SUBROUTINE MB02QY( M, N, NRHS, RANK, A, LDA, JPVT, B, LDB, TAU, $ DWORK, LDWORK, INFO ) C C RELEASE 4.0, WGS COPYRIGHT 1999. C C PURPOSE C C To determine the minimum-norm solution to a real linear least C squares problem: C C minimize || A * X - B ||, C C using the rank-revealing QR factorization of a real general C M-by-N matrix A, computed by SLICOT Library routine MB03OD. C C ARGUMENTS C C Input/Output Parameters C C M (input) INTEGER C The number of rows of the matrices A and B. M >= 0. C C N (input) INTEGER C The number of columns of the matrix A. N >= 0. C C NRHS (input) INTEGER C The number of columns of the matrix B. NRHS >= 0. C C RANK (input) INTEGER C The effective rank of A, as returned by SLICOT Library C routine MB03OD. min(M,N) >= RANK >= 0. C C A (input/output) DOUBLE PRECISION array, dimension C ( LDA, N ) C On entry, the leading min(M,N)-by-N upper trapezoidal C part of this array contains the triangular factor R, as C returned by SLICOT Library routine MB03OD. The strict C lower trapezoidal part of A is not referenced. C On exit, if RANK < N, the leading RANK-by-RANK upper C triangular part of this array contains the upper C triangular matrix R of the complete orthogonal C factorization of A, and the submatrix (1:RANK,RANK+1:N) C of this array, with the array TAU, represent the C orthogonal matrix Z (of the complete orthogonal C factorization of A), as a product of RANK elementary C reflectors. C On exit, if RANK = N, this array is unchanged. C C LDA INTEGER C The leading dimension of the array A. LDA >= max(1,M). C C JPVT (input) INTEGER array, dimension ( N ) C The recorded permutations performed by SLICOT Library C routine MB03OD; if JPVT(i) = k, then the i-th column C of A*P was the k-th column of the original matrix A. C C B (input/output) DOUBLE PRECISION array, dimension C ( LDB, NRHS ) C On entry, if NRHS > 0, the leading M-by-NRHS part of C this array must contain the matrix B (corresponding to C the transformed matrix A, returned by SLICOT Library C routine MB03OD). C On exit, if NRHS > 0, the leading N-by-NRHS part of this C array contains the solution matrix X. C If M >= N and RANK = N, the residual sum-of-squares C for the solution in the i-th column is given by the sum C of squares of elements N+1:M in that column. C If NRHS = 0, the array B is not referenced. C C LDB INTEGER C The leading dimension of the array B. C LDB >= max(1,M,N), if NRHS > 0. C LDB >= 1, if NRHS = 0. C C TAU (output) DOUBLE PRECISION array, dimension ( min(M,N) ) C The scalar factors of the elementary reflectors. C If RANK = N, the array TAU is not referenced. C C Workspace C C DWORK DOUBLE PRECISION array, dimension ( LDWORK ) C On exit, if INFO = 0, DWORK(1) returns the optimal value C of LDWORK. C C LDWORK INTEGER C The length of the array DWORK. C LDWORK >= max( 1, N, NRHS ). C For good performance, LDWORK should sometimes be larger. C C Error Indicator C C INFO INTEGER C = 0: successful exit; C < 0: if INFO = -i, the i-th argument had an illegal C value. C C METHOD C C The routine uses a QR factorization with column pivoting: C C A * P = Q * R = Q * [ R11 R12 ], C [ 0 R22 ] C C where R11 is an upper triangular submatrix of estimated rank C RANK, the effective rank of A. The submatrix R22 can be C considered as negligible. C C If RANK < N, then R12 is annihilated by orthogonal C transformations from the right, arriving at the complete C orthogonal factorization: C C A * P = Q * [ T11 0 ] * Z. C [ 0 0 ] C C The minimum-norm solution is then C C X = P * Z' [ inv(T11)*Q1'*B ], C [ 0 ] C C where Q1 consists of the first RANK columns of Q. C C The input data for MB02QY are the transformed matrices Q' * A C (returned by SLICOT Library routine MB03OD) and Q' * B. C Matrix Q is not needed. C C NUMERICAL ASPECTS C C The implemented method is numerically stable. C C CONTRIBUTOR C C V. Sima, Research Institute for Informatics, Bucharest, Aug. 1999. C C REVISIONS C C - C C KEYWORDS C C Least squares solutions; QR decomposition. C C ****************************************************************** C C .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) C .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LDWORK, M, N, NRHS, RANK C .. Array Arguments .. INTEGER JPVT( * ) DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ), TAU( * ) C .. Local Scalars .. INTEGER I, IASCL, IBSCL, J, MN DOUBLE PRECISION ANRM, BIGNUM, BNRM, MAXWRK, SMLNUM C .. External Functions .. DOUBLE PRECISION DLAMCH, DLANGE, DLANTR EXTERNAL DLAMCH, DLANGE, DLANTR C .. External Subroutines .. EXTERNAL DCOPY, DLABAD, DLASCL, DLASET, DORMRZ, DTRSM, $ DTZRZF, XERBLA C .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN C .. C .. Executable Statements .. C MN = MIN( M, N ) C C Test the input scalar arguments. C INFO = 0 IF( M.LT.0 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( NRHS.LT.0 ) THEN INFO = -3 ELSE IF( RANK.LT.0 .OR. RANK.GT.MN ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -6 ELSE IF( LDB.LT.1 .OR. ( NRHS.GT.0 .AND. LDB.LT.MAX( M, N ) ) ) $ THEN INFO = -9 ELSE IF( LDWORK.LT.MAX( 1, N, NRHS ) ) THEN INFO = -12 END IF C IF( INFO.NE.0 ) THEN CALL XERBLA( 'MB02QY', -INFO ) RETURN END IF C C Quick return if possible. C IF( MIN( MN, NRHS ).EQ.0 ) THEN DWORK( 1 ) = ONE RETURN END IF C C Logically partition R = [ R11 R12 ], C [ 0 R22 ] C C where R11 = R(1:RANK,1:RANK). If RANK = N, let T11 = R11. C MAXWRK = DBLE( N ) IF( RANK.LT.N ) THEN C C Get machine parameters. C SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) C C Scale A, B if max entries outside range [SMLNUM,BIGNUM]. C ANRM = DLANTR( 'MaxNorm', 'Upper', 'Non-unit', RANK, N, A, LDA, $ DWORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN C C Scale matrix norm up to SMLNUM. C CALL DLASCL( 'Upper', 0, 0, ANRM, SMLNUM, RANK, N, A, LDA, $ INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN C C Scale matrix norm down to BIGNUM. C CALL DLASCL( 'Upper', 0, 0, ANRM, BIGNUM, RANK, N, A, LDA, $ INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN C C Matrix all zero. Return zero solution. C CALL DLASET( 'Full', N, NRHS, ZERO, ZERO, B, LDB ) DWORK( 1 ) = ONE RETURN END IF C BNRM = DLANGE( 'MaxNorm', M, NRHS, B, LDB, DWORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN C C Scale matrix norm up to SMLNUM. C CALL DLASCL( 'General', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, $ INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN C C Scale matrix norm down to BIGNUM. C CALL DLASCL( 'General', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, $ INFO ) IBSCL = 2 END IF C C [R11,R12] = [ T11, 0 ] * Z. C Details of Householder rotations are stored in TAU. C Workspace need RANK, prefer RANK*NB. C CALL DTZRZF( RANK, N, A, LDA, TAU, DWORK, LDWORK, INFO ) MAXWRK = MAX( MAXWRK, DWORK( 1 ) ) END IF C C B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS). C CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK, $ NRHS, ONE, A, LDA, B, LDB ) C IF( RANK.LT.N ) THEN C CALL DLASET( 'Full', N-RANK, NRHS, ZERO, ZERO, B( RANK+1, 1 ), $ LDB ) C C B(1:N,1:NRHS) := Z' * B(1:N,1:NRHS). C Workspace need NRHS, prefer NRHS*NB. C CALL DORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A, $ LDA, TAU, B, LDB, DWORK, LDWORK, INFO ) MAXWRK = MAX( MAXWRK, DWORK( 1 ) ) C C Undo scaling. C IF( IASCL.EQ.1 ) THEN CALL DLASCL( 'General', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, $ INFO ) CALL DLASCL( 'Upper', 0, 0, SMLNUM, ANRM, RANK, RANK, A, $ LDA, INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL DLASCL( 'General', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, $ INFO ) CALL DLASCL( 'Upper', 0, 0, BIGNUM, ANRM, RANK, RANK, A, $ LDA, INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL DLASCL( 'General', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, $ INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL DLASCL( 'General', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, $ INFO ) END IF END IF C C B(1:N,1:NRHS) := P * B(1:N,1:NRHS). C Workspace N. C DO 20 J = 1, NRHS C DO 10 I = 1, N DWORK( JPVT( I ) ) = B( I, J ) 10 CONTINUE C CALL DCOPY( N, DWORK, 1, B( 1, J ), 1 ) 20 CONTINUE C DWORK( 1 ) = MAXWRK RETURN C C *** Last line of MB02QY *** END