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SUBROUTINE MB03UD( JOBQ, JOBP, N, A, LDA, Q, LDQ, SV, DWORK,
$ LDWORK, INFO )
C
C RELEASE 4.0, WGS COPYRIGHT 1999.
C
C PURPOSE
C
C To compute all, or part, of the singular value decomposition of a
C real upper triangular matrix.
C
C The N-by-N upper triangular matrix A is factored as A = Q*S*P',
C where Q and P are N-by-N orthogonal matrices and S is an
C N-by-N diagonal matrix with non-negative diagonal elements,
C SV(1), SV(2), ..., SV(N), ordered such that
C
C SV(1) >= SV(2) >= ... >= SV(N) >= 0.
C
C The columns of Q are the left singular vectors of A, the diagonal
C elements of S are the singular values of A and the columns of P
C are the right singular vectors of A.
C
C Either or both of Q and P' may be requested.
C When P' is computed, it is returned in A.
C
C ARGUMENTS
C
C Mode Parameters
C
C JOBQ CHARACTER*1
C Specifies whether the user wishes to compute the matrix Q
C of left singular vectors as follows:
C = 'V': Left singular vectors are computed;
C = 'N': No left singular vectors are computed.
C
C JOBP CHARACTER*1
C Specifies whether the user wishes to compute the matrix P'
C of right singular vectors as follows:
C = 'V': Right singular vectors are computed;
C = 'N': No right singular vectors are computed.
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the matrix A. N >= 0.
C
C A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
C On entry, the leading N-by-N upper triangular part of this
C array must contain the upper triangular matrix A.
C On exit, if JOBP = 'V', the leading N-by-N part of this
C array contains the N-by-N orthogonal matrix P'; otherwise
C the N-by-N upper triangular part of A is used as internal
C workspace. The strictly lower triangular part of A is set
C internally to zero before the reduction to bidiagonal form
C is performed.
C
C LDA INTEGER
C The leading dimension of array A. LDA >= MAX(1,N).
C
C Q (output) DOUBLE PRECISION array, dimension (LDQ,N)
C If JOBQ = 'V', the leading N-by-N part of this array
C contains the orthogonal matrix Q.
C If JOBQ = 'N', Q is not referenced.
C
C LDQ INTEGER
C The leading dimension of array Q.
C LDQ >= 1, and when JOBQ = 'V', LDQ >= MAX(1,N).
C
C SV (output) DOUBLE PRECISION array, dimension (N)
C The N singular values of the matrix A, sorted in
C descending order.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK;
C if INFO > 0, DWORK(2:N) contains the unconverged
C superdiagonal elements of an upper bidiagonal matrix B
C whose diagonal is in SV (not necessarily sorted).
C B satisfies A = Q*B*P', so it has the same singular
C values as A, and singular vectors related by Q and P'.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX(1,5*N).
C For optimum performance LDWORK should be larger.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C > 0: the QR algorithm has failed to converge. In this
C case INFO specifies how many superdiagonals did not
C converge (see the description of DWORK).
C This failure is not likely to occur.
C
C METHOD
C
C The routine reduces A to bidiagonal form by means of elementary
C reflectors and then uses the QR algorithm on the bidiagonal form.
C
C CONTRIBUTOR
C
C V. Sima, Research Institute of Informatics, Bucharest, and
C A. Varga, German Aerospace Center, DLR Oberpfaffenhofen,
C March 1998. Based on the RASP routine DTRSVD.
C
C REVISIONS
C
C V. Sima, Feb. 2000.
C
C KEYWORDS
C
C Bidiagonalization, orthogonal transformation, singular value
C decomposition, singular values, triangular form.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
C .. Scalar Arguments ..
CHARACTER JOBP, JOBQ
INTEGER INFO, LDA, LDQ, LDWORK, N
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), DWORK(*), Q(LDQ,*), SV(*)
C .. Local Scalars ..
LOGICAL WANTQ, WANTP
INTEGER I, IE, ISCL, ITAUP, ITAUQ, JWORK, MAXWRK,
$ MINWRK, NCOLP, NCOLQ
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
C .. Local Arrays ..
DOUBLE PRECISION DUM(1)
C .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANTR
EXTERNAL DLAMCH, DLANTR, ILAENV, LSAME
C .. External Subroutines ..
EXTERNAL DBDSQR, DGEBRD, DLACPY, DLASCL, DLASET, DORGBR,
$ XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
C .. Executable Statements ..
C
C Check the input scalar arguments.
C
INFO = 0
WANTQ = LSAME( JOBQ, 'V' )
WANTP = LSAME( JOBP, 'V' )
MINWRK = 1
IF( .NOT.WANTQ .AND. .NOT.LSAME( JOBQ, 'N' ) ) THEN
INFO = -1
ELSE IF( .NOT.WANTP .AND. .NOT.LSAME( JOBP, 'N' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( ( WANTQ .AND. LDQ.LT.MAX( 1, N ) ) .OR.
$ ( .NOT.WANTQ .AND. LDQ.LT.1 ) ) THEN
INFO = -7
END IF
C
C Compute workspace
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of workspace needed at that point in the code,
C as well as the preferred amount for good performance.
C NB refers to the optimal block size for the immediately following
C subroutine, as returned by ILAENV.)
C
IF( INFO.EQ.0 .AND. LDWORK.GE.1 .AND. N.GT.0 ) THEN
MAXWRK = 3*N+2*N*ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 )
IF( WANTQ )
$ MAXWRK = MAX( MAXWRK, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'Q', N, N, N, -1 ) )
IF( WANTP )
$ MAXWRK = MAX( MAXWRK, 3*N+N*
$ ILAENV( 1, 'DORGBR', 'P', N, N, N, -1 ) )
MINWRK = 5*N
MAXWRK = MAX( MAXWRK, MINWRK )
DWORK(1) = MAXWRK
END IF
C
IF( LDWORK.LT.MINWRK ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MB03UD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 ) THEN
DWORK(1) = ONE
RETURN
END IF
C
C Get machine constants.
C
EPS = DLAMCH( 'P' )
SMLNUM = SQRT( DLAMCH( 'S' ) ) / EPS
BIGNUM = ONE / SMLNUM
C
C Scale A if max entry outside range [SMLNUM,BIGNUM].
C
ANRM = DLANTR( 'Max', 'Upper', 'Non-unit', N, N, A, LDA, DUM )
ISCL = 0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ISCL = 1
CALL DLASCL( 'Upper', 0, 0, ANRM, SMLNUM, N, N, A, LDA, INFO )
ELSE IF( ANRM.GT.BIGNUM ) THEN
ISCL = 1
CALL DLASCL( 'Upper', 0, 0, ANRM, BIGNUM, N, N, A, LDA, INFO )
END IF
C
C Zero out below.
C
IF ( N.GT.1 )
$ CALL DLASET( 'Lower', N-1, N-1, ZERO, ZERO, A(2,1), LDA )
C
C Find the singular values and optionally the singular vectors
C of the upper triangular matrix A.
C
IE = 1
ITAUQ = IE + N
ITAUP = ITAUQ + N
JWORK = ITAUP + N
C
C First reduce the matrix to bidiagonal form. The diagonal
C elements will be in SV and the superdiagonals in DWORK(IE).
C (Workspace: need 4*N, prefer 3*N+2*N*NB)
C
CALL DGEBRD( N, N, A, LDA, SV, DWORK(IE), DWORK(ITAUQ),
$ DWORK(ITAUP), DWORK(JWORK), LDWORK-JWORK+1, INFO )
IF( WANTQ ) THEN
C
C Generate the transformation matrix Q corresponding to the
C left singular vectors.
C (Workspace: need 4*N, prefer 3*N+N*NB)
C
NCOLQ = N
CALL DLACPY( 'Lower', N, N, A, LDA, Q, LDQ )
CALL DORGBR( 'Q', N, N, N, Q, LDQ, DWORK(ITAUQ), DWORK(JWORK),
$ LDWORK-JWORK+1, INFO )
ELSE
NCOLQ = 0
END IF
IF( WANTP ) THEN
C
C Generate the transformation matrix P' corresponding to the
C right singular vectors.
C (Workspace: need 4*N, prefer 3*N+N*NB)
C
NCOLP = N
CALL DORGBR( 'P', N, N, N, A, LDA, DWORK(ITAUP), DWORK(JWORK),
$ LDWORK-JWORK+1, INFO )
ELSE
NCOLP = 0
END IF
JWORK = IE + N
C
C Perform bidiagonal QR iteration, to obtain all or part of the
C singular value decomposition of A.
C (Workspace: need 5*N)
C
CALL DBDSQR( 'U', N, NCOLP, NCOLQ, 0, SV, DWORK(IE), A, LDA,
$ Q, LDQ, DUM, 1, DWORK(JWORK), INFO )
C
C If DBDSQR failed to converge, copy unconverged superdiagonals
C to DWORK(2:N).
C
IF( INFO.NE.0 ) THEN
DO 10 I = N - 1, 1, -1
DWORK(I+1) = DWORK(I+IE-1)
10 CONTINUE
END IF
C
C Undo scaling if necessary.
C
IF( ISCL.EQ.1 ) THEN
IF( ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, N, 1, SV, N, INFO )
IF( INFO.NE.0 .AND. ANRM.GT.BIGNUM )
$ CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, N-1, 1, DWORK(2), N,
$ INFO )
IF( ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, N, 1, SV, N, INFO )
IF( INFO.NE.0 .AND. ANRM.LT.SMLNUM )
$ CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, N-1, 1, DWORK(2), N,
$ INFO )
END IF
C
C Return optimal workspace in DWORK(1).
C
DWORK(1) = MAXWRK
C
RETURN
C *** Last line of MB03UD ***
END
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