RQ Factorization of a General Matrix

Calling Sequence

CALL DGERQF	(M, N, DA, LDA, DTAU, DWORK, LDWORK, INFO)
CALL SGERQF	(M, N, SA, LDA, STAU, SWORK, LDWORK, INFO)
CALL ZGERQF	(M, N, ZA, LDA, ZTAU, ZWORK, LDWORK, INFO)
CALL CGERQF	(M, N, CA, LDA, CTAU, CWORK, LDWORK, INFO)
void dgerqf	(int m, int n, double *da, int lda, double *dtau, int *info)
void sgerqf	(int m, int n, float *sa, int lda, float *stau, int *info)
void zgerqf	(int m, int n, doublecomplex *za, int lda, doublecomplex *ztau, int *info)
void cgerqf	(int m, int n, complex *ca, int lda, complex *ctau, int *info)

Arguments

M	Number of rows of the matrix A. M 0.
N	Number of columns of the matrix A. N 0.
xA	On entry, the matrix A. On exit, A and TAU contain an RQ factorization of the matrix A.
LDA	Leading dimension of the array A as specified in a dimension or type statement. LDA max(1, M).
xTAU	On exit, the min(M, N) scalar factors of the elementary reflectors. The elementary reflectors in xTAU together with xA contain an RQ factorization of the matrix A that can be used by other LAPACK subroutines that operate on the matrix A.
xWORK	Scratch array with a dimension of LDWORK.
LDWORK	Leading dimension of the array WORK as specified in a dimension or type statement. LDWORK max(1, M).
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDR, LDTAU, LDWORK, M, N

PARAMETER (M = 5)

PARAMETER (N = 4)

PARAMETER (LDA = M)

PARAMETER (LDR = M)

PARAMETER (LDTAU = LDA)

PARAMETER (LDWORK = M)

C

DOUBLE PRECISION A(LDA,N), R(LDR,N), TAU(LDTAU)

DOUBLE PRECISION WORK(LDWORK)

INTEGER ICOL, INFO, IROW

C

EXTERNAL DCOPY, DGERQF, DORGRQ

INTRINSIC ABS, MAX

C

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

C

C 1 1 0 0

C 1 1 1 1

C A = 1 1 -1 -1

C 1 -1 1 -1

C 1 -1 -1 1

C

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

$ 1.0D0, 1.0D0, 1.0D0, -1.0D0, -1.0D0,

$ 0.0D0, 1.0D0, -1.0D0, 1.0D0, -1.0D0,

$ 0.0D0, 1.0D0, -1.0D0, -1.0D0, 1.0D0 /

C

C Print the initial value of the array A.

C

PRINT 1000

DO 100, IROW = 1, M

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

100 CONTINUE

C

C Compute the RQ factorization of A.

C

CALL DGERQF (M, N, A, LDA, TAU, WORK, LDWORK, INFO)

IF (INFO .NE. 0) THEN

PRINT 1020, ABS(INFO)

STOP 1

END IF

DO 110, ICOL = 1, N

CALL DCOPY (M, A(1,ICOL), 1, R(1,ICOL), 1)

110 CONTINUE

C

C Compute and print the matrix Q used in the RQ factorization

C of A.

C

CALL DORGRQ (N, N, N, A(M-N+1,1), LDA, TAU, WORK, LDWORK,

$ INFO)

IF (INFO .NE. 0) THEN

PRINT 1030, ABS(INFO)

STOP 2

END IF

PRINT 1040

DO 120, IROW = 1, M

PRINT 1010, (0.0D0, ICOL = 1, IROW - (M - N) - 1),

$ (R(IROW,ICOL), ICOL = MAX(1,IROW - (M - N)),N)

120 CONTINUE

PRINT 1050

DO 130, IROW = M - N + 1, M

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

130 CONTINUE

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (5(3X, F8.4))

1020 FORMAT (1X, 'Illegal value for argument #',

$ I1, ' in DGERQF.')

1030 FORMAT (1X, 'Illegal value for argument #',

$ I1, ' in DORGRQ.')

1040 FORMAT (/1X, 'R:')

1050 FORMAT (/1X, 'Q:')

C

END

Sample Output

A:
1.0000 1.0000 0.0000 0.0000
1.0000 1.0000 1.0000 1.0000
1.0000 1.0000 -1.0000 -1.0000
1.0000 -1.0000 1.0000 -1.0000
1.0000 -1.0000 -1.0000 1.0000
R:
-1.0000 -1.0000 0.0000 0.0000
-2.0000 0.0000 0.0000 0.0000
0.0000 -2.0000 0.0000 0.0000
0.0000 0.0000 -2.0000 0.0000
0.0000 0.0000 0.0000 -2.0000
Q:
-0.5000 -0.5000 -0.5000 -0.5000
-0.5000 -0.5000 0.5000 0.5000
-0.5000 0.5000 -0.5000 0.5000
-0.5000 0.5000 0.5000 -0.5000