--- rpl/lapack/lapack/dgesvj.f 2011/07/22 07:38:05 1.6
+++ rpl/lapack/lapack/dgesvj.f 2017/06/17 11:06:17 1.17
@@ -1,22 +1,347 @@
- SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
- $ LDV, WORK, LWORK, INFO )
+*> \brief \b DGESVJ
+*
+* =========== DOCUMENTATION ===========
*
-* -- LAPACK routine (version 3.3.1) --
+* Online html documentation available at
+* http://www.netlib.org/lapack/explore-html/
*
-* -- Contributed by Zlatko Drmac of the University of Zagreb and --
-* -- Kresimir Veselic of the Fernuniversitaet Hagen --
-* -- April 2011 --
+*> \htmlonly
+*> Download DGESVJ + dependencies
+*>
+*> [TGZ]
+*>
+*> [ZIP]
+*>
+*> [TXT]
+*> \endhtmlonly
+*
+* Definition:
+* ===========
+*
+* SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
+* LDV, WORK, LWORK, INFO )
+*
+* .. Scalar Arguments ..
+* INTEGER INFO, LDA, LDV, LWORK, M, MV, N
+* CHARACTER*1 JOBA, JOBU, JOBV
+* ..
+* .. Array Arguments ..
+* DOUBLE PRECISION A( LDA, * ), SVA( N ), V( LDV, * ),
+* $ WORK( LWORK )
+* ..
+*
+*
+*> \par Purpose:
+* =============
+*>
+*> \verbatim
+*>
+*> DGESVJ computes the singular value decomposition (SVD) of a real
+*> M-by-N matrix A, where M >= N. The SVD of A is written as
+*> [++] [xx] [x0] [xx]
+*> A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx]
+*> [++] [xx]
+*> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
+*> matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
+*> of SIGMA are the singular values of A. The columns of U and V are the
+*> left and the right singular vectors of A, respectively.
+*> DGESVJ can sometimes compute tiny singular values and their singular vectors much
+*> more accurately than other SVD routines, see below under Further Details.
+*> \endverbatim
+*
+* Arguments:
+* ==========
+*
+*> \param[in] JOBA
+*> \verbatim
+*> JOBA is CHARACTER* 1
+*> Specifies the structure of A.
+*> = 'L': The input matrix A is lower triangular;
+*> = 'U': The input matrix A is upper triangular;
+*> = 'G': The input matrix A is general M-by-N matrix, M >= N.
+*> \endverbatim
+*>
+*> \param[in] JOBU
+*> \verbatim
+*> JOBU is CHARACTER*1
+*> Specifies whether to compute the left singular vectors
+*> (columns of U):
+*> = 'U': The left singular vectors corresponding to the nonzero
+*> singular values are computed and returned in the leading
+*> columns of A. See more details in the description of A.
+*> The default numerical orthogonality threshold is set to
+*> approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
+*> = 'C': Analogous to JOBU='U', except that user can control the
+*> level of numerical orthogonality of the computed left
+*> singular vectors. TOL can be set to TOL = CTOL*EPS, where
+*> CTOL is given on input in the array WORK.
+*> No CTOL smaller than ONE is allowed. CTOL greater
+*> than 1 / EPS is meaningless. The option 'C'
+*> can be used if M*EPS is satisfactory orthogonality
+*> of the computed left singular vectors, so CTOL=M could
+*> save few sweeps of Jacobi rotations.
+*> See the descriptions of A and WORK(1).
+*> = 'N': The matrix U is not computed. However, see the
+*> description of A.
+*> \endverbatim
+*>
+*> \param[in] JOBV
+*> \verbatim
+*> JOBV is CHARACTER*1
+*> Specifies whether to compute the right singular vectors, that
+*> is, the matrix V:
+*> = 'V' : the matrix V is computed and returned in the array V
+*> = 'A' : the Jacobi rotations are applied to the MV-by-N
+*> array V. In other words, the right singular vector
+*> matrix V is not computed explicitly, instead it is
+*> applied to an MV-by-N matrix initially stored in the
+*> first MV rows of V.
+*> = 'N' : the matrix V is not computed and the array V is not
+*> referenced
+*> \endverbatim
+*>
+*> \param[in] M
+*> \verbatim
+*> M is INTEGER
+*> The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
+*> \endverbatim
+*>
+*> \param[in] N
+*> \verbatim
+*> N is INTEGER
+*> The number of columns of the input matrix A.
+*> M >= N >= 0.
+*> \endverbatim
+*>
+*> \param[in,out] A
+*> \verbatim
+*> A is DOUBLE PRECISION array, dimension (LDA,N)
+*> On entry, the M-by-N matrix A.
+*> On exit :
+*> If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C' :
+*> If INFO .EQ. 0 :
+*> RANKA orthonormal columns of U are returned in the
+*> leading RANKA columns of the array A. Here RANKA <= N
+*> is the number of computed singular values of A that are
+*> above the underflow threshold DLAMCH('S'). The singular
+*> vectors corresponding to underflowed or zero singular
+*> values are not computed. The value of RANKA is returned
+*> in the array WORK as RANKA=NINT(WORK(2)). Also see the
+*> descriptions of SVA and WORK. The computed columns of U
+*> are mutually numerically orthogonal up to approximately
+*> TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
+*> see the description of JOBU.
+*> If INFO .GT. 0 :
+*> the procedure DGESVJ did not converge in the given number
+*> of iterations (sweeps). In that case, the computed
+*> columns of U may not be orthogonal up to TOL. The output
+*> U (stored in A), SIGMA (given by the computed singular
+*> values in SVA(1:N)) and V is still a decomposition of the
+*> input matrix A in the sense that the residual
+*> ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
+*>
+*> If JOBU .EQ. 'N' :
+*> If INFO .EQ. 0 :
+*> Note that the left singular vectors are 'for free' in the
+*> one-sided Jacobi SVD algorithm. However, if only the
+*> singular values are needed, the level of numerical
+*> orthogonality of U is not an issue and iterations are
+*> stopped when the columns of the iterated matrix are
+*> numerically orthogonal up to approximately M*EPS. Thus,
+*> on exit, A contains the columns of U scaled with the
+*> corresponding singular values.
+*> If INFO .GT. 0 :
+*> the procedure DGESVJ did not converge in the given number
+*> of iterations (sweeps).
+*> \endverbatim
+*>
+*> \param[in] LDA
+*> \verbatim
+*> LDA is INTEGER
+*> The leading dimension of the array A. LDA >= max(1,M).
+*> \endverbatim
+*>
+*> \param[out] SVA
+*> \verbatim
+*> SVA is DOUBLE PRECISION array, dimension (N)
+*> On exit :
+*> If INFO .EQ. 0 :
+*> depending on the value SCALE = WORK(1), we have:
+*> If SCALE .EQ. ONE :
+*> SVA(1:N) contains the computed singular values of A.
+*> During the computation SVA contains the Euclidean column
+*> norms of the iterated matrices in the array A.
+*> If SCALE .NE. ONE :
+*> The singular values of A are SCALE*SVA(1:N), and this
+*> factored representation is due to the fact that some of the
+*> singular values of A might underflow or overflow.
+*> If INFO .GT. 0 :
+*> the procedure DGESVJ did not converge in the given number of
+*> iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
+*> \endverbatim
+*>
+*> \param[in] MV
+*> \verbatim
+*> MV is INTEGER
+*> If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ
+*> is applied to the first MV rows of V. See the description of JOBV.
+*> \endverbatim
+*>
+*> \param[in,out] V
+*> \verbatim
+*> V is DOUBLE PRECISION array, dimension (LDV,N)
+*> If JOBV = 'V', then V contains on exit the N-by-N matrix of
+*> the right singular vectors;
+*> If JOBV = 'A', then V contains the product of the computed right
+*> singular vector matrix and the initial matrix in
+*> the array V.
+*> If JOBV = 'N', then V is not referenced.
+*> \endverbatim
+*>
+*> \param[in] LDV
+*> \verbatim
+*> LDV is INTEGER
+*> The leading dimension of the array V, LDV .GE. 1.
+*> If JOBV .EQ. 'V', then LDV .GE. max(1,N).
+*> If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .
+*> \endverbatim
+*>
+*> \param[in,out] WORK
+*> \verbatim
+*> WORK is DOUBLE PRECISION array, dimension MAX(6,M+N).
+*> On entry :
+*> If JOBU .EQ. 'C' :
+*> WORK(1) = CTOL, where CTOL defines the threshold for convergence.
+*> The process stops if all columns of A are mutually
+*> orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
+*> It is required that CTOL >= ONE, i.e. it is not
+*> allowed to force the routine to obtain orthogonality
+*> below EPS.
+*> On exit :
+*> WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
+*> are the computed singular values of A.
+*> (See description of SVA().)
+*> WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
+*> singular values.
+*> WORK(3) = NINT(WORK(3)) is the number of the computed singular
+*> values that are larger than the underflow threshold.
+*> WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
+*> rotations needed for numerical convergence.
+*> WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
+*> This is useful information in cases when DGESVJ did
+*> not converge, as it can be used to estimate whether
+*> the output is stil useful and for post festum analysis.
+*> WORK(6) = the largest absolute value over all sines of the
+*> Jacobi rotation angles in the last sweep. It can be
+*> useful for a post festum analysis.
+*> \endverbatim
+*>
+*> \param[in] LWORK
+*> \verbatim
+*> LWORK is INTEGER
+*> length of WORK, WORK >= MAX(6,M+N)
+*> \endverbatim
+*>
+*> \param[out] INFO
+*> \verbatim
+*> INFO is INTEGER
+*> = 0 : successful exit.
+*> < 0 : if INFO = -i, then the i-th argument had an illegal value
+*> > 0 : DGESVJ did not converge in the maximal allowed number (30)
+*> of sweeps. The output may still be useful. See the
+*> description of WORK.
+*> \endverbatim
+*
+* Authors:
+* ========
+*
+*> \author Univ. of Tennessee
+*> \author Univ. of California Berkeley
+*> \author Univ. of Colorado Denver
+*> \author NAG Ltd.
+*
+*> \date December 2016
+*
+*> \ingroup doubleGEcomputational
+*
+*> \par Further Details:
+* =====================
+*>
+*> \verbatim
+*>
+*> The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
+*> rotations. The rotations are implemented as fast scaled rotations of
+*> Anda and Park [1]. In the case of underflow of the Jacobi angle, a
+*> modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
+*> column interchanges of de Rijk [2]. The relative accuracy of the computed
+*> singular values and the accuracy of the computed singular vectors (in
+*> angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
+*> The condition number that determines the accuracy in the full rank case
+*> is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
+*> spectral condition number. The best performance of this Jacobi SVD
+*> procedure is achieved if used in an accelerated version of Drmac and
+*> Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
+*> Some tunning parameters (marked with [TP]) are available for the
+*> implementer.
+*> The computational range for the nonzero singular values is the machine
+*> number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
+*> denormalized singular values can be computed with the corresponding
+*> gradual loss of accurate digits.
+*> \endverbatim
+*
+*> \par Contributors:
+* ==================
+*>
+*> \verbatim
+*>
+*> ============
+*>
+*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
+*> \endverbatim
+*
+*> \par References:
+* ================
+*>
+*> \verbatim
+*>
+*> [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
+*> SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
+*> [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
+*> singular value decomposition on a vector computer.
+*> SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
+*> [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
+*> [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
+*> value computation in floating point arithmetic.
+*> SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
+*> [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
+*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
+*> LAPACK Working note 169.
+*> [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
+*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
+*> LAPACK Working note 170.
+*> [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
+*> QSVD, (H,K)-SVD computations.
+*> Department of Mathematics, University of Zagreb, 2008.
+*> \endverbatim
+*
+*> \par Bugs, examples and comments:
+* =================================
+*>
+*> \verbatim
+*> ===========================
+*> Please report all bugs and send interesting test examples and comments to
+*> drmac@math.hr. Thank you.
+*> \endverbatim
+*>
+* =====================================================================
+ SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
+ $ LDV, WORK, LWORK, INFO )
*
+* -- LAPACK computational 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..--
+* December 2016
*
-* This routine is also part of SIGMA (version 1.23, October 23. 2008.)
-* SIGMA is a library of algorithms for highly accurate algorithms for
-* computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the
-* eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.
-*
- IMPLICIT NONE
-* ..
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDV, LWORK, M, MV, N
CHARACTER*1 JOBA, JOBU, JOBV
@@ -26,237 +351,11 @@
$ WORK( LWORK )
* ..
*
-* Purpose
-* =======
-*
-* DGESVJ computes the singular value decomposition (SVD) of a real
-* M-by-N matrix A, where M >= N. The SVD of A is written as
-* [++] [xx] [x0] [xx]
-* A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx]
-* [++] [xx]
-* where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
-* matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
-* of SIGMA are the singular values of A. The columns of U and V are the
-* left and the right singular vectors of A, respectively.
-*
-* Further Details
-* ~~~~~~~~~~~~~~~
-* The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
-* rotations. The rotations are implemented as fast scaled rotations of
-* Anda and Park [1]. In the case of underflow of the Jacobi angle, a
-* modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses
-* column interchanges of de Rijk [2]. The relative accuracy of the computed
-* singular values and the accuracy of the computed singular vectors (in
-* angle metric) is as guaranteed by the theory of Demmel and Veselic [3].
-* The condition number that determines the accuracy in the full rank case
-* is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
-* spectral condition number. The best performance of this Jacobi SVD
-* procedure is achieved if used in an accelerated version of Drmac and
-* Veselic [5,6], and it is the kernel routine in the SIGMA library [7].
-* Some tunning parameters (marked with [TP]) are available for the
-* implementer.
-* The computational range for the nonzero singular values is the machine
-* number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
-* denormalized singular values can be computed with the corresponding
-* gradual loss of accurate digits.
-*
-* Contributors
-* ~~~~~~~~~~~~
-* Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
-*
-* References
-* ~~~~~~~~~~
-* [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
-* SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.
-* [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
-* singular value decomposition on a vector computer.
-* SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
-* [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
-* [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular
-* value computation in floating point arithmetic.
-* SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
-* [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
-* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
-* LAPACK Working note 169.
-* [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
-* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
-* LAPACK Working note 170.
-* [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
-* QSVD, (H,K)-SVD computations.
-* Department of Mathematics, University of Zagreb, 2008.
-*
-* Bugs, Examples and Comments
-* ~~~~~~~~~~~~~~~~~~~~~~~~~~~
-* Please report all bugs and send interesting test examples and comments to
-* drmac@math.hr. Thank you.
-*
-* Arguments
-* =========
-*
-* JOBA (input) CHARACTER* 1
-* Specifies the structure of A.
-* = 'L': The input matrix A is lower triangular;
-* = 'U': The input matrix A is upper triangular;
-* = 'G': The input matrix A is general M-by-N matrix, M >= N.
-*
-* JOBU (input) CHARACTER*1
-* Specifies whether to compute the left singular vectors
-* (columns of U):
-* = 'U': The left singular vectors corresponding to the nonzero
-* singular values are computed and returned in the leading
-* columns of A. See more details in the description of A.
-* The default numerical orthogonality threshold is set to
-* approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E').
-* = 'C': Analogous to JOBU='U', except that user can control the
-* level of numerical orthogonality of the computed left
-* singular vectors. TOL can be set to TOL = CTOL*EPS, where
-* CTOL is given on input in the array WORK.
-* No CTOL smaller than ONE is allowed. CTOL greater
-* than 1 / EPS is meaningless. The option 'C'
-* can be used if M*EPS is satisfactory orthogonality
-* of the computed left singular vectors, so CTOL=M could
-* save few sweeps of Jacobi rotations.
-* See the descriptions of A and WORK(1).
-* = 'N': The matrix U is not computed. However, see the
-* description of A.
-*
-* JOBV (input) CHARACTER*1
-* Specifies whether to compute the right singular vectors, that
-* is, the matrix V:
-* = 'V' : the matrix V is computed and returned in the array V
-* = 'A' : the Jacobi rotations are applied to the MV-by-N
-* array V. In other words, the right singular vector
-* matrix V is not computed explicitly, instead it is
-* applied to an MV-by-N matrix initially stored in the
-* first MV rows of V.
-* = 'N' : the matrix V is not computed and the array V is not
-* referenced
-*
-* M (input) INTEGER
-* The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
-*
-* N (input) INTEGER
-* The number of columns of the input matrix A.
-* M >= N >= 0.
-*
-* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
-* On entry, the M-by-N matrix A.
-* On exit :
-* If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C' :
-* If INFO .EQ. 0 :
-* RANKA orthonormal columns of U are returned in the
-* leading RANKA columns of the array A. Here RANKA <= N
-* is the number of computed singular values of A that are
-* above the underflow threshold DLAMCH('S'). The singular
-* vectors corresponding to underflowed or zero singular
-* values are not computed. The value of RANKA is returned
-* in the array WORK as RANKA=NINT(WORK(2)). Also see the
-* descriptions of SVA and WORK. The computed columns of U
-* are mutually numerically orthogonal up to approximately
-* TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
-* see the description of JOBU.
-* If INFO .GT. 0 :
-* the procedure DGESVJ did not converge in the given number
-* of iterations (sweeps). In that case, the computed
-* columns of U may not be orthogonal up to TOL. The output
-* U (stored in A), SIGMA (given by the computed singular
-* values in SVA(1:N)) and V is still a decomposition of the
-* input matrix A in the sense that the residual
-* ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small.
-*
-* If JOBU .EQ. 'N' :
-* If INFO .EQ. 0 :
-* Note that the left singular vectors are 'for free' in the
-* one-sided Jacobi SVD algorithm. However, if only the
-* singular values are needed, the level of numerical
-* orthogonality of U is not an issue and iterations are
-* stopped when the columns of the iterated matrix are
-* numerically orthogonal up to approximately M*EPS. Thus,
-* on exit, A contains the columns of U scaled with the
-* corresponding singular values.
-* If INFO .GT. 0 :
-* the procedure DGESVJ did not converge in the given number
-* of iterations (sweeps).
-*
-* LDA (input) INTEGER
-* The leading dimension of the array A. LDA >= max(1,M).
-*
-* SVA (workspace/output) DOUBLE PRECISION array, dimension (N)
-* On exit :
-* If INFO .EQ. 0 :
-* depending on the value SCALE = WORK(1), we have:
-* If SCALE .EQ. ONE :
-* SVA(1:N) contains the computed singular values of A.
-* During the computation SVA contains the Euclidean column
-* norms of the iterated matrices in the array A.
-* If SCALE .NE. ONE :
-* The singular values of A are SCALE*SVA(1:N), and this
-* factored representation is due to the fact that some of the
-* singular values of A might underflow or overflow.
-* If INFO .GT. 0 :
-* the procedure DGESVJ did not converge in the given number of
-* iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
-*
-* MV (input) INTEGER
-* If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ
-* is applied to the first MV rows of V. See the description of JOBV.
-*
-* V (input/output) DOUBLE PRECISION array, dimension (LDV,N)
-* If JOBV = 'V', then V contains on exit the N-by-N matrix of
-* the right singular vectors;
-* If JOBV = 'A', then V contains the product of the computed right
-* singular vector matrix and the initial matrix in
-* the array V.
-* If JOBV = 'N', then V is not referenced.
-*
-* LDV (input) INTEGER
-* The leading dimension of the array V, LDV .GE. 1.
-* If JOBV .EQ. 'V', then LDV .GE. max(1,N).
-* If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .
-*
-* WORK (input/workspace/output) DOUBLE PRECISION array, dimension max(4,M+N).
-* On entry :
-* If JOBU .EQ. 'C' :
-* WORK(1) = CTOL, where CTOL defines the threshold for convergence.
-* The process stops if all columns of A are mutually
-* orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
-* It is required that CTOL >= ONE, i.e. it is not
-* allowed to force the routine to obtain orthogonality
-* below EPS.
-* On exit :
-* WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
-* are the computed singular values of A.
-* (See description of SVA().)
-* WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
-* singular values.
-* WORK(3) = NINT(WORK(3)) is the number of the computed singular
-* values that are larger than the underflow threshold.
-* WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
-* rotations needed for numerical convergence.
-* WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
-* This is useful information in cases when DGESVJ did
-* not converge, as it can be used to estimate whether
-* the output is stil useful and for post festum analysis.
-* WORK(6) = the largest absolute value over all sines of the
-* Jacobi rotation angles in the last sweep. It can be
-* useful for a post festum analysis.
-*
-* LWORK (input) INTEGER
-* length of WORK, WORK >= MAX(6,M+N)
-*
-* INFO (output) INTEGER
-* = 0 : successful exit.
-* < 0 : if INFO = -i, then the i-th argument had an illegal value
-* > 0 : DGESVJ did not converge in the maximal allowed number (30)
-* of sweeps. The output may still be useful. See the
-* description of WORK.
-*
* =====================================================================
*
* .. Local Parameters ..
- DOUBLE PRECISION ZERO, HALF, ONE, TWO
- PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0,
- $ TWO = 2.0D0 )
+ DOUBLE PRECISION ZERO, HALF, ONE
+ PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0)
INTEGER NSWEEP
PARAMETER ( NSWEEP = 30 )
* ..
@@ -277,7 +376,7 @@
DOUBLE PRECISION FASTR( 5 )
* ..
* .. Intrinsic Functions ..
- INTRINSIC DABS, DMAX1, DMIN1, DBLE, MIN0, DSIGN, DSQRT
+ INTRINSIC DABS, MAX, MIN, DBLE, DSIGN, DSQRT
* ..
* .. External Functions ..
* ..
@@ -331,7 +430,7 @@
INFO = -11
ELSE IF( UCTOL .AND. ( WORK( 1 ).LE.ONE ) ) THEN
INFO = -12
- ELSE IF( LWORK.LT.MAX0( M+N, 6 ) ) THEN
+ ELSE IF( LWORK.LT.MAX( M+N, 6 ) ) THEN
INFO = -13
ELSE
INFO = 0
@@ -497,8 +596,8 @@
AAPP = ZERO
AAQQ = BIG
DO 4781 p = 1, N
- IF( SVA( p ).NE.ZERO )AAQQ = DMIN1( AAQQ, SVA( p ) )
- AAPP = DMAX1( AAPP, SVA( p ) )
+ IF( SVA( p ).NE.ZERO )AAQQ = MIN( AAQQ, SVA( p ) )
+ AAPP = MAX( AAPP, SVA( p ) )
4781 CONTINUE
*
* #:) Quick return for zero matrix
@@ -539,19 +638,19 @@
TEMP1 = DSQRT( BIG / DBLE( N ) )
IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR.
$ ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN
- TEMP1 = DMIN1( BIG, TEMP1 / AAPP )
+ TEMP1 = MIN( BIG, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN
- TEMP1 = DMIN1( SN / AAQQ, BIG / ( AAPP*DSQRT( DBLE( N ) ) ) )
+ TEMP1 = MIN( SN / AAQQ, BIG / ( AAPP*DSQRT( DBLE( N ) ) ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
- TEMP1 = DMAX1( SN / AAQQ, TEMP1 / AAPP )
+ TEMP1 = MAX( SN / AAQQ, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
- TEMP1 = DMIN1( SN / AAQQ, BIG / ( DSQRT( DBLE( N ) )*AAPP ) )
+ TEMP1 = MIN( SN / AAQQ, BIG / ( DSQRT( DBLE( N ) )*AAPP ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE
@@ -592,7 +691,7 @@
* The boundaries are determined dynamically, based on the number of
* pivots above a threshold.
*
- KBL = MIN0( 8, N )
+ KBL = MIN( 8, N )
*[TP] KBL is a tuning parameter that defines the tile size in the
* tiling of the p-q loops of pivot pairs. In general, an optimal
* value of KBL depends on the matrix dimensions and on the
@@ -604,7 +703,7 @@
BLSKIP = KBL**2
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
*
- ROWSKIP = MIN0( 5, KBL )
+ ROWSKIP = MIN( 5, KBL )
*[TP] ROWSKIP is a tuning parameter.
*
LKAHEAD = 1
@@ -615,7 +714,7 @@
* invokes cubic convergence. Big part of this cycle is done inside
* canonical subspaces of dimensions less than M.
*
- IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX0( 64, 4*KBL ) ) ) THEN
+ IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX( 64, 4*KBL ) ) ) THEN
*[TP] The number of partition levels and the actual partition are
* tuning parameters.
N4 = N / 4
@@ -713,11 +812,11 @@
*
igl = ( ibr-1 )*KBL + 1
*
- DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr )
+ DO 1002 ir1 = 0, MIN( LKAHEAD, NBL-ibr )
*
igl = igl + ir1*KBL
*
- DO 2001 p = igl, MIN0( igl+KBL-1, N-1 )
+ DO 2001 p = igl, MIN( igl+KBL-1, N-1 )
*
* .. de Rijk's pivoting
*
@@ -766,7 +865,7 @@
*
PSKIPPED = 0
*
- DO 2002 q = p + 1, MIN0( igl+KBL-1, N )
+ DO 2002 q = p + 1, MIN( igl+KBL-1, N )
*
AAQQ = SVA( q )
*
@@ -805,7 +904,7 @@
END IF
END IF
*
- MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
+ MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
@@ -838,11 +937,11 @@
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, DABS( T ) )
+ MXSINJ = MAX( MXSINJ, DABS( T ) )
*
ELSE
*
@@ -854,10 +953,10 @@
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
*
- MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ MXSINJ = MAX( MXSINJ, DABS( SN ) )
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = WORK( p ) / WORK( q )
@@ -971,9 +1070,9 @@
$ A( 1, q ), 1 )
CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M,
$ 1, A( 1, q ), LDA, IERR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, SFMIN )
+ MXSINJ = MAX( MXSINJ, SFMIN )
END IF
* END IF ROTOK THEN ... ELSE
*
@@ -1039,7 +1138,7 @@
ELSE
SVA( p ) = AAPP
IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) )
- $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p
+ $ NOTROT = NOTROT + MIN( igl+KBL-1, N ) - p
END IF
*
2001 CONTINUE
@@ -1059,14 +1158,14 @@
* doing the block at ( ibr, jbc )
*
IJBLSK = 0
- DO 2100 p = igl, MIN0( igl+KBL-1, N )
+ DO 2100 p = igl, MIN( igl+KBL-1, N )
*
AAPP = SVA( p )
IF( AAPP.GT.ZERO ) THEN
*
PSKIPPED = 0
*
- DO 2200 q = jgl, MIN0( jgl+KBL-1, N )
+ DO 2200 q = jgl, MIN( jgl+KBL-1, N )
*
AAQQ = SVA( q )
IF( AAQQ.GT.ZERO ) THEN
@@ -1116,7 +1215,7 @@
END IF
END IF
*
- MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) )
+ MXAAPQ = MAX( MXAAPQ, DABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
@@ -1144,11 +1243,11 @@
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, DABS( T ) )
+ MXSINJ = MAX( MXSINJ, DABS( T ) )
ELSE
*
* .. choose correct signum for THETA and rotate
@@ -1159,10 +1258,10 @@
$ DSQRT( ONE+THETA*THETA ) )
CS = DSQRT( ONE / ( ONE+T*T ) )
SN = T*CS
- MXSINJ = DMAX1( MXSINJ, DABS( SN ) )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ MXSINJ = MAX( MXSINJ, DABS( SN ) )
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
- AAPP = AAPP*DSQRT( DMAX1( ZERO,
+ AAPP = AAPP*DSQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = WORK( p ) / WORK( q )
@@ -1279,9 +1378,9 @@
CALL DLASCL( 'G', 0, 0, ONE, AAQQ,
$ M, 1, A( 1, q ), LDA,
$ IERR )
- SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO,
+ SVA( q ) = AAQQ*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, SFMIN )
+ MXSINJ = MAX( MXSINJ, SFMIN )
ELSE
CALL DCOPY( M, A( 1, q ), 1,
$ WORK( N+1 ), 1 )
@@ -1297,9 +1396,9 @@
CALL DLASCL( 'G', 0, 0, ONE, AAPP,
$ M, 1, A( 1, p ), LDA,
$ IERR )
- SVA( p ) = AAPP*DSQRT( DMAX1( ZERO,
+ SVA( p ) = AAPP*DSQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
- MXSINJ = DMAX1( MXSINJ, SFMIN )
+ MXSINJ = MAX( MXSINJ, SFMIN )
END IF
END IF
* END IF ROTOK THEN ... ELSE
@@ -1369,7 +1468,7 @@
ELSE
*
IF( AAPP.EQ.ZERO )NOTROT = NOTROT +
- $ MIN0( jgl+KBL-1, N ) - jgl + 1
+ $ MIN( jgl+KBL-1, N ) - jgl + 1
IF( AAPP.LT.ZERO )NOTROT = 0
*
END IF
@@ -1380,7 +1479,7 @@
* end of the jbc-loop
2011 CONTINUE
*2011 bailed out of the jbc-loop
- DO 2012 p = igl, MIN0( igl+KBL-1, N )
+ DO 2012 p = igl, MIN( igl+KBL-1, N )
SVA( p ) = DABS( SVA( p ) )
2012 CONTINUE
***
@@ -1476,11 +1575,11 @@
END IF
*
* Undo scaling, if necessary (and possible).
- IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG /
- $ SKL) ) ) .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( N2 ).GT.
+ IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG / SKL) ) )
+ $ .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( MAX( N2, 1 ) ) .GT.
$ ( SFMIN / SKL) ) ) ) THEN
DO 2400 p = 1, N
- SVA( p ) = SKL*SVA( p )
+ SVA( P ) = SKL*SVA( P )
2400 CONTINUE
SKL= ONE
END IF