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EIGENALS.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME EIGENALS
* Problem :
* --------
* Solving symmetric eigenvalue problems as systems of
* nonlinear equations.
* The problem is, given a symmetric matrix A, to find an orthogonal
* matrix Q and diagonal matrix D such that A = Q(T) D Q.
* Example A: a diagonal matrix with eigenvales 1, .... , N.
* Source: An idea by Nick Gould
* Least-squares version
* SIF input: Nick Gould, Nov 1992.
* classification SUR2-AN-V-0
* The dimension of the matrix.
*IE N 2 $-PARAMETER
*IE N 10 $-PARAMETER original value
IE N 50 $-PARAMETER
* other parameter definitions
IE 1 1
* Define the upper triangular part of the matrix.
DO J 1 N
RI RJ J
A= A(J,J) RJ
IA J-1 J -1
DO I 1 J-1
AE A(I,J) 0.0
ND
VARIABLES
DO J 1 N
* Define the eigenvalues
X D(J)
DO I 1 N
* Define the eigenvectors
X Q(I,J)
ND
GROUPS
DO J 1 N
DO I 1 J
* Introduce the eigen-equations Q(T) D Q - A = 0.
XN E(I,J)
* Introduce the orthogonality-equations Q(T) Q - I = 0.
XN O(I,J)
ND
CONSTANTS
DO J 1 N
X EIGENALS O(J,J) 1.0
DO I 1 J
Z EIGENALS E(I,J) A(I,J)
ND
BOUNDS
FR EIGENALS 'DEFAULT'
START POINT
XV EIGENALS 'DEFAULT' 0.0
DO J 1 N
XV EIGENALS D(J) 1.0
XV EIGENALS Q(J,J) 1.0
ND
ELEMENT TYPE
EV 2PROD Q1 Q2
EV 3PROD Q1 Q2
EV 3PROD D
ELEMENT USES
DO J 1 N
DO I 1 J
DO K 1 N
XT E(I,J,K) 3PROD
ZV E(I,J,K) Q1 Q(K,I)
ZV E(I,J,K) Q2 Q(K,J)
ZV E(I,J,K) D D(K)
XT O(I,J,K) 2PROD
ZV O(I,J,K) Q1 Q(K,I)
ZV O(I,J,K) Q2 Q(K,J)
ND
GROUP TYPE
GV L2 GVAR
GROUP USES
XT 'DEFAULT' L2
DO J 1 N
DO I 1 J
DO K 1 N
XE E(I,J) E(I,J,K)
XE O(I,J) O(I,J,K)
ND
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS EIGENALS
INDIVIDUALS
T 2PROD
F Q1 * Q2
G Q1 Q2
G Q2 Q1
H Q1 Q2 1.0D+0
T 3PROD
F Q1 * Q2 * D
G Q1 Q2 * D
G Q2 Q1 * D
G D Q1 * Q2
H Q1 Q2 D
H Q1 D Q2
H Q2 D Q1
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS EIGENALS
INDIVIDUALS
T L2
F GVAR * GVAR
G GVAR + GVAR
H 2.0D+0
ENDATA