SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, \$ WORK, LWORK, IWORK, INFO ) * * -- LAPACK driver routine (instrumented to count ops, version 3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * September 26, 2002 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK REAL RCOND * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * ) * .. * Common blocks to return operation counts and timings. * .. Common blocks .. COMMON / LATIME / OPS, ITCNT COMMON / LSTIME / OPCNT, TIMNG * .. * .. Scalars in Common .. REAL ITCNT, OPS * .. * .. Arrays in Common .. REAL OPCNT( 6 ), TIMNG( 6 ) * .. * * Purpose * ======= * * SGELSD computes the minimum-norm solution to a real linear least * squares problem: * minimize 2-norm(| b - A*x |) * using the singular value decomposition (SVD) of A. A is an M-by-N * matrix which may be rank-deficient. * * 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. * * The problem is solved in three steps: * (1) Reduce the coefficient matrix A to bidiagonal form with * Householder transformations, reducing the original problem * into a "bidiagonal least squares problem" (BLS) * (2) Solve the BLS using a divide and conquer approach. * (3) Apply back all the Householder tranformations to solve * the original least squares problem. * * The effective rank of A is determined by treating as zero those * singular values which are less than RCOND times the largest singular * value. * * The divide and conquer algorithm makes very mild assumptions about * floating point arithmetic. It will work on machines with a guard * digit in add/subtract, or on those binary machines without guard * digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or * Cray-2. It could conceivably fail on hexadecimal or decimal machines * without guard digits, but we know of none. * * Arguments * ========= * * M (input) INTEGER * The number of rows of A. M >= 0. * * N (input) INTEGER * The number of columns of 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) REAL array, dimension (LDA,N) * On entry, the M-by-N matrix A. * On exit, A has been destroyed. * * 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 M-by-NRHS right hand side matrix B. * On exit, B is overwritten by the N-by-NRHS solution * matrix X. If m >= n and RANK = n, the residual * sum-of-squares for the solution in the i-th column is given * by the sum of squares of elements n+1:m in that column. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,max(M,N)). * * S (output) REAL array, dimension (min(M,N)) * The singular values of A in decreasing order. * The condition number of A in the 2-norm = S(1)/S(min(m,n)). * * RCOND (input) REAL * RCOND is used to determine the effective rank of A. * Singular values S(i) <= RCOND*S(1) are treated as zero. * If RCOND < 0, machine precision is used instead. * * RANK (output) INTEGER * The effective rank of A, i.e., the number of singular values * which are greater than RCOND*S(1). * * WORK (workspace/output) REAL array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK must be at least 1. * The exact minimum amount of workspace needed depends on M, * N and NRHS. As long as LWORK is at least * 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, * if M is greater than or equal to N or * 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, * if M is less than N, the code will execute correctly. * SMLSIZ is returned by ILAENV and is equal to the maximum * size of the subproblems at the bottom of the computation * tree (usually about 25), and * NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) * For good performance, LWORK should generally be larger. * * 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. * * IWORK (workspace) INTEGER array, dimension (LIWORK) * LIWORK >= 3 * MINMN * NLVL + 11 * MINMN, * where MINMN = MIN( M,N ). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: the algorithm for computing the SVD failed to converge; * if INFO = i, i off-diagonal elements of an intermediate * bidiagonal form did not converge to zero. * * Further Details * =============== * * Based on contributions by * Ming Gu and Ren-Cang Li, Computer Science Division, University of * California at Berkeley, USA * Osni Marques, LBNL/NERSC, USA * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TWO PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 ) * .. * .. Local Scalars .. LOGICAL LQUERY INTEGER GEBRD, GELQF, GELSD, GEQRF, IASCL, IBSCL, IE, \$ IL, ITAU, ITAUP, ITAUQ, LALSD, LDWORK, MAXMN, \$ MAXWRK, MINMN, MINWRK, MM, MNTHR, NB, NLVL, \$ NWORK, ORMBR, ORMLQ, ORMQR, SMLSIZ, WLALSD REAL ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM, T1, T2 * .. * .. External Subroutines .. EXTERNAL SGEBRD, SGELQF, SGEQRF, SLABAD, SLACPY, SLALSD, \$ SLASCL, SLASET, SORMBR, SORMLQ, SORMQR, XERBLA * .. * .. External Functions .. INTEGER ILAENV REAL SECOND, SLAMCH, SLANGE, SOPLA, SOPLA2 EXTERNAL ILAENV, SECOND, SLAMCH, SLANGE, SOPLA, SOPLA2 * .. * .. Intrinsic Functions .. INTRINSIC INT, LOG, MAX, MIN, REAL * .. * .. Data statements .. DATA GEBRD / 3 / , GELQF / 2 / , GELSD / 1 / , \$ GEQRF / 2 / , LALSD / 5 / , ORMBR / 4 / , \$ ORMLQ / 4 / , ORMQR / 2 / * .. * .. Executable Statements .. * * Test the input arguments. * INFO = 0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNTHR = ILAENV( 6, 'SGELSD', ' ', M, N, NRHS, -1 ) LQUERY = ( LWORK.EQ.-1 ) 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( LDA.LT.MAX( 1, M ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN INFO = -7 END IF * SMLSIZ = ILAENV( 9, 'SGELSD', ' ', 0, 0, 0, 0 ) * * Compute workspace. * (Note: Comments in the code beginning "Workspace:" describe the * minimal amount of workspace needed at that point in the code, * as well as the preferred amount for good performance. * NB refers to the optimal block size for the immediately * following subroutine, as returned by ILAENV.) * MINWRK = 1 MINMN = MAX( 1, MINMN ) NLVL = MAX( INT( LOG( REAL( MINMN ) / REAL( SMLSIZ+1 ) ) / \$ LOG( TWO ) )+1, 0 ) * IF( INFO.EQ.0 ) THEN MAXWRK = 0 MM = M IF( M.GE.N .AND. M.GE.MNTHR ) THEN * * Path 1a - overdetermined, with many more rows than columns. * MM = N MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'SGEQRF', ' ', M, N, \$ -1, -1 ) ) MAXWRK = MAX( MAXWRK, N+NRHS* \$ ILAENV( 1, 'SORMQR', 'LT', M, NRHS, N, -1 ) ) END IF IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined. * MAXWRK = MAX( MAXWRK, 3*N+( MM+N )* \$ ILAENV( 1, 'SGEBRD', ' ', MM, N, -1, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+NRHS* \$ ILAENV( 1, 'SORMBR', 'QLT', MM, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*N+( N-1 )* \$ ILAENV( 1, 'SORMBR', 'PLN', N, NRHS, N, -1 ) ) WLALSD = 9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + \$ ( SMLSIZ+1 )**2 MAXWRK = MAX( MAXWRK, 3*N+WLALSD ) MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD ) END IF IF( N.GT.M ) THEN WLALSD = 9*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + \$ ( SMLSIZ+1 )**2 IF( N.GE.MNTHR ) THEN * * Path 2a - underdetermined, with many more columns * than rows. * MAXWRK = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) MAXWRK = MAX( MAXWRK, M*M+4*M+2*M* \$ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS* \$ ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )* \$ ILAENV( 1, 'SORMBR', 'PLN', M, NRHS, M, -1 ) ) IF( NRHS.GT.1 ) THEN MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS ) ELSE MAXWRK = MAX( MAXWRK, M*M+2*M ) END IF MAXWRK = MAX( MAXWRK, M+NRHS* \$ ILAENV( 1, 'SORMLQ', 'LT', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD ) ELSE * * Path 2 - remaining underdetermined cases. * MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'SGEBRD', ' ', M, N, \$ -1, -1 ) MAXWRK = MAX( MAXWRK, 3*M+NRHS* \$ ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, N, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+M* \$ ILAENV( 1, 'SORMBR', 'PLN', N, NRHS, M, -1 ) ) MAXWRK = MAX( MAXWRK, 3*M+WLALSD ) END IF MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD ) END IF MINWRK = MIN( MINWRK, MAXWRK ) WORK( 1 ) = MAXWRK IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGELSD', -INFO ) RETURN ELSE IF( LQUERY ) THEN GO TO 10 END IF * * Quick return if possible. * IF( M.EQ.0 .OR. N.EQ.0 ) THEN RANK = 0 RETURN END IF * * Get machine parameters. * EPS = SLAMCH( 'P' ) SFMIN = SLAMCH( 'S' ) OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( 2 ) SMLNUM = SFMIN / EPS BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) * * Scale A if max entry outside range [SMLNUM,BIGNUM]. * ANRM = SLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM. * OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( M*N ) 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. * OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( M*N ) 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 ) CALL SLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 ) RANK = 0 GO TO 10 END IF * * Scale B if max entry outside range [SMLNUM,BIGNUM]. * BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM. * OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( M*NRHS ) CALL SLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM. * OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( M*NRHS ) CALL SLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO ) IBSCL = 2 END IF * * If M < N make sure certain entries of B are zero. * IF( M.LT.N ) \$ CALL SLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) * * Overdetermined case. * IF( M.GE.N ) THEN * * Path 1 - overdetermined or exactly determined. * MM = M IF( M.GE.MNTHR ) THEN * * Path 1a - overdetermined, with many more rows than columns. * MM = N ITAU = 1 NWORK = ITAU + N * * Compute A=Q*R. * (Workspace: need 2*N, prefer N+N*NB) * NB = ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 ) OPCNT( GEQRF ) = OPCNT( GEQRF ) + \$ SOPLA( 'SGEQRF', M, N, 0, 0, NB ) T1 = SECOND( ) CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( GEQRF ) = TIMNG( GEQRF ) + ( T2-T1 ) * * Multiply B by transpose(Q). * (Workspace: need N+NRHS, prefer N+NRHS*NB) * NB = ILAENV( 1, 'SORMQR', 'LT', M, NRHS, N, -1 ) OPCNT( ORMQR ) = OPCNT( ORMQR ) + \$ SOPLA( 'SORMQR', M, NRHS, N, 0, NB ) T1 = SECOND( ) CALL SORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMQR ) = TIMNG( ORMQR ) + ( T2-T1 ) * * Zero out below R. * IF( N.GT.1 ) THEN CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA ) END IF END IF * IE = 1 ITAUQ = IE + N ITAUP = ITAUQ + N NWORK = ITAUP + N * * Bidiagonalize R in A. * (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB) * NB = ILAENV( 1, 'SGEBRD', ' ', MM, N, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ SOPLA( 'SGEBRD', MM, N, 0, 0, NB ) T1 = SECOND( ) CALL SGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, \$ INFO ) T2 = SECOND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors of R. * (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB) * NB = ILAENV( 1, 'SORMBR', 'QLT', MM, NRHS, N, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'QLT', MM, NRHS, N, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * * Solve the bidiagonal least squares problem. * OPS = ZERO T1 = SECOND( ) CALL SLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB, \$ RCOND, RANK, WORK( NWORK ), IWORK, INFO ) T2 = SECOND( ) TIMNG( LALSD ) = TIMNG( LALSD ) + ( T2-T1 ) OPCNT( LALSD ) = OPCNT( LALSD ) + OPS IF( INFO.NE.0 ) THEN GO TO 10 END IF * * Multiply B by right bidiagonalizing vectors of R. * NB = ILAENV( 1, 'SORMBR', 'PLN', N, NRHS, N, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'PLN', N, NRHS, N, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+ \$ MAX( M, 2*M-4, NRHS, N-3*M ) ) THEN * * Path 2a - underdetermined, with many more columns than rows * and sufficient workspace for an efficient algorithm. * LDWORK = M IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ), \$ M*LDA+M+M*NRHS ) )LDWORK = LDA ITAU = 1 NWORK = M + 1 * * Compute A=L*Q. * (Workspace: need 2*M, prefer M+M*NB) * NB = ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 ) OPCNT( GELQF ) = OPCNT( GELQF ) + \$ SOPLA( 'SGELQF', M, N, 0, 0, NB ) T1 = SECOND( ) CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( GELQF ) = TIMNG( GELQF ) + ( T2-T1 ) IL = NWORK * * Copy L to WORK(IL), zeroing out above its diagonal. * CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK ) CALL SLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ), \$ LDWORK ) IE = IL + LDWORK*M ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M * * Bidiagonalize L in WORK(IL). * (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB) * NB = ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ SOPLA( 'SGEBRD', M, M, 0, 0, NB ) T1 = SECOND( ) CALL SGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ), \$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors of L. * (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB) * NB = ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, M, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'QLT', M, NRHS, M, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK, \$ WORK( ITAUQ ), B, LDB, WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * * Solve the bidiagonal least squares problem. * OPS = ZERO T1 = SECOND( ) CALL SLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, \$ RCOND, RANK, WORK( NWORK ), IWORK, INFO ) T2 = SECOND( ) TIMNG( LALSD ) = TIMNG( LALSD ) + ( T2-T1 ) OPCNT( LALSD ) = OPCNT( LALSD ) + OPS IF( INFO.NE.0 ) THEN GO TO 10 END IF * * Multiply B by right bidiagonalizing vectors of L. * NB = ILAENV( 1, 'SORMBR', 'PLN', M, NRHS, M, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'PLN', M, NRHS, M, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK, \$ WORK( ITAUP ), B, LDB, WORK( NWORK ), \$ LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * * Zero out below first M rows of B. * CALL SLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB ) NWORK = ITAU + M * * Multiply transpose(Q) by B. * (Workspace: need M+NRHS, prefer M+NRHS*NB) * NB = ILAENV( 1, 'SORMLQ', 'LT', N, NRHS, M, -1 ) OPCNT( ORMLQ ) = OPCNT( ORMLQ ) + \$ SOPLA( 'SORMLQ', N, NRHS, M, 0, NB ) T1 = SECOND( ) CALL SORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B, \$ LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMLQ ) = TIMNG( ORMLQ ) + ( T2-T1 ) * ELSE * * Path 2 - remaining underdetermined cases. * IE = 1 ITAUQ = IE + M ITAUP = ITAUQ + M NWORK = ITAUP + M * * Bidiagonalize A. * (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB) * NB = ILAENV( 1, 'SGEBRD', ' ', M, N, -1, -1 ) OPCNT( GEBRD ) = OPCNT( GEBRD ) + \$ SOPLA( 'SGEBRD', M, N, 0, 0, NB ) T1 = SECOND( ) CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ), \$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1, \$ INFO ) T2 = SECOND( ) TIMNG( GEBRD ) = TIMNG( GEBRD ) + ( T2-T1 ) * * Multiply B by transpose of left bidiagonalizing vectors. * (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB) * NB = ILAENV( 1, 'SORMBR', 'QLT', M, NRHS, N, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'QLT', M, NRHS, N, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * * Solve the bidiagonal least squares problem. * OPS = ZERO T1 = SECOND( ) CALL SLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB, \$ RCOND, RANK, WORK( NWORK ), IWORK, INFO ) T2 = SECOND( ) TIMNG( LALSD ) = TIMNG( LALSD ) + ( T2-T1 ) OPCNT( LALSD ) = OPCNT( LALSD ) + OPS IF( INFO.NE.0 ) THEN GO TO 10 END IF * * Multiply B by right bidiagonalizing vectors of A. * NB = ILAENV( 1, 'SORMBR', 'PLN', N, NRHS, M, -1 ) OPCNT( ORMBR ) = OPCNT( ORMBR ) + \$ SOPLA2( 'SORMBR', 'PLN', N, NRHS, M, 0, NB ) T1 = SECOND( ) CALL SORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ), \$ B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO ) T2 = SECOND( ) TIMNG( ORMBR ) = TIMNG( ORMBR ) + ( T2-T1 ) * END IF * * Undo scaling. * IF( IASCL.EQ.1 ) THEN OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( N*NRHS+MINMN ) CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO ) CALL SLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN, \$ INFO ) ELSE IF( IASCL.EQ.2 ) THEN OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( N*NRHS+MINMN ) CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO ) CALL SLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN, \$ INFO ) END IF IF( IBSCL.EQ.1 ) THEN OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( N*NRHS ) CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO ) ELSE IF( IBSCL.EQ.2 ) THEN OPCNT( GELSD ) = OPCNT( GELSD ) + REAL( N*NRHS ) CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO ) END IF * 10 CONTINUE WORK( 1 ) = MAXWRK RETURN * * End of SGELSD * END