SUBROUTINE SGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, \$ INFO ) * * -- LAPACK driver routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS * .. * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * Purpose * ======= * * SGELS solves overdetermined or underdetermined real linear systems * involving an M-by-N matrix A, or its transpose, using a QR or LQ * factorization of A. It is assumed that A has full rank. * * The following options are provided: * * 1. If TRANS = 'N' and m >= n: find the least squares solution of * an overdetermined system, i.e., solve the least squares problem * minimize || B - A*X ||. * * 2. If TRANS = 'N' and m < n: find the minimum norm solution of * an underdetermined system A * X = B. * * 3. If TRANS = 'T' and m >= n: find the minimum norm solution of * an undetermined system A**T * X = B. * * 4. If TRANS = 'T' and m < n: find the least squares solution of * an overdetermined system, i.e., solve the least squares problem * minimize || B - A**T * X ||. * * Several right hand side vectors b and solution vectors x can be * handled in a single call; they are stored as the columns of the * M-by-NRHS right hand side matrix B and the N-by-NRHS solution * matrix X. * * Arguments * ========= * * TRANS (input) CHARACTER*1 * = 'N': the linear system involves A; * = 'T': the linear system involves A**T. * * M (input) INTEGER * The number of rows of the matrix A. M >= 0. * * N (input) INTEGER * The number of columns of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of * columns of the matrices B and X. NRHS >=0. * * A (input/output) REAL array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, * if M >= N, A is overwritten by details of its QR * factorization as returned by SGEQRF; * if M < N, A is overwritten by details of its LQ * factorization as returned by SGELQF. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,M). * * B (input/output) REAL array, dimension (LDB,NRHS) * On entry, the matrix B of right hand side vectors, stored * columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS * if TRANS = 'T'. * On exit, if INFO = 0, B is overwritten by the solution * vectors, stored columnwise: * if TRANS = 'N' and m >= n, rows 1 to n of B contain the least * squares solution vectors; the residual sum of squares for the * solution in each column is given by the sum of squares of * elements N+1 to M in that column; * if TRANS = 'N' and m < n, rows 1 to N of B contain the * minimum norm solution vectors; * if TRANS = 'T' and m >= n, rows 1 to M of B contain the * minimum norm solution vectors; * if TRANS = 'T' and m < n, rows 1 to M of B contain the * least squares solution vectors; the residual sum of squares * for the solution in each column is given by the sum of * squares of elements M+1 to N in that column. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= MAX(1,M,N). * * WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. * LWORK >= max( 1, MN + max( MN, NRHS ) ). * For optimal performance, * LWORK >= max( 1, MN + max( MN, NRHS )*NB ). * where MN = min(M,N) and NB is the optimum block size. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the i-th diagonal element of the * triangular factor of A is zero, so that A does not have * full rank; the least squares solution could not be * computed. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, TPSD INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE REAL ANRM, BIGNUM, BNRM, SMLNUM * .. * .. Local Arrays .. REAL RWORK( 1 ) * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV REAL SLAMCH, SLANGE EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE * .. * .. External Subroutines .. EXTERNAL SGELQF, SGEQRF, SLABAD, SLASCL, SLASET, SORMLQ, \$ SORMQR, STRTRS, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL * .. * .. Executable Statements .. * * Test the input arguments. * INFO = 0 MN = MIN( M, N ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'T' ) ) ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN INFO = -8 ELSE IF( LWORK.LT.MAX( 1, MN + MAX( MN, NRHS ) ) .AND. \$ .NOT.LQUERY ) THEN INFO = -10 END IF * * Figure out optimal block size * IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN * TPSD = .TRUE. IF( LSAME( TRANS, 'N' ) ) \$ TPSD = .FALSE. * IF( M.GE.N ) THEN NB = ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 ) IF( TPSD ) THEN NB = MAX( NB, ILAENV( 1, 'SORMQR', 'LN', M, NRHS, N, \$ -1 ) ) ELSE NB = MAX( NB, ILAENV( 1, 'SORMQR', 'LT', M, NRHS, N, \$ -1 ) ) END IF ELSE NB = ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) IF( TPSD ) THEN NB = MAX( NB, ILAENV( 1, 'SORMLQ', 'LT', N, NRHS, M, \$ -1 ) ) ELSE NB = MAX( NB, ILAENV( 1, 'SORMLQ', 'LN', N, NRHS, M, \$ -1 ) ) END IF END IF * WSIZE = MAX( 1, MN + MAX( MN, NRHS )*NB ) WORK( 1 ) = REAL( WSIZE ) * END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGELS ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( MIN( M, N, NRHS ).EQ.0 ) THEN CALL SLASET( 'Full', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) RETURN END IF * * Get machine parameters * SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' ) BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) * * Scale A, B if max element outside range [SMLNUM,BIGNUM] * ANRM = SLANGE( 'M', M, N, A, LDA, RWORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN * * Matrix all zero. Return zero solution. * CALL SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB ) GO TO 50 END IF * BROW = M IF( TPSD ) \$ BROW = N BNRM = SLANGE( 'M', BROW, NRHS, B, LDB, RWORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB, \$ INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB, \$ INFO ) IBSCL = 2 END IF * IF( M.GE.N ) THEN * * compute QR factorization of A * CALL SGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN, \$ INFO ) * * workspace at least N, optimally N*NB * IF( .NOT.TPSD ) THEN * * Least-Squares Problem min || A * X - B || * * B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS) * CALL SORMQR( 'Left', 'Transpose', M, NRHS, N, A, LDA, \$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, \$ INFO ) * * workspace at least NRHS, optimally NRHS*NB * * B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS) * CALL STRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS, \$ A, LDA, B, LDB, INFO ) * IF( INFO.GT.0 ) THEN RETURN END IF * SCLLEN = N * ELSE * * Overdetermined system of equations A**T * X = B * * B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS) * CALL STRTRS( 'Upper', 'Transpose', 'Non-unit', N, NRHS, \$ A, LDA, B, LDB, INFO ) * IF( INFO.GT.0 ) THEN RETURN END IF * * B(N+1:M,1:NRHS) = ZERO * DO 20 J = 1, NRHS DO 10 I = N + 1, M B( I, J ) = ZERO 10 CONTINUE 20 CONTINUE * * B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS) * CALL SORMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA, \$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, \$ INFO ) * * workspace at least NRHS, optimally NRHS*NB * SCLLEN = M * END IF * ELSE * * Compute LQ factorization of A * CALL SGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN, \$ INFO ) * * workspace at least M, optimally M*NB. * IF( .NOT.TPSD ) THEN * * underdetermined system of equations A * X = B * * B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS) * CALL STRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS, \$ A, LDA, B, LDB, INFO ) * IF( INFO.GT.0 ) THEN RETURN END IF * * B(M+1:N,1:NRHS) = 0 * DO 40 J = 1, NRHS DO 30 I = M + 1, N B( I, J ) = ZERO 30 CONTINUE 40 CONTINUE * * B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS) * CALL SORMLQ( 'Left', 'Transpose', N, NRHS, M, A, LDA, \$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, \$ INFO ) * * workspace at least NRHS, optimally NRHS*NB * SCLLEN = N * ELSE * * overdetermined system min || A**T * X - B || * * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) * CALL SORMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA, \$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN, \$ INFO ) * * workspace at least NRHS, optimally NRHS*NB * * B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS) * CALL STRTRS( 'Lower', 'Transpose', 'Non-unit', M, NRHS, \$ A, LDA, B, LDB, INFO ) * IF( INFO.GT.0 ) THEN RETURN END IF * SCLLEN = M * END IF * END IF * * Undo scaling * IF( IASCL.EQ.1 ) THEN CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB, \$ INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB, \$ INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB, \$ INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB, \$ INFO ) END IF * 50 CONTINUE WORK( 1 ) = REAL( WSIZE ) * RETURN * * End of SGELS * END