ 1.0, 3.0, -4.0], $
[-4.0, -4.0, 8.0]]
; Compute the tridiagonal form of A:
TRIRED, A, D, E
; Compute the eigenvalues (returned in vector D) and the
; eigenvectors (returned in the rows of the array A):
TRIQL, D, E, A
; Print eigenvalues:
PRINT, 'Eigenvalues:'
PRINT, D
; Print eigenvectors:
PRINT, 'Eigenvectors:'
PRINT, A
IDL prints:
Eigenvalues:
2.00000 4.76837e-7 12.0000
Eigenvectors:
0.707107 -0.707107 0.00000
-0.577350 -0.577350 -0.577350
-0.408248 -0.408248 0.816497
The exact eigenvalues are:
[2.0, 0.0, 12.0]
The exact eigenvectors are:
[ 1.0/sqrt(2.0), -1.0/sqrt(2.0), 0.0/sqrt(2.0)],
[-1.0/sqrt(3.0), -1.0/sqrt(3.0), -1.0/sqrt(3.0)],
[-1.0/sqrt(6.0), -1.0/sqrt(6.0), 2.0/sqrt(6.0)]
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4.0 |
Introduced |