SUBROUTINE DGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT,
$ WORK, LWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOBU, JOBVT
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), S( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DGESVD computes the singular value decomposition (SVD) of a real
* M-by-N matrix A, optionally computing the left and/or right singular
* vectors. The SVD is written
*
* A = U * SIGMA * transpose(V)
*
* where SIGMA is an M-by-N matrix which is zero except for its
* min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and
* V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA
* are the singular values of A; they are real and non-negative, and
* are returned in descending order. The first min(m,n) columns of
* U and V are the left and right singular vectors of A.
*
* Note that the routine returns V**T, not V.
*
* Arguments
* =========
*
* JOBU (input) CHARACTER*1
* Specifies options for computing all or part of the matrix U:
* = 'A': all M columns of U are returned in array U:
* = 'S': the first min(m,n) columns of U (the left singular
* vectors) are returned in the array U;
* = 'O': the first min(m,n) columns of U (the left singular
* vectors) are overwritten on the array A;
* = 'N': no columns of U (no left singular vectors) are
* computed.
*
* JOBVT (input) CHARACTER*1
* Specifies options for computing all or part of the matrix
* V**T:
* = 'A': all N rows of V**T are returned in the array VT;
* = 'S': the first min(m,n) rows of V**T (the right singular
* vectors) are returned in the array VT;
* = 'O': the first min(m,n) rows of V**T (the right singular
* vectors) are overwritten on the array A;
* = 'N': no rows of V**T (no right singular vectors) are
* computed.
*
* JOBVT and JOBU cannot both be 'O'.
*
* M (input) INTEGER
* The number of rows of the input matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the input matrix A. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* On exit,
* if JOBU = 'O', A is overwritten with the first min(m,n)
* columns of U (the left singular vectors,
* stored columnwise);
* if JOBVT = 'O', A is overwritten with the first min(m,n)
* rows of V**T (the right singular vectors,
* stored rowwise);
* if JOBU .ne. 'O' and JOBVT .ne. 'O', the contents of A
* are destroyed.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* S (output) DOUBLE PRECISION array, dimension (min(M,N))
* The singular values of A, sorted so that S(i) >= S(i+1).
*
* U (output) DOUBLE PRECISION array, dimension (LDU,UCOL)
* (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.
* If JOBU = 'A', U contains the M-by-M orthogonal matrix U;
* if JOBU = 'S', U contains the first min(m,n) columns of U
* (the left singular vectors, stored columnwise);
* if JOBU = 'N' or 'O', U is not referenced.
*
* LDU (input) INTEGER
* The leading dimension of the array U. LDU >= 1; if
* JOBU = 'S' or 'A', LDU >= M.
*
* VT (output) DOUBLE PRECISION array, dimension (LDVT,N)
* If JOBVT = 'A', VT contains the N-by-N orthogonal matrix
* V**T;
* if JOBVT = 'S', VT contains the first min(m,n) rows of
* V**T (the right singular vectors, stored rowwise);
* if JOBVT = 'N' or 'O', VT is not referenced.
*
* LDVT (input) INTEGER
* The leading dimension of the array VT. LDVT >= 1; if
* JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
*
* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK;
* if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged
* superdiagonal elements of an upper bidiagonal matrix B
* whose diagonal is in S (not necessarily sorted). B
* satisfies A = U * B * VT, so it has the same singular values
* as A, and singular vectors related by U and VT.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)).
* 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.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
* > 0: if DBDSQR did not converge, INFO specifies how many
* superdiagonals of an intermediate bidiagonal form B
* did not converge to zero. See the description of WORK
* above for details.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, WNTUA, WNTUAS, WNTUN, WNTUO, WNTUS,
$ WNTVA, WNTVAS, WNTVN, WNTVO, WNTVS
INTEGER BDSPAC, BLK, CHUNK, I, IE, IERR, IR, ISCL,
$ ITAU, ITAUP, ITAUQ, IU, IWORK, LDWRKR, LDWRKU,
$ MAXWRK, MINMN, MINWRK, MNTHR, NCU, NCVT, NRU,
$ NRVT, WRKBL
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
DOUBLE PRECISION DUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSQR, DGEBRD, DGELQF, DGEMM, DGEQRF, DLACPY,
$ DLASCL, DLASET, DORGBR, DORGLQ, DORGQR, DORMBR,
$ XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, SQRT
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
INFO = 0
MINMN = MIN( M, N )
WNTUA = LSAME( JOBU, 'A' )
WNTUS = LSAME( JOBU, 'S' )
WNTUAS = WNTUA .OR. WNTUS
WNTUO = LSAME( JOBU, 'O' )
WNTUN = LSAME( JOBU, 'N' )
WNTVA = LSAME( JOBVT, 'A' )
WNTVS = LSAME( JOBVT, 'S' )
WNTVAS = WNTVA .OR. WNTVS
WNTVO = LSAME( JOBVT, 'O' )
WNTVN = LSAME( JOBVT, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.( WNTUA .OR. WNTUS .OR. WNTUO .OR. WNTUN ) ) THEN
INFO = -1
ELSE IF( .NOT.( WNTVA .OR. WNTVS .OR. WNTVO .OR. WNTVN ) .OR.
$ ( WNTVO .AND. WNTUO ) ) THEN
INFO = -2
ELSE IF( M.LT.0 ) THEN
INFO = -3
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDU.LT.1 .OR. ( WNTUAS .AND. LDU.LT.M ) ) THEN
INFO = -9
ELSE IF( LDVT.LT.1 .OR. ( WNTVA .AND. LDVT.LT.N ) .OR.
$ ( WNTVS .AND. LDVT.LT.MINMN ) ) THEN
INFO = -11
END IF
*
* 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.)
*
IF( INFO.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSQR
*
MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
BDSPAC = 5*N
IF( M.GE.MNTHR ) THEN
IF( WNTUN ) THEN
*
* Path 1 (M much larger than N, JOBU='N')
*
MAXWRK = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1,
$ -1 )
MAXWRK = MAX( MAXWRK, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
IF( WNTVO .OR. WNTVAS )
$ MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 4*N, BDSPAC )
ELSE IF( WNTUO .AND. WNTVN ) THEN
*
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
$ N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N )
MINWRK = MAX( 3*N+M, BDSPAC )
ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or
* 'A')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
$ N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+( N-1 )*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N )
MINWRK = MAX( 3*N+M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVN ) THEN
*
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
$ N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N+M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVO ) THEN
*
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
$ N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+( N-1 )*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*N*N + WRKBL
MINWRK = MAX( 3*N+M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVAS ) THEN
*
* Path 6 (M much larger than N, JOBU='S', JOBVT='S' or
* 'A')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
$ N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+( N-1 )*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N+M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVN ) THEN
*
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
$ M, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N+M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVO ) THEN
*
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
$ M, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+( N-1 )*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*N*N + WRKBL
MINWRK = MAX( 3*N+M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVAS ) THEN
*
* Path 9 (M much larger than N, JOBU='A', JOBVT='S' or
* 'A')
*
WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
$ M, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+2*N*
$ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, 3*N+( N-1 )*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
MINWRK = MAX( 3*N+M, BDSPAC )
END IF
ELSE
*
* Path 10 (M at least N, but not much larger)
*
MAXWRK = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N,
$ -1, -1 )
IF( WNTUS .OR. WNTUO )
$ MAXWRK = MAX( MAXWRK, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', M, N, N, -1 ) )
IF( WNTUA )
$ MAXWRK = MAX( MAXWRK, 3*N+M*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, N, -1 ) )
IF( .NOT.WNTVN )
$ MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 3*N+M, BDSPAC )
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSQR
*
MNTHR = ILAENV( 6, 'DGESVD', JOBU // JOBVT, M, N, 0, 0 )
BDSPAC = 5*M
IF( N.GE.MNTHR ) THEN
IF( WNTVN ) THEN
*
* Path 1t(N much larger than M, JOBVT='N')
*
MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
$ -1 )
MAXWRK = MAX( MAXWRK, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
IF( WNTUO .OR. WNTUAS )
$ MAXWRK = MAX( MAXWRK, 3*M+M*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 4*M, BDSPAC )
ELSE IF( WNTVO .AND. WNTUN ) THEN
*
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M )
MINWRK = MAX( 3*M+N, BDSPAC )
ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
* Path 3t(N much larger than M, JOBU='S' or 'A',
* JOBVT='O')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+M*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M )
MINWRK = MAX( 3*M+N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUN ) THEN
*
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M+N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUO ) THEN
*
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+M*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*M*M + WRKBL
MINWRK = MAX( 3*M+N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUAS ) THEN
*
* Path 6t(N much larger than M, JOBU='S' or 'A',
* JOBVT='S')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+M*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M+N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUN ) THEN
*
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M+N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUO ) THEN
*
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+M*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*M*M + WRKBL
MINWRK = MAX( 3*M+N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUAS ) THEN
*
* Path 9t(N much larger than M, JOBU='S' or 'A',
* JOBVT='A')
*
WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
$ N, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+2*M*
$ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
WRKBL = MAX( WRKBL, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'P', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, 3*M+M*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
MINWRK = MAX( 3*M+N, BDSPAC )
END IF
ELSE
*
* Path 10t(N greater than M, but not much larger)
*
MAXWRK = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N,
$ -1, -1 )
IF( WNTVS .OR. WNTVO )
$ MAXWRK = MAX( MAXWRK, 3*M+M*
$ ILAENV( 1, 'DORGBR', 'P', M, N, M, -1 ) )
IF( WNTVA )
$ MAXWRK = MAX( MAXWRK, 3*M+N*
$ ILAENV( 1, 'DORGBR', 'P', N, N, M, -1 ) )
IF( .NOT.WNTUN )
$ MAXWRK = MAX( MAXWRK, 3*M+( M-1 )*
$ ILAENV( 1, 'DORGBR', 'Q', M, M, M, -1 ) )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 3*M+N, BDSPAC )
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
WORK( 1 ) = MAXWRK
*
IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGESVD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', M, N, A, LDA, DUM )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, IERR )
ELSE IF( ANRM.GT.BIGNUM ) THEN
ISCL = 1
CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, IERR )
END IF
*
IF( M.GE.N ) THEN
*
* A has at least as many rows as columns. If A has sufficiently
* more rows than columns, first reduce using the QR
* decomposition (if sufficient workspace available)
*
IF( M.GE.MNTHR ) THEN
*
IF( WNTUN ) THEN
*
* Path 1 (M much larger than N, JOBU='N')
* No left singular vectors to be computed
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Zero out below R
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
NCVT = 0
IF( WNTVO .OR. WNTVAS ) THEN
*
* If right singular vectors desired, generate P'.
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
NCVT = N
END IF
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in A if desired
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, 0, 0, S, WORK( IE ), A, LDA,
$ DUM, 1, DUM, 1, WORK( IWORK ), INFO )
*
* If right singular vectors desired in VT, copy them there
*
IF( WNTVAS )
$ CALL DLACPY( 'F', N, N, A, LDA, VT, LDVT )
*
ELSE IF( WNTUO .AND. WNTVN ) THEN
*
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
* N left singular vectors to be overwritten on A and
* no right singular vectors to be computed
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+LDA*N ) THEN
*
* WORK(IU) is LDA by N, WORK(IR) is LDA by N
*
LDWRKU = LDA
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+N*N ) THEN
*
* WORK(IU) is LDA by N, WORK(IR) is N by N
*
LDWRKU = LDA
LDWRKR = N
ELSE
*
* WORK(IU) is LDWRKU by N, WORK(IR) is N by N
*
LDWRKU = ( LWORK-N*N-N ) / N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IR) and zero out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
$ LDWRKR )
*
* Generate Q in A
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing R
* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
* (Workspace: need N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM, 1,
$ WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + N
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in WORK(IU) and copying to A
* (Workspace: need N*N+2*N, prefer N*N+M*N+N)
*
DO 10 I = 1, M, LDWRKU
CHUNK = MIN( M-I+1, LDWRKU )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IR ), LDWRKR, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
$ A( I, 1 ), LDA )
10 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize A
* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing A
* (Workspace: need 4*N, prefer 3*N+N*NB)
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM, 1,
$ A, LDA, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or 'A')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+LDA*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
LDWRKU = LDA
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+N*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
LDWRKU = LDA
LDWRKR = N
ELSE
*
* WORK(IU) is LDWRKU by N and WORK(IR) is N by N
*
LDWRKU = ( LWORK-N*N-N ) / N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
*
* Generate Q in A
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT, copying result to WORK(IR)
* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', N, N, VT, LDVT, WORK( IR ), LDWRKR )
*
* Generate left vectors bidiagonalizing R in WORK(IR)
* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in VT
* (Workspace: need N*N+4*N-1, prefer N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR) and computing right
* singular vectors of R in VT
* (Workspace: need N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT, LDVT,
$ WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + N
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in WORK(IU) and copying to A
* (Workspace: need N*N+2*N, prefer N*N+M*N+N)
*
DO 20 I = 1, M, LDWRKU
CHUNK = MIN( M-I+1, LDWRKU )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IR ), LDWRKR, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', CHUNK, N, WORK( IU ), LDWRKU,
$ A( I, 1 ), LDA )
20 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
*
* Generate Q in A
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in A by left vectors bidiagonalizing R
* (Workspace: need 3*N+M, prefer 3*N+M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), A, LDA, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in VT
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in A and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT, LDVT,
$ A, LDA, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTUS ) THEN
*
IF( WNTVN ) THEN
*
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
* N left singular vectors to be computed in U and
* no right singular vectors to be computed
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IR) is LDA by N
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is N by N
*
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IR+1 ), LDWRKR )
*
* Generate Q in A
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing R in WORK(IR)
* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
* (Workspace: need N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
$ 1, WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
$ WORK( IR ), LDWRKR, ZERO, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left vectors bidiagonalizing R
* (Workspace: need 3*N+M, prefer 3*N+M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM,
$ 1, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVO ) THEN
*
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
* N left singular vectors to be computed in U and
* N right singular vectors to be overwritten on A
*
IF( LWORK.GE.2*N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = N
ELSE
*
* WORK(IU) is N by N and WORK(IR) is N by N
*
LDWRKU = N
IR = IU + LDWRKU*N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
*
* Generate Q in A
* (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*N*N+4*N,
* prefer 2*N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*N*N+4*N, prefer 2*N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*N*N+4*N-1,
* prefer 2*N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in WORK(IR)
* (Workspace: need 2*N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, WORK( IU ),
$ LDWRKU, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in U
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
$ WORK( IU ), LDWRKU, ZERO, U, LDU )
*
* Copy right singular vectors of R to A
* (Workspace: need N*N)
*
CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left vectors bidiagonalizing R
* (Workspace: need 3*N+M, prefer 3*N+M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in A
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
$ LDA, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVAS ) THEN
*
* Path 6 (M much larger than N, JOBU='S', JOBVT='S'
* or 'A')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IF( LWORK.GE.N*N+MAX( 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IU) is LDA by N
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is N by N
*
LDWRKU = N
END IF
ITAU = IU + LDWRKU*N
IWORK = ITAU + N
*
* Compute A=Q*R
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
*
* Generate Q in A
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to VT
* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
$ LDVT )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need N*N+4*N-1,
* prefer N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in VT
* (Workspace: need N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in U
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA,
$ WORK( IU ), LDWRKU, ZERO, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in VT
* (Workspace: need 3*N+M, prefer 3*N+M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
ELSE IF( WNTUA ) THEN
*
IF( WNTVN ) THEN
*
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
* M left singular vectors to be computed in U and
* no right singular vectors to be computed
*
IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IR) is LDA by N
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is N by N
*
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IR+1 ), LDWRKR )
*
* Generate Q in U
* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
* (Workspace: need N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
$ 1, WORK( IR ), LDWRKR, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IR), storing result in A
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
$ WORK( IR ), LDWRKR, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N+M, prefer N+M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in A
* (Workspace: need 3*N+M, prefer 3*N+M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, 0, M, 0, S, WORK( IE ), DUM,
$ 1, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVO ) THEN
*
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
* M left singular vectors to be computed in U and
* N right singular vectors to be overwritten on A
*
IF( LWORK.GE.2*N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = N
ELSE
*
* WORK(IU) is N by N and WORK(IR) is N by N
*
LDWRKU = N
IR = IU + LDWRKU*N
LDWRKR = N
END IF
ITAU = IR + LDWRKR*N
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need 2*N*N+N+M, prefer 2*N*N+N+M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*N*N+4*N,
* prefer 2*N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*N*N+4*N, prefer 2*N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*N*N+4*N-1,
* prefer 2*N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in WORK(IR)
* (Workspace: need 2*N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, WORK( IU ),
$ LDWRKU, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
$ WORK( IU ), LDWRKU, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
* Copy right singular vectors of R from WORK(IR) to A
*
CALL DLACPY( 'F', N, N, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N+M, prefer N+M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Zero out below R in A
*
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ),
$ LDA )
*
* Bidiagonalize R in A
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in A
* (Workspace: need 3*N+M, prefer 3*N+M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in A
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), A,
$ LDA, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTVAS ) THEN
*
* Path 9 (M much larger than N, JOBU='A', JOBVT='S'
* or 'A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
IF( LWORK.GE.N*N+MAX( N+M, 4*N, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*N ) THEN
*
* WORK(IU) is LDA by N
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is N by N
*
LDWRKU = N
END IF
ITAU = IU + LDWRKU*N
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R to WORK(IU), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ WORK( IU+1 ), LDWRKU )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to VT
* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', N, N, WORK( IU ), LDWRKU, VT,
$ LDVT )
*
* Generate left bidiagonalizing vectors in WORK(IU)
* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need N*N+4*N-1,
* prefer N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in VT
* (Workspace: need N*N+BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
* (Workspace: need N*N)
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU,
$ WORK( IU ), LDWRKU, ZERO, A, LDA )
*
* Copy left singular vectors of A from A to U
*
CALL DLACPY( 'F', M, N, A, LDA, U, LDU )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
* (Workspace: need 2*N, prefer N+N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
* (Workspace: need N+M, prefer N+M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy R from A to VT, zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
IF( N.GT.1 )
$ CALL DLASET( 'L', N-1, N-1, ZERO, ZERO,
$ VT( 2, 1 ), LDVT )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
* (Workspace: need 4*N, prefer 3*N+2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left bidiagonalizing vectors
* in VT
* (Workspace: need 3*N+M, prefer 3*N+M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + N
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
END IF
*
ELSE
*
* M .LT. MNTHR
*
* Path 10 (M at least N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
*
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
IWORK = ITAUP + N
*
* Bidiagonalize A
* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
IF( WNTUAS ) THEN
*
* If left singular vectors desired in U, copy result to U
* and generate left bidiagonalizing vectors in U
* (Workspace: need 3*N+NCU, prefer 3*N+NCU*NB)
*
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
IF( WNTUS )
$ NCU = N
IF( WNTUA )
$ NCU = M
CALL DORGBR( 'Q', M, NCU, N, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVAS ) THEN
*
* If right singular vectors desired in VT, copy result to
* VT and generate right bidiagonalizing vectors in VT
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTUO ) THEN
*
* If left singular vectors desired in A, generate left
* bidiagonalizing vectors in A
* (Workspace: need 4*N, prefer 3*N+N*NB)
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVO ) THEN
*
* If right singular vectors desired in A, generate right
* bidiagonalizing vectors in A
* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IWORK = IE + N
IF( WNTUAS .OR. WNTUO )
$ NRU = M
IF( WNTUN )
$ NRU = 0
IF( WNTVAS .OR. WNTVO )
$ NCVT = N
IF( WNTVN )
$ NCVT = 0
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), A, LDA,
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in A and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', N, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
END IF
*
END IF
*
ELSE
*
* A has more columns than rows. If A has sufficiently more
* columns than rows, first reduce using the LQ decomposition (if
* sufficient workspace available)
*
IF( N.GE.MNTHR ) THEN
*
IF( WNTVN ) THEN
*
* Path 1t(N much larger than M, JOBVT='N')
* No right singular vectors to be computed
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Zero out above L
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ), LDA )
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
IF( WNTUO .OR. WNTUAS ) THEN
*
* If left singular vectors desired, generate Q
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IWORK = IE + M
NRU = 0
IF( WNTUO .OR. WNTUAS )
$ NRU = M
*
* Perform bidiagonal QR iteration, computing left singular
* vectors of A in A if desired
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, 0, NRU, 0, S, WORK( IE ), DUM, 1, A,
$ LDA, DUM, 1, WORK( IWORK ), INFO )
*
* If left singular vectors desired in U, copy them there
*
IF( WNTUAS )
$ CALL DLACPY( 'F', M, M, A, LDA, U, LDU )
*
ELSE IF( WNTVO .AND. WNTUN ) THEN
*
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
* M right singular vectors to be overwritten on A and
* no left singular vectors to be computed
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+LDA*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+M*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is M by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = M
ELSE
*
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
*
LDWRKU = M
CHUNK = ( LWORK-M*M-M ) / M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IR) and zero out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ), LDWRKR )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in A
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing L
* (Workspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + M
*
* Multiply right singular vectors of L in WORK(IR) by Q
* in A, storing result in WORK(IU) and copying to A
* (Workspace: need M*M+2*M, prefer M*M+M*N+M)
*
DO 30 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IR ),
$ LDWRKR, A( 1, I ), LDA, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
$ A( 1, I ), LDA )
30 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize A
* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing A
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, N, 0, 0, S, WORK( IE ), A, LDA,
$ DUM, 1, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
* Path 3t(N much larger than M, JOBU='S' or 'A', JOBVT='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+LDA*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = LDA
ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+M*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is M by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = M
ELSE
*
* WORK(IU) is M by CHUNK and WORK(IR) is M by M
*
LDWRKU = M
CHUNK = ( LWORK-M*M-M ) / M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing about above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
$ LDU )
*
* Generate Q in A
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U, copying result to WORK(IR)
* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, M, U, LDU, WORK( IR ), LDWRKR )
*
* Generate right vectors bidiagonalizing L in WORK(IR)
* (Workspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing L in U
* (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U, and computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, U, LDU, DUM, 1,
$ WORK( IWORK ), INFO )
IU = IE + M
*
* Multiply right singular vectors of L in WORK(IR) by Q
* in A, storing result in WORK(IU) and copying to A
* (Workspace: need M*M+2*M, prefer M*M+M*N+M))
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IR ),
$ LDWRKR, A( 1, I ), LDA, ZERO,
$ WORK( IU ), LDWRKU )
CALL DLACPY( 'F', M, BLK, WORK( IU ), LDWRKU,
$ A( 1, I ), LDA )
40 CONTINUE
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
$ LDU )
*
* Generate Q in A
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in A
* (Workspace: need 3*M+N, prefer 3*M+N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), A, LDA, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing L in U
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), A, LDA,
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
*
END IF
*
ELSE IF( WNTVS ) THEN
*
IF( WNTUN ) THEN
*
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
* M right singular vectors to be computed in VT and
* no left singular vectors to be computed
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IR) is LDA by M
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is M by M
*
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IR), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in A
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing L in
* WORK(IR)
* (Workspace: need M*M+4*M, prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IR) by
* Q in A, storing result in VT
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
$ LDWRKR, A, LDA, ZERO, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy result to VT
*
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in VT
* (Workspace: need 3*M+N, prefer 3*M+N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
$ LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUO ) THEN
*
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be overwritten on A
*
IF( LWORK.GE.2*M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is M by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = M
ELSE
*
* WORK(IU) is M by M and WORK(IR) is M by M
*
LDWRKU = M
IR = IU + LDWRKU*M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out below it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
*
* Generate Q in A
* (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*M*M+4*M,
* prefer 2*M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*M*M+4*M-1,
* prefer 2*M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*M*M+4*M, prefer 2*M*M+3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in WORK(IR) and computing
* right singular vectors of L in WORK(IU)
* (Workspace: need 2*M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, WORK( IR ),
$ LDWRKR, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in A, storing result in VT
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, A, LDA, ZERO, VT, LDVT )
*
* Copy left singular vectors of L to A
* (Workspace: need M*M)
*
CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in VT
* (Workspace: need 3*M+N, prefer 3*M+N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors of L in A
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, compute left
* singular vectors of A in A and compute right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUAS ) THEN
*
* Path 6t(N much larger than M, JOBU='S' or 'A',
* JOBVT='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IF( LWORK.GE.M*M+MAX( 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IU) is LDA by N
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is LDA by M
*
LDWRKU = M
END IF
ITAU = IU + LDWRKU*M
IWORK = ITAU + M
*
* Compute A=L*Q
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
*
* Generate Q in A
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to U
* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
$ LDU )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need M*M+4*M-1,
* prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U and computing right
* singular vectors of L in WORK(IU)
* (Workspace: need M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in A, storing result in VT
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, A, LDA, ZERO, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
$ LDU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in U by Q
* in VT
* (Workspace: need 3*M+N, prefer 3*M+N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
ELSE IF( WNTVA ) THEN
*
IF( WNTUN ) THEN
*
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
* N right singular vectors to be computed in VT and
* no left singular vectors to be computed
*
IF( LWORK.GE.M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IR = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IR) is LDA by M
*
LDWRKR = LDA
ELSE
*
* WORK(IR) is M by M
*
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Copy L to WORK(IR), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IR ),
$ LDWRKR )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in VT
* (Workspace: need M*M+M+N, prefer M*M+M+N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
* (Workspace: need M*M+4*M-1,
* prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
* (Workspace: need M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IR) by
* Q in VT, storing result in A
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IR ),
$ LDWRKR, VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M+N, prefer M+N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in A by Q
* in VT
* (Workspace: need 3*M+N, prefer 3*M+N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, 0, 0, S, WORK( IE ), VT,
$ LDVT, DUM, 1, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUO ) THEN
*
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be overwritten on A
*
IF( LWORK.GE.2*M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+2*LDA*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is LDA by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = LDA
ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is M by M
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = M
ELSE
*
* WORK(IU) is M by M and WORK(IR) is M by M
*
LDWRKU = M
IR = IU + LDWRKU*M
LDWRKR = M
END IF
ITAU = IR + LDWRKR*M
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need 2*M*M+M+N, prefer 2*M*M+M+N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to
* WORK(IR)
* (Workspace: need 2*M*M+4*M,
* prefer 2*M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU,
$ WORK( IR ), LDWRKR )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need 2*M*M+4*M-1,
* prefer 2*M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
* (Workspace: need 2*M*M+4*M, prefer 2*M*M+3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in WORK(IR) and computing
* right singular vectors of L in WORK(IU)
* (Workspace: need 2*M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, WORK( IR ),
$ LDWRKR, DUM, 1, WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in VT, storing result in A
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
* Copy left singular vectors of A from WORK(IR) to A
*
CALL DLACPY( 'F', M, M, WORK( IR ), LDWRKR, A,
$ LDA )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M+N, prefer M+N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Zero out above L in A
*
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, A( 1, 2 ),
$ LDA )
*
* Bidiagonalize L in A
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in A by Q
* in VT
* (Workspace: need 3*M+N, prefer 3*M+N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in A
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in A and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
ELSE IF( WNTUAS ) THEN
*
* Path 9t(N much larger than M, JOBU='S' or 'A',
* JOBVT='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
IF( LWORK.GE.M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IU = 1
IF( LWORK.GE.WRKBL+LDA*M ) THEN
*
* WORK(IU) is LDA by M
*
LDWRKU = LDA
ELSE
*
* WORK(IU) is M by M
*
LDWRKU = M
END IF
ITAU = IU + LDWRKU*M
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M*M+M+N, prefer M*M+M+N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to WORK(IU), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IU ),
$ LDWRKU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
$ WORK( IU+LDWRKU ), LDWRKU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to U
* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, M, WORK( IU ), LDWRKU, U,
$ LDU )
*
* Generate right bidiagonalizing vectors in WORK(IU)
* (Workspace: need M*M+4*M, prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U and computing right
* singular vectors of L in WORK(IU)
* (Workspace: need M*M+BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
$ WORK( IWORK ), INFO )
*
* Multiply right singular vectors of L in WORK(IU) by
* Q in VT, storing result in A
* (Workspace: need M*M)
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IU ),
$ LDWRKU, VT, LDVT, ZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
CALL DLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
ELSE
*
* Insufficient workspace for a fast algorithm
*
ITAU = 1
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
* (Workspace: need 2*M, prefer M+M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
* (Workspace: need M+N, prefer M+N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Copy L to U, zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, U( 1, 2 ),
$ LDU )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize L in U
* (Workspace: need 4*M, prefer 3*M+2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right bidiagonalizing vectors in U by Q
* in VT
* (Workspace: need 3*M+N, prefer 3*M+N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = IE + M
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of A in U and computing right
* singular vectors of A in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'U', M, N, M, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ),
$ INFO )
*
END IF
*
END IF
*
END IF
*
ELSE
*
* N .LT. MNTHR
*
* Path 10t(N greater than M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
*
IE = 1
ITAUQ = IE + M
ITAUP = ITAUQ + M
IWORK = ITAUP + M
*
* Bidiagonalize A
* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
$ IERR )
IF( WNTUAS ) THEN
*
* If left singular vectors desired in U, copy result to U
* and generate left bidiagonalizing vectors in U
* (Workspace: need 4*M-1, prefer 3*M+(M-1)*NB)
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DORGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVAS ) THEN
*
* If right singular vectors desired in VT, copy result to
* VT and generate right bidiagonalizing vectors in VT
* (Workspace: need 3*M+NRVT, prefer 3*M+NRVT*NB)
*
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
IF( WNTVA )
$ NRVT = N
IF( WNTVS )
$ NRVT = M
CALL DORGBR( 'P', NRVT, N, M, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTUO ) THEN
*
* If left singular vectors desired in A, generate left
* bidiagonalizing vectors in A
* (Workspace: need 4*M-1, prefer 3*M+(M-1)*NB)
*
CALL DORGBR( 'Q', M, M, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IF( WNTVO ) THEN
*
* If right singular vectors desired in A, generate right
* bidiagonalizing vectors in A
* (Workspace: need 4*M, prefer 3*M+M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
IWORK = IE + M
IF( WNTUAS .OR. WNTUO )
$ NRU = M
IF( WNTUN )
$ NRU = 0
IF( WNTVAS .OR. WNTVO )
$ NCVT = N
IF( WNTVN )
$ NCVT = 0
IF( ( .NOT.WNTUO ) .AND. ( .NOT.WNTVO ) ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE IF( ( .NOT.WNTUO ) .AND. WNTVO ) THEN
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in U and computing right singular
* vectors in A
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), A, LDA,
$ U, LDU, DUM, 1, WORK( IWORK ), INFO )
ELSE
*
* Perform bidiagonal QR iteration, if desired, computing
* left singular vectors in A and computing right singular
* vectors in VT
* (Workspace: need BDSPAC)
*
CALL DBDSQR( 'L', M, NCVT, NRU, 0, S, WORK( IE ), VT,
$ LDVT, A, LDA, DUM, 1, WORK( IWORK ), INFO )
END IF
*
END IF
*
END IF
*
* If DBDSQR failed to converge, copy unconverged superdiagonals
* to WORK( 2:MINMN )
*
IF( INFO.NE.0 ) THEN
IF( IE.GT.2 ) THEN
DO 50 I = 1, MINMN - 1
WORK( I+1 ) = WORK( I+IE-1 )
50 CONTINUE
END IF
IF( IE.LT.2 ) THEN
DO 60 I = MINMN - 1, 1, -1
WORK( I+1 ) = WORK( I+IE-1 )
60 CONTINUE
END IF
END IF
*
* Undo scaling if necessary
*
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
IF( INFO.NE.0 .AND. ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN-1, 1, WORK( 2 ),
$ MINMN, IERR )
IF( ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
$ IERR )
IF( INFO.NE.0 .AND. ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN-1, 1, WORK( 2 ),
$ MINMN, IERR )
END IF
*
* Return optimal workspace in WORK(1)
*
WORK( 1 ) = MAXWRK
*
RETURN
*
* End of DGESVD
*
END