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QR3DBD.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME QR3DBD
* Problem :
* *********
* Find the QR factorization of a tridiagonal matrix A.
* The problem is formulated as a system of quadratic equations
* whose unknowns are the elements of the orthogonal matrix Q and of
* the upper triangular matrix R. In this version of the problem,
* the banded structure of R is imposed. As a consequence, there are
* less variables than equations.
* See problem QR3D for the case where this structure is ignored.
* The problem is non-convex.
* Source:
* Ph. Toint, private communication.
* SIF input: Ph. Toint, Nov 1993
* classification NQR2-AN-V-V
* Define the matrix order M ( M >= 3 ).
* There are n = M * M + 3 * ( M - 1 ) variables
* and m = M * ( 3 * M + 1 ) equations.
*IE M 5 $-PARAMETER n= 37, m=40
*IE M 10 $-PARAMETER n=129, m=155 original value
IE M 20 $-PARAMETER n=457, m=610
* Define useful parameters
IE 1 1
IE 2 2
IE 3 3
IA M-1 M -1
IA M-2 M -2
RI RM M
* Define the matrix A to factorize
* a(i,i-1)=a(i,i+1)= (1-i)/m, a(i,i) = 2i/m
RD 2/M RM 2.0
A= A(1,1) 2/M
AE A(1,2) 0.0
DO I 2 M-1
IA I+1 I 1
IA I-1 I -1
IM 1-I I-1 -1
RI R1-I 1-I
R/ 1-I/M R1-I RM
IM 2I I 2
RI R2I 2I
R/ 2I/M R2I RM
A= A(I,I-1) 1-I/M
A= A(I,I) 2I/M
A= A(I,I+1) 1-I/M
OD I
RI RM-1 M-1
RM 1-M RM-1 -1.0
R/ 1-M/M 1-M RM
RM 2M RM 2.0
A= A(M,M-1) 1-M/M
A= A(M,M) 2M
VARIABLES
* The orthogonal matrix Q
DO I 1 M
DO J 1 M
X Q(I,J)
OD J
OD I
* The upper triangular matrix R
DO I 1 M-2
IA I+1 I 1
IA I+2 I 2
X R(I,I)
X R(I,I+1)
X R(I,I+2)
OD I
X R(M-1,M-1)
X R(M-1,M)
X R(M,M)
GROUPS
* Orthogonality of the Q matrix
DO I 1 M
DO J I M
XE O(I,J)
OD J
OD I
* Factorization equations A= QR
DO I 1 M
DO J 1 M
XE F(I,J)
OD J
OD I
CONSTANTS
* Orthogonality constraints
DO I 1 M
X QR3DBD O(I,I) 1.0
OD J
* Factorization constraints
Z QR3DBD F(1,1) A(1,1)
Z QR3DBD F(1,2) A(1,2)
DO I 2 M-1
IA I-1 I -1
IA I+1 I 1
Z QR3DBD F(I,I-1) A(I,I-1)
Z QR3DBD F(I,I) A(I,I)
Z QR3DBD F(I,I+1) A(I,I+1)
OD I
Z QR3DBD F(M,M-1) A(M,M-1)
Z QR3DBD F(M,M) A(M,M)
BOUNDS
* All variables are free, except the diagonals of R
* which are nonnegative
FR QR3DBD 'DEFAULT'
DO I 1 M
XL QR3DBD R(I,I) 0.0
OD I
START POINT
* Initialize Q to the identity
DO I 1 M
XV QR3DBD Q(I,I) 1.0
OD I
* Initialize R to the upper triangular part of A
DO I 1 M-1
IA I+1 I 1
ZV QR3DBD R(I,I) A(I,I)
ZV QR3DBD R(I,I+1) A(I,I+1)
OD I
ZV QR3DBD R(M,M) A(M,M)
ELEMENT TYPE
EV 2PR V1 V2
ELEMENT USES
XT 'DEFAULT' 2PR
DO I 1 M
DO J I M
DO K 1 M
ZV C(I,J,K) V1 Q(I,K)
ZV C(I,J,K) V2 Q(J,K)
OD K
OD J
OD I
DO I 1 M
ZV B(I,1,1) V1 Q(I,1)
ZV B(I,1,1) V2 R(1,1)
ZV B(I,2,1) V1 Q(I,1)
ZV B(I,2,1) V2 R(1,2)
ZV B(I,2,2) V1 Q(I,2)
ZV B(I,2,2) V2 R(2,2)
DO J 3 M
IA J-2 J -2
DO K J-2 J
ZV B(I,J,K) V1 Q(I,K)
ZV B(I,J,K) V2 R(K,J)
OD K
OD J
OD I
GROUP USES
* Orthogonality constraints
DO I 1 M
DO J I M
DO K 1 M
XE O(I,J) C(I,J,K)
OD K
OD J
OD I
* Factorization constraints
DO I 1 M
XE F(I,1) B(I,1,1)
XE F(I,2) B(I,2,1) B(I,2,2)
DO J 3 M
IA J-2 J -2
DO K J-2 J
XE F(I,J) B(I,J,K)
OD K
OD J
OD I
OBJECT BOUND
* Solution
*LO SOLTN 0.0
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS QR3DBD
INDIVIDUALS
T 2PR
F V1 * V2
G V1 V2
G V2 V1
H V1 V2 1.0
ENDATA