QR Factorization of a General Matrix

Calling Sequence

CALL DQRDC	(DA, LDA, N, P, DQRAUX, IPIVOT, DWORK, JOB)
CALL SQRDC	(SA, LDA, N, P, SQRAUX, IPIVOT, SWORK, JOB)
CALL ZQRDC	(ZA, LDA, N, P, ZQRAUX, IPIVOT, ZWORK, JOB)
CALL CQRDC	(CA, LDA, N, P, CQRAUX, IPIVOT, CWORK, JOB)
void dqrdc	(double *da, int lda, int n, int p, double *dqraux, int *ipivot, int job)
void sqrdc	(float *sa, int lda, int n, int p, float *sqraux, int *ipivot, int job)
void zqrdc	(doublecomplex *za, int lda, int n, int p, doublecomplex *zqraux, int *ipivot, int job)
void cqrdc	(complex *ca, int lda, int n, int p, complex *cqraux, int *ipivot, int job)

Arguments

xA	On entry, the matrix A. On exit, the upper triangle of A contains the matrix R and the strict lower triangle of A contains information that will allow construction of the matrix Q. If pivoting was requested, A contains the factorization of the original matrix A permuted by the requested pivots.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1,N).
N	Number of rows in the matrix A. N 0.
P	Number of columns in the matrix A. P 0.
xQRAUX	On exit, contains information required to construct the orthogonal matrix Q.
IPIVOT	If JOB selected no pivoting, then IPIVOT is not referenced. If JOB selected pivoting, then on entry to the subroutines, IPIVOT contains integers representing array indices that control the selection of pivot elements from the diagonal of A according to the system below:
	IPIVOT(k) > 0	A(k,k) is an initial element.
	IPIVOT(k) = 0	A(k,k) is a free element.
	IPIVOT(k) < 0	A(k,k) is a final element.
	Before the decomposition is computed, symmetric row and column interchanges are used to move initial elements to the beginning of A and final elements to the end of A. During the computation, symmetric row and column interchanges are used to move the largest remaining free diagonal element into the pivot position.
	On exit, IPIVOT(k) contains the index of the diagonal element of A that was moved into the kth position.
xWORK	Scratch array with a dimension of N. WORK is not referenced if JOB = 0.
JOB	Determines how the factorization is done:
	0	with no pivoting
	not 0	with pivoting

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER IDOB, IDORSD, IDOXB, LDA, N, NCOLA, NOPIV

INTEGER NROWA

PARAMETER (IDOB = 100)

PARAMETER (IDORSD = 10)

PARAMETER (IDOXB = 1)

PARAMETER (N = 3)

PARAMETER (LDA = N)

PARAMETER (NCOLA = 2)

PARAMETER (NOPIV = 0)

PARAMETER (NROWA = N)

C

DOUBLE PRECISION A(LDA,NCOLA), B(NCOLA), NULL(N), QRAUX(N)

DOUBLE PRECISION RESID(N), WORK(N), Y(N)

INTEGER ICOL, INFO, IROW, JOB, JPIVOT

C

EXTERNAL DQRDC, DQRSL

C

C Initialize the array A to store the matrix A shown below.

C Initialize the array Y to store the vector y shown below.

C

C 1 1 1

C A = 1 0 y = 0

C 0 1 -5

C

DATA A / 1.0D0, 1.0D0, 0.0D0, 1.0D0, 0.0D0, 1.0D0 /

DATA Y / 1.0D0, 0.0D0, -5.0D0 /

C

PRINT 1000

PRINT 1010, ((A(IROW,ICOL), ICOL = 1, NCOLA), IROW = 1,

$ NROWA)

PRINT 1020

PRINT 1030, Y

JOB = NOPIV

CALL DQRDC (A, LDA, NROWA, NCOLA, QRAUX, JPIVOT, WORK, JOB)

JOB = IDOB + IDORSD + IDOXB

CALL DQRSL (A, LDA, NROWA, NCOLA, QRAUX, Y, NULL, NULL, B,

$ RESID, NULL, JOB, INFO)

IF (INFO .EQ. 0) THEN

PRINT 1040

PRINT 1050, B

PRINT 1060

PRINT 1050, RESID

ELSE

PRINT 1070

END IF

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (2(3X, F4.1))

1020 FORMAT (/1X, 'y:')

1030 FORMAT (3X, F4.1)

1040 FORMAT (/1X, 'Least squares solution:')

1050 FORMAT (3X, F4.1)

1060 FORMAT (/1X, 'Residual:')

1070 FORMAT (1X, 'A is singular.')

C

END

Sample Output

A:
1.0 1.0
1.0 0.0
0.0 1.0
y:
1.0
0.0
-5.0
Least squares solution:
2.0
-3.0
Residual:
2.0
-2.0
-2.0