--- rpl/lapack/lapack/zgesdd.f 2010/08/13 21:04:03 1.7
+++ rpl/lapack/lapack/zgesdd.f 2017/06/17 11:06:43 1.18
@@ -1,11 +1,236 @@
- SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
- $ LWORK, RWORK, IWORK, INFO )
+*> \brief \b ZGESDD
*
-* -- LAPACK driver routine (version 3.2.2) --
+* =========== DOCUMENTATION ===========
+*
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
+*
+*> \htmlonly
+*> Download ZGESDD + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+* WORK, LWORK, RWORK, IWORK, INFO )
+*
+* .. Scalar Arguments ..
+* CHARACTER JOBZ
+* INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
+* ..
+* .. Array Arguments ..
+* INTEGER IWORK( * )
+* DOUBLE PRECISION RWORK( * ), S( * )
+* COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ),
+* $ WORK( * )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> ZGESDD computes the singular value decomposition (SVD) of a complex
+*> M-by-N matrix A, optionally computing the left and/or right singular
+*> vectors, by using divide-and-conquer method. The SVD is written
+*>
+*> A = U * SIGMA * conjugate-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 unitary matrix, and
+*> V is an N-by-N unitary 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 VT = V**H, not V.
+*>
+*> 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.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] JOBZ
+*> \verbatim
+*> JOBZ is CHARACTER*1
+*> Specifies options for computing all or part of the matrix U:
+*> = 'A': all M columns of U and all N rows of V**H are
+*> returned in the arrays U and VT;
+*> = 'S': the first min(M,N) columns of U and the first
+*> min(M,N) rows of V**H are returned in the arrays U
+*> and VT;
+*> = 'O': If M >= N, the first N columns of U are overwritten
+*> in the array A and all rows of V**H are returned in
+*> the array VT;
+*> otherwise, all columns of U are returned in the
+*> array U and the first M rows of V**H are overwritten
+*> in the array A;
+*> = 'N': no columns of U or rows of V**H are computed.
+*> \endverbatim
+*>
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> The number of rows of the input matrix A. M >= 0.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The number of columns of the input matrix A. N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is COMPLEX*16 array, dimension (LDA,N)
+*> On entry, the M-by-N matrix A.
+*> On exit,
+*> if JOBZ = 'O', A is overwritten with the first N columns
+*> of U (the left singular vectors, stored
+*> columnwise) if M >= N;
+*> A is overwritten with the first M rows
+*> of V**H (the right singular vectors, stored
+*> rowwise) otherwise.
+*> if JOBZ .ne. 'O', the contents of A are destroyed.
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,M).
+*> \endverbatim
+*>
+*> \param[out] S
+*> \verbatim
+*> S is DOUBLE PRECISION array, dimension (min(M,N))
+*> The singular values of A, sorted so that S(i) >= S(i+1).
+*> \endverbatim
+*>
+*> \param[out] U
+*> \verbatim
+*> U is COMPLEX*16 array, dimension (LDU,UCOL)
+*> UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
+*> UCOL = min(M,N) if JOBZ = 'S'.
+*> If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
+*> unitary matrix U;
+*> if JOBZ = 'S', U contains the first min(M,N) columns of U
+*> (the left singular vectors, stored columnwise);
+*> if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
+*> \endverbatim
+*>
+*> \param[in] LDU
+*> \verbatim
+*> LDU is INTEGER
+*> The leading dimension of the array U. LDU >= 1;
+*> if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
+*> \endverbatim
+*>
+*> \param[out] VT
+*> \verbatim
+*> VT is COMPLEX*16 array, dimension (LDVT,N)
+*> If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
+*> N-by-N unitary matrix V**H;
+*> if JOBZ = 'S', VT contains the first min(M,N) rows of
+*> V**H (the right singular vectors, stored rowwise);
+*> if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
+*> \endverbatim
+*>
+*> \param[in] LDVT
+*> \verbatim
+*> LDVT is INTEGER
+*> The leading dimension of the array VT. LDVT >= 1;
+*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
+*> if JOBZ = 'S', LDVT >= min(M,N).
+*> \endverbatim
+*>
+*> \param[out] WORK
+*> \verbatim
+*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
+*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is INTEGER
+*> The dimension of the array WORK. LWORK >= 1.
+*> If LWORK = -1, a workspace query is assumed. The optimal
+*> size for the WORK array is calculated and stored in WORK(1),
+*> and no other work except argument checking is performed.
+*>
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LWORK >= 2*mn + mx.
+*> If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
+*> If JOBZ = 'S', LWORK >= mn*mn + 3*mn.
+*> If JOBZ = 'A', LWORK >= mn*mn + 2*mn + mx.
+*> These are not tight minimums in all cases; see comments inside code.
+*> For good performance, LWORK should generally be larger;
+*> a query is recommended.
+*> \endverbatim
+*>
+*> \param[out] RWORK
+*> \verbatim
+*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
+*> else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
+*> else LRWORK >= max( 5*mn*mn + 5*mn,
+*> 2*mx*mn + 2*mn*mn + mn ).
+*> \endverbatim
+*>
+*> \param[out] IWORK
+*> \verbatim
+*> IWORK is INTEGER array, dimension (8*min(M,N))
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0: successful exit.
+*> < 0: if INFO = -i, the i-th argument had an illegal value.
+*> > 0: The updating process of DBDSDC did not converge.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date June 2016
+*
+*> \ingroup complex16GEsing
+*
+*> \par Contributors:
+* ==================
+*>
+*> Ming Gu and Huan Ren, Computer Science Division, University of
+*> California at Berkeley, USA
+*>
+* =====================================================================
+ SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+ $ WORK, LWORK, RWORK, IWORK, INFO )
+ implicit none
+*
+* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
-* June 2010
-* 8-15-00: Improve consistency of WS calculations (eca)
+* June 2016
*
* .. Scalar Arguments ..
CHARACTER JOBZ
@@ -18,137 +243,9 @@
$ WORK( * )
* ..
*
-* Purpose
-* =======
-*
-* ZGESDD computes the singular value decomposition (SVD) of a complex
-* M-by-N matrix A, optionally computing the left and/or right singular
-* vectors, by using divide-and-conquer method. The SVD is written
-*
-* A = U * SIGMA * conjugate-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 unitary matrix, and
-* V is an N-by-N unitary 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 VT = V**H, not V.
-*
-* 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
-* =========
-*
-* JOBZ (input) CHARACTER*1
-* Specifies options for computing all or part of the matrix U:
-* = 'A': all M columns of U and all N rows of V**H are
-* returned in the arrays U and VT;
-* = 'S': the first min(M,N) columns of U and the first
-* min(M,N) rows of V**H are returned in the arrays U
-* and VT;
-* = 'O': If M >= N, the first N columns of U are overwritten
-* in the array A and all rows of V**H are returned in
-* the array VT;
-* otherwise, all columns of U are returned in the
-* array U and the first M rows of V**H are overwritten
-* in the array A;
-* = 'N': no columns of U or rows of V**H are computed.
-*
-* 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) COMPLEX*16 array, dimension (LDA,N)
-* On entry, the M-by-N matrix A.
-* On exit,
-* if JOBZ = 'O', A is overwritten with the first N columns
-* of U (the left singular vectors, stored
-* columnwise) if M >= N;
-* A is overwritten with the first M rows
-* of V**H (the right singular vectors, stored
-* rowwise) otherwise.
-* if JOBZ .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) COMPLEX*16 array, dimension (LDU,UCOL)
-* UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;
-* UCOL = min(M,N) if JOBZ = 'S'.
-* If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M
-* unitary matrix U;
-* if JOBZ = 'S', U contains the first min(M,N) columns of U
-* (the left singular vectors, stored columnwise);
-* if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.
-*
-* LDU (input) INTEGER
-* The leading dimension of the array U. LDU >= 1; if
-* JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
-*
-* VT (output) COMPLEX*16 array, dimension (LDVT,N)
-* If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the
-* N-by-N unitary matrix V**H;
-* if JOBZ = 'S', VT contains the first min(M,N) rows of
-* V**H (the right singular vectors, stored rowwise);
-* if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.
-*
-* LDVT (input) INTEGER
-* The leading dimension of the array VT. LDVT >= 1; if
-* JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
-* if JOBZ = 'S', LDVT >= min(M,N).
-*
-* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
-* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
-*
-* LWORK (input) INTEGER
-* The dimension of the array WORK. LWORK >= 1.
-* if JOBZ = 'N', LWORK >= 2*min(M,N)+max(M,N).
-* if JOBZ = 'O',
-* LWORK >= 2*min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
-* if JOBZ = 'S' or 'A',
-* LWORK >= min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
-* For good performance, LWORK should generally be larger.
-*
-* If LWORK = -1, a workspace query is assumed. The optimal
-* size for the WORK array is calculated and stored in WORK(1),
-* and no other work except argument checking is performed.
-*
-* RWORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
-* If JOBZ = 'N', LRWORK >= 5*min(M,N).
-* Otherwise,
-* LRWORK >= min(M,N)*max(5*min(M,N)+7,2*max(M,N)+2*min(M,N)+1)
-*
-* IWORK (workspace) INTEGER array, dimension (8*min(M,N))
-*
-* INFO (output) INTEGER
-* = 0: successful exit.
-* < 0: if INFO = -i, the i-th argument had an illegal value.
-* > 0: The updating process of DBDSDC did not converge.
-*
-* Further Details
-* ===============
-*
-* Based on contributions by
-* Ming Gu and Huan Ren, Computer Science Division, University of
-* California at Berkeley, USA
-*
* =====================================================================
*
* .. Parameters ..
- INTEGER LQUERV
- PARAMETER ( LQUERV = -1 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
@@ -156,16 +253,27 @@
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
- LOGICAL WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
+ LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
INTEGER BLK, CHUNK, I, IE, IERR, IL, IR, IRU, IRVT,
$ ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR1, MNTHR2, NRWORK, NWORK, WRKBL
+ INTEGER LWORK_ZGEBRD_MN, LWORK_ZGEBRD_MM,
+ $ LWORK_ZGEBRD_NN, LWORK_ZGELQF_MN,
+ $ LWORK_ZGEQRF_MN,
+ $ LWORK_ZUNGBR_P_MN, LWORK_ZUNGBR_P_NN,
+ $ LWORK_ZUNGBR_Q_MN, LWORK_ZUNGBR_Q_MM,
+ $ LWORK_ZUNGLQ_MN, LWORK_ZUNGLQ_NN,
+ $ LWORK_ZUNGQR_MM, LWORK_ZUNGQR_MN,
+ $ LWORK_ZUNMBR_PRC_MM, LWORK_ZUNMBR_QLN_MM,
+ $ LWORK_ZUNMBR_PRC_MN, LWORK_ZUNMBR_QLN_MN,
+ $ LWORK_ZUNMBR_PRC_NN, LWORK_ZUNMBR_QLN_NN
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DUM( 1 )
+ COMPLEX*16 CDUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSDC, DLASCL, XERBLA, ZGEBRD, ZGELQF, ZGEMM,
@@ -174,9 +282,8 @@
* ..
* .. External Functions ..
LOGICAL LSAME
- INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANGE
- EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
+ EXTERNAL LSAME, DLAMCH, ZLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
@@ -185,15 +292,16 @@
*
* Test the input arguments
*
- INFO = 0
- MINMN = MIN( M, N )
+ INFO = 0
+ MINMN = MIN( M, N )
MNTHR1 = INT( MINMN*17.0D0 / 9.0D0 )
MNTHR2 = INT( MINMN*5.0D0 / 3.0D0 )
- WNTQA = LSAME( JOBZ, 'A' )
- WNTQS = LSAME( JOBZ, 'S' )
+ WNTQA = LSAME( JOBZ, 'A' )
+ WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
- WNTQO = LSAME( JOBZ, 'O' )
- WNTQN = LSAME( JOBZ, 'N' )
+ WNTQO = LSAME( JOBZ, 'O' )
+ WNTQN = LSAME( JOBZ, 'N' )
+ LQUERY = ( LWORK.EQ.-1 )
MINWRK = 1
MAXWRK = 1
*
@@ -215,244 +323,296 @@
END IF
*
* Compute workspace
-* (Note: Comments in the code beginning "Workspace:" describe the
-* minimal amount of workspace needed at that point in the code,
+* Note: Comments in the code beginning "Workspace:" describe the
+* minimal amount of workspace allocated at that point in the code,
* as well as the preferred amount for good performance.
* CWorkspace refers to complex workspace, and RWorkspace to
* real workspace. NB refers to the optimal block size for the
* immediately following subroutine, as returned by ILAENV.)
*
- IF( INFO.EQ.0 .AND. M.GT.0 .AND. N.GT.0 ) THEN
- IF( M.GE.N ) THEN
+ IF( INFO.EQ.0 ) THEN
+ MINWRK = 1
+ MAXWRK = 1
+ IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
* There is no complex work space needed for bidiagonal SVD
-* The real work space needed for bidiagonal SVD is BDSPAC
-* for computing singular values and singular vectors; BDSPAN
-* for computing singular values only.
-* BDSPAC = 5*N*N + 7*N
-* BDSPAN = MAX(7*N+4, 3*N+2+SMLSIZ*(SMLSIZ+8))
+* The real work space needed for bidiagonal SVD (dbdsdc) is
+* BDSPAC = 3*N*N + 4*N for singular values and vectors;
+* BDSPAC = 4*N for singular values only;
+* not including e, RU, and RVT matrices.
+*
+* Compute space preferred for each routine
+ CALL ZGEBRD( M, N, CDUM(1), M, DUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_MN = INT( CDUM(1) )
+*
+ CALL ZGEBRD( N, N, CDUM(1), N, DUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_NN = INT( CDUM(1) )
+*
+ CALL ZGEQRF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEQRF_MN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'P', N, N, N, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_P_NN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_Q_MM = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'Q', M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_Q_MN = INT( CDUM(1) )
+*
+ CALL ZUNGQR( M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGQR_MM = INT( CDUM(1) )
+*
+ CALL ZUNGQR( M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGQR_MN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', N, N, N, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_NN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_MM = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', M, N, N, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_MN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', N, N, N, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_NN = INT( CDUM(1) )
*
IF( M.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
*
- MAXWRK = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1,
- $ -1 )
- MAXWRK = MAX( MAXWRK, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
+ MAXWRK = N + LWORK_ZGEQRF_MN
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZGEBRD_NN )
MINWRK = 3*N
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
+* Path 2 (M >> N, JOBZ='O')
*
- WRKBL = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'ZUNGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
+ WRKBL = N + LWORK_ZGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_ZUNGQR_MN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_PRC_NN )
MAXWRK = M*N + N*N + WRKBL
MINWRK = 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
+* Path 3 (M >> N, JOBZ='S')
*
- WRKBL = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'ZUNGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
+ WRKBL = N + LWORK_ZGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_ZUNGQR_MN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
MINWRK = N*N + 3*N
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
+* Path 4 (M >> N, JOBZ='A')
*
- WRKBL = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'ZUNGQR', ' ', M,
- $ M, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
+ WRKBL = N + LWORK_ZGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_ZUNGQR_MM )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
- MINWRK = N*N + 2*N + M
+ MINWRK = N*N + MAX( 3*N, N + M )
END IF
ELSE IF( M.GE.MNTHR2 ) THEN
*
-* Path 5 (M much larger than N, but not as much as MNTHR1)
+* Path 5 (M >> N, but not as much as MNTHR1)
*
- MAXWRK = 2*N + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*N + LWORK_ZGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, N, N, -1 ) )
+* Path 5o (M >> N, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_Q_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, N, N, -1 ) )
+* Path 5s (M >> N, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_Q_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5a (M >> N, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_Q_MM )
END IF
ELSE
*
-* Path 6 (M at least N, but not much larger)
+* Path 6 (M >= N, but not much larger)
*
- MAXWRK = 2*N + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*N + LWORK_ZGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, N, N, -1 ) )
+* Path 6o (M >= N, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_PRC_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_QLN_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, N, N, -1 ) )
+* Path 6s (M >= N, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_QLN_MN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_PRC_NN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+M*
- $ ILAENV( 1, 'ZUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6a (M >= N, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_PRC_NN )
END IF
END IF
- ELSE
+ ELSE IF( MINMN.GT.0 ) THEN
*
* There is no complex work space needed for bidiagonal SVD
-* The real work space needed for bidiagonal SVD is BDSPAC
-* for computing singular values and singular vectors; BDSPAN
-* for computing singular values only.
-* BDSPAC = 5*M*M + 7*M
-* BDSPAN = MAX(7*M+4, 3*M+2+SMLSIZ*(SMLSIZ+8))
+* The real work space needed for bidiagonal SVD (dbdsdc) is
+* BDSPAC = 3*M*M + 4*M for singular values and vectors;
+* BDSPAC = 4*M for singular values only;
+* not including e, RU, and RVT matrices.
+*
+* Compute space preferred for each routine
+ CALL ZGEBRD( M, N, CDUM(1), M, DUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_MN = INT( CDUM(1) )
+*
+ CALL ZGEBRD( M, M, CDUM(1), M, DUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_MM = INT( CDUM(1) )
+*
+ CALL ZGELQF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGELQF_MN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'P', M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_P_MN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'P', N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_P_NN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_Q_MM = INT( CDUM(1) )
+*
+ CALL ZUNGLQ( M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGLQ_MN = INT( CDUM(1) )
+*
+ CALL ZUNGLQ( N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGLQ_NN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', M, M, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_MM = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', M, N, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_MN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', N, N, M, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_NN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_MM = INT( CDUM(1) )
*
IF( N.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
*
- MAXWRK = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1,
- $ -1 )
- MAXWRK = MAX( MAXWRK, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
+ MAXWRK = M + LWORK_ZGELQF_MN
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZGEBRD_MM )
MINWRK = 3*M
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
+* Path 2t (N >> M, JOBZ='O')
*
- WRKBL = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'ZUNGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, M, -1 ) )
+ WRKBL = M + LWORK_ZGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_ZUNGLQ_MN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_PRC_MM )
MAXWRK = M*N + M*M + WRKBL
MINWRK = 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
+* Path 3t (N >> M, JOBZ='S')
*
- WRKBL = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'ZUNGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, M, -1 ) )
+ WRKBL = M + LWORK_ZGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_ZUNGLQ_MN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
MINWRK = M*M + 3*M
ELSE IF( WNTQA ) THEN
*
-* Path 4t (N much larger than M, JOBZ='A')
+* Path 4t (N >> M, JOBZ='A')
*
- WRKBL = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'ZUNGLQ', ' ', N,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, M, -1 ) )
+ WRKBL = M + LWORK_ZGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_ZUNGLQ_NN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
- MINWRK = M*M + 2*M + N
+ MINWRK = M*M + MAX( 3*M, M + N )
END IF
ELSE IF( N.GE.MNTHR2 ) THEN
*
-* Path 5t (N much larger than M, but not as much as MNTHR1)
+* Path 5t (N >> M, but not as much as MNTHR1)
*
- MAXWRK = 2*M + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*M + LWORK_ZGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'P', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5to (N >> M, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_P_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'P', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5ts (N >> M, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_P_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5ta (N >> M, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_P_NN )
END IF
ELSE
*
-* Path 6t (N greater than M, but not much larger)
+* Path 6t (N > M, but not much larger)
*
- MAXWRK = 2*M + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*M + LWORK_ZGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, N, -1 ) )
+* Path 6to (N > M, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_PRC_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'PRC', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6ts (N > M, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_PRC_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+N*
- $ ILAENV( 1, 'ZUNGBR', 'PRC', N, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6ta (N > M, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_PRC_NN )
END IF
END IF
END IF
@@ -460,18 +620,20 @@
END IF
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = MAXWRK
- IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV )
- $ INFO = -13
+ IF( LWORK.LT.MINWRK .AND. .NOT. LQUERY ) THEN
+ INFO = -12
+ END IF
END IF
*
-* Quick returns
-*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGESDD', -INFO )
RETURN
+ ELSE IF( LQUERY ) THEN
+ RETURN
END IF
- IF( LWORK.EQ.LQUERV )
- $ RETURN
+*
+* Quick return if possible
+*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
@@ -504,15 +666,16 @@
*
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: need 0)
+* CWorkspace: need N [tau] + N [work]
+* CWorkspace: prefer N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
@@ -527,8 +690,9 @@
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -536,15 +700,15 @@
NRWORK = IE + N
*
* Perform bidiagonal SVD, compute singular values only
-* (CWorkspace: 0)
-* (RWorkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
+* Path 2 (M >> N, JOBZ='O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
@@ -554,20 +718,21 @@
*
LDWRKU = N
IR = IU + LDWRKU*N
- IF( LWORK.GE.M*N+N*N+3*N ) THEN
+ IF( LWORK .GE. M*N + N*N + 3*N ) THEN
*
* WORK(IR) is M by N
*
LDWRKR = M
ELSE
- LDWRKR = ( LWORK-N*N-3*N ) / N
+ LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need N*N+2*N, prefer M*N+N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
@@ -579,8 +744,9 @@
$ LDWRKR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
@@ -590,8 +756,9 @@
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
@@ -600,8 +767,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of R in WORK(IRU) and computing right singular vectors
* of R in WORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
@@ -612,8 +779,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by the left singular vectors of R
-* (CWorkspace: need 2*N*N+3*N, prefer M*N+N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
@@ -623,8 +791,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by the right singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
@@ -633,8 +802,9 @@
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
-* (CWorkspace: need 2*N*N, prefer N*N+M*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R]
+* CWorkspace: prefer N*N [U] + M*N [R]
+* RWorkspace: need 0
*
DO 10 I = 1, M, LDWRKR
CHUNK = MIN( M-I+1, LDWRKR )
@@ -647,7 +817,7 @@
*
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
+* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
@@ -660,8 +830,9 @@
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
@@ -673,8 +844,9 @@
$ LDWRKR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
@@ -684,8 +856,9 @@
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
@@ -694,8 +867,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
@@ -706,8 +879,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
@@ -716,8 +890,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
@@ -726,8 +901,8 @@
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
-* (CWorkspace: need N*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R]
+* RWorkspace: need 0
*
CALL ZLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL ZGEMM( 'N', 'N', M, N, N, CONE, A, LDA, WORK( IR ),
@@ -735,7 +910,7 @@
*
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
+* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
@@ -748,16 +923,18 @@
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (CWorkspace: need N+M, prefer N+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N [tau] + M [work]
+* CWorkspace: prefer N*N [U] + N [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
@@ -772,8 +949,9 @@
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -785,8 +963,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
@@ -794,8 +972,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by left singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
@@ -805,8 +984,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
@@ -815,8 +995,8 @@
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
-* (CWorkspace: need N*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U]
+* RWorkspace: need 0
*
CALL ZGEMM( 'N', 'N', M, N, N, CONE, U, LDU, WORK( IU ),
$ LDWRKU, CZERO, A, LDA )
@@ -831,7 +1011,7 @@
*
* MNTHR2 <= M < MNTHR1
*
-* Path 5 (M much larger than N, but not as much as MNTHR1)
+* Path 5 (M >> N, but not as much as MNTHR1)
* Reduce to bidiagonal form without QR decomposition, use
* ZUNGBR and matrix multiplication to compute singular vectors
*
@@ -842,19 +1022,21 @@
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 5n (M >> N, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IU = NWORK
@@ -862,22 +1044,25 @@
IRVT = IRU + N*N
NRWORK = IRVT + N*N
*
+* Path 5o (M >> N, JOBZ='O')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
* WORK( IU ) is M by N
*
@@ -886,15 +1071,15 @@
*
* WORK(IU) is LDWRKU by N
*
- LDWRKU = ( LWORK-3*N ) / N
+ LDWRKU = ( LWORK - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
@@ -902,8 +1087,8 @@
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in WORK(IU), copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL ZLARCM( N, N, RWORK( IRVT ), N, VT, LDVT,
$ WORK( IU ), LDWRKU, RWORK( NRWORK ) )
@@ -911,8 +1096,10 @@
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need N*N, prefer M*N)
-* (Rworkspace: need 3*N*N, prefer N*N+2*M*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U]
+* RWorkspace: need N [e] + N*N [RU] + 2*N*N [rwork]
+* RWorkspace: prefer N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
DO 20 I = 1, M, LDWRKU
@@ -925,17 +1112,20 @@
*
ELSE IF( WNTQS ) THEN
*
+* Path 5s (M >> N, JOBZ='S')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to U, generate Q
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, N, N, U, LDU, WORK( ITAUQ ),
@@ -944,8 +1134,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
@@ -956,8 +1146,8 @@
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL ZLARCM( N, N, RWORK( IRVT ), N, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
@@ -965,8 +1155,8 @@
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need N*N+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
CALL ZLACRM( M, N, U, LDU, RWORK( IRU ), N, A, LDA,
@@ -974,17 +1164,20 @@
CALL ZLACPY( 'F', M, N, A, LDA, U, LDU )
ELSE
*
+* Path 5a (M >> N, JOBZ='A')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to U, generate Q
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
@@ -993,8 +1186,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
@@ -1005,8 +1198,8 @@
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL ZLARCM( N, N, RWORK( IRVT ), N, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
@@ -1014,8 +1207,8 @@
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
CALL ZLACRM( M, N, U, LDU, RWORK( IRU ), N, A, LDA,
@@ -1027,7 +1220,7 @@
*
* M .LT. MNTHR2
*
-* Path 6 (M at least N, but not much larger)
+* Path 6 (M >= N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
* Use ZUNMBR to compute singular vectors
*
@@ -1038,26 +1231,28 @@
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 6n (M >= N, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IU = NWORK
IRU = NRWORK
IRVT = IRU + N*N
NRWORK = IRVT + N*N
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
* WORK( IU ) is M by N
*
@@ -1066,15 +1261,16 @@
*
* WORK( IU ) is LDWRKU by N
*
- LDWRKU = ( LWORK-3*N ) / N
+ LDWRKU = ( LWORK - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
+* Path 6o (M >= N, JOBZ='O')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
@@ -1082,21 +1278,24 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*N [U] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
-* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
-* Overwrite WORK(IU) by left singular vectors of A, copying
-* to A
-* (Cworkspace: need M*N+2*N, prefer M*N+N+N*NB)
-* (Rworkspace: need 0)
+* Path 6o-fast
+* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
+* Overwrite WORK(IU) by left singular vectors of A, copying
+* to A
+* CWorkspace: need 2*N [tauq, taup] + M*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU]
*
CALL ZLASET( 'F', M, N, CZERO, CZERO, WORK( IU ),
$ LDWRKU )
@@ -1108,17 +1307,21 @@
CALL ZLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
ELSE
*
+* Path 6o-slow
* Generate Q in A
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*N [U] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need N*N, prefer M*N)
-* (Rworkspace: need 3*N*N, prefer N*N+2*M*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U]
+* RWorkspace: need N [e] + N*N [RU] + 2*N*N [rwork]
+* RWorkspace: prefer N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
DO 30 I = 1, M, LDWRKU
@@ -1133,11 +1336,12 @@
*
ELSE IF( WNTQS ) THEN
*
+* Path 6s (M >= N, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
@@ -1148,8 +1352,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLASET( 'F', M, N, CZERO, CZERO, U, LDU )
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
@@ -1159,8 +1364,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
@@ -1168,11 +1374,12 @@
$ LWORK-NWORK+1, IERR )
ELSE
*
+* Path 6a (M >= N, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
@@ -1191,8 +1398,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
@@ -1201,8 +1409,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
@@ -1222,15 +1431,16 @@
*
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M [tau] + M [work]
+* CWorkspace: prefer M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
@@ -1245,8 +1455,9 @@
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -1254,15 +1465,15 @@
NRWORK = IE + M
*
* Perform bidiagonal SVD, compute singular values only
-* (CWorkspace: 0)
-* (RWorkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
+* Path 2t (N >> M, JOBZ='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
@@ -1272,7 +1483,7 @@
* WORK(IVT) is M by M
*
IL = IVT + LDWKVT*M
- IF( LWORK.GE.M*N+M*M+3*M ) THEN
+ IF( LWORK .GE. M*N + M*M + 3*M ) THEN
*
* WORK(IL) M by N
*
@@ -1283,14 +1494,15 @@
* WORK(IL) is M by CHUNK
*
LDWRKL = M
- CHUNK = ( LWORK-M*M-3*M ) / M
+ CHUNK = ( LWORK - M*M - 3*M ) / M
END IF
ITAU = IL + LDWRKL*CHUNK
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
@@ -1302,8 +1514,9 @@
$ WORK( IL+LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
@@ -1313,8 +1526,9 @@
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, WORK( IL ), LDWRKL, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
@@ -1323,8 +1537,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
@@ -1335,8 +1549,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by the left singular vectors of L
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
@@ -1345,8 +1560,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
* Overwrite WORK(IVT) by the right singular vectors of L
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
@@ -1356,8 +1572,9 @@
*
* Multiply right singular vectors of L in WORK(IL) by Q
* in A, storing result in WORK(IL) and copying to A
-* (CWorkspace: need 2*M*M, prefer M*M+M*N))
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L]
+* CWorkspace: prefer M*M [VT] + M*N [L]
+* RWorkspace: need 0
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
@@ -1370,9 +1587,9 @@
*
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
-* M right singular vectors to be computed in VT and
-* M left singular vectors to be computed in U
+* Path 3t (N >> M, JOBZ='S')
+* M right singular vectors to be computed in VT and
+* M left singular vectors to be computed in U
*
IL = 1
*
@@ -1383,8 +1600,9 @@
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
@@ -1396,8 +1614,9 @@
$ WORK( IL+LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
@@ -1407,8 +1626,9 @@
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, WORK( IL ), LDWRKL, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
@@ -1417,8 +1637,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
@@ -1429,8 +1649,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
@@ -1439,8 +1660,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by left singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', M, M, M, WORK( IL ), LDWRKL,
@@ -1449,8 +1671,8 @@
*
* Copy VT to WORK(IL), multiply right singular vectors of L
* in WORK(IL) by Q in A, storing result in VT
-* (CWorkspace: need M*M)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L]
+* RWorkspace: need 0
*
CALL ZLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IL ), LDWRKL,
@@ -1458,7 +1680,7 @@
*
ELSE IF( WNTQA ) THEN
*
-* Path 9t (N much larger than M, JOBZ='A')
+* Path 4t (N >> M, JOBZ='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
@@ -1471,16 +1693,18 @@
NWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (CWorkspace: need M+N, prefer M+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M [tau] + N [work]
+* CWorkspace: prefer M*M [VT] + M [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
@@ -1495,8 +1719,9 @@
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -1505,8 +1730,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
@@ -1517,8 +1742,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of L
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
@@ -1527,8 +1753,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
* Overwrite WORK(IVT) by right singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
@@ -1538,8 +1765,8 @@
*
* Multiply right singular vectors of L in WORK(IVT) by
* Q in VT, storing result in A
-* (CWorkspace: need M*M)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT]
+* RWorkspace: need 0
*
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IVT ), LDWKVT,
$ VT, LDVT, CZERO, A, LDA )
@@ -1554,11 +1781,10 @@
*
* MNTHR2 <= N < MNTHR1
*
-* Path 5t (N much larger than M, but not as much as MNTHR1)
+* Path 5t (N >> M, but not as much as MNTHR1)
* Reduce to bidiagonal form without QR decomposition, use
* ZUNGBR and matrix multiplication to compute singular vectors
*
-*
IE = 1
NRWORK = IE + M
ITAUQ = 1
@@ -1566,8 +1792,9 @@
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
-* (RWorkspace: M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
@@ -1575,11 +1802,12 @@
*
IF( WNTQN ) THEN
*
+* Path 5tn (N >> M, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IRVT = NRWORK
@@ -1587,23 +1815,26 @@
NRWORK = IRU + M*M
IVT = NWORK
*
+* Path 5to (N >> M, JOBZ='O')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Generate P**H in A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
LDWKVT = M
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
* WORK( IVT ) is M by N
*
@@ -1613,15 +1844,15 @@
*
* WORK( IVT ) is M by CHUNK
*
- CHUNK = ( LWORK-3*M ) / M
+ CHUNK = ( LWORK - 3*M ) / M
NWORK = IVT + LDWKVT*CHUNK
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
CALL DBDSDC( 'L', 'I', M, S, RWORK( IE ), RWORK( IRU ),
$ M, RWORK( IRVT ), M, DUM, IDUM,
@@ -1629,8 +1860,8 @@
*
* Multiply Q in U by real matrix RWORK(IRVT)
* storing the result in WORK(IVT), copying to U
-* (Cworkspace: need 0)
-* (Rworkspace: need 2*M*M)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL ZLACRM( M, M, U, LDU, RWORK( IRU ), M, WORK( IVT ),
$ LDWKVT, RWORK( NRWORK ) )
@@ -1638,8 +1869,10 @@
*
* Multiply RWORK(IRVT) by P**H in A, storing the
* result in WORK(IVT), copying to A
-* (CWorkspace: need M*M, prefer M*N)
-* (Rworkspace: need 2*M*M, prefer 2*M*N)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT]
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*M [rwork]
+* RWorkspace: prefer M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
DO 50 I = 1, N, CHUNK
@@ -1651,17 +1884,20 @@
50 CONTINUE
ELSE IF( WNTQS ) THEN
*
+* Path 5ts (N >> M, JOBZ='S')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', M, N, M, VT, LDVT, WORK( ITAUP ),
@@ -1670,8 +1906,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
@@ -1682,8 +1918,8 @@
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need 3*M*M)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL ZLACRM( M, M, U, LDU, RWORK( IRU ), M, A, LDA,
$ RWORK( NRWORK ) )
@@ -1691,8 +1927,8 @@
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need M*M+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
CALL ZLARCM( M, N, RWORK( IRVT ), M, VT, LDVT, A, LDA,
@@ -1700,17 +1936,20 @@
CALL ZLACPY( 'F', M, N, A, LDA, VT, LDVT )
ELSE
*
+* Path 5ta (N >> M, JOBZ='A')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, M, VT, LDVT, WORK( ITAUP ),
@@ -1719,8 +1958,8 @@
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
@@ -1731,8 +1970,8 @@
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need 3*M*M)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL ZLACRM( M, M, U, LDU, RWORK( IRU ), M, A, LDA,
$ RWORK( NRWORK ) )
@@ -1740,9 +1979,10 @@
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need M*M+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
+ NRWORK = IRU
CALL ZLARCM( M, N, RWORK( IRVT ), M, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
CALL ZLACPY( 'F', M, N, A, LDA, VT, LDVT )
@@ -1752,7 +1992,7 @@
*
* N .LT. MNTHR2
*
-* Path 6t (N greater than M, but not much larger)
+* Path 6t (N > M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
* Use ZUNMBR to compute singular vectors
*
@@ -1763,24 +2003,27 @@
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
-* (RWorkspace: M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 6tn (N > M, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
+* Path 6to (N > M, JOBZ='O')
LDWKVT = M
IVT = NWORK
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
* WORK( IVT ) is M by N
*
@@ -1791,15 +2034,15 @@
*
* WORK( IVT ) is M by CHUNK
*
- CHUNK = ( LWORK-3*M ) / M
+ CHUNK = ( LWORK - 3*M ) / M
NWORK = IVT + LDWKVT*CHUNK
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
@@ -1810,21 +2053,24 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*M [VT] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
-* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
-* Overwrite WORK(IVT) by right singular vectors of A,
-* copying to A
-* (Cworkspace: need M*N+2*M, prefer M*N+M+M*NB)
-* (Rworkspace: need 0)
+* Path 6to-fast
+* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
+* Overwrite WORK(IVT) by right singular vectors of A,
+* copying to A
+* CWorkspace: need 2*M [tauq, taup] + M*N [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
@@ -1834,17 +2080,21 @@
CALL ZLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
ELSE
*
+* Path 6to-slow
* Generate P**H in A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*M [VT] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need M*M, prefer M*N)
-* (Rworkspace: need 3*M*M, prefer M*M+2*M*N)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT]
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*M [rwork]
+* RWorkspace: prefer M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
DO 60 I = 1, N, CHUNK
@@ -1858,11 +2108,12 @@
END IF
ELSE IF( WNTQS ) THEN
*
+* Path 6ts (N > M, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
@@ -1873,8 +2124,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
@@ -1883,8 +2135,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL ZLASET( 'F', M, N, CZERO, CZERO, VT, LDVT )
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
@@ -1893,11 +2146,12 @@
$ LWORK-NWORK+1, IERR )
ELSE
*
+* Path 6ta (N > M, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
@@ -1909,8 +2163,9 @@
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
@@ -1923,8 +2178,9 @@
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + N*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, M, A, LDA,