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cgejsv.f
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*> \brief \b CGEJSV
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGEJSV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgejsv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgejsv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgejsv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
* M, N, A, LDA, SVA, U, LDU, V, LDV,
* CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* IMPLICIT NONE
* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
* ..
* .. Array Arguments ..
* COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
* REAL SVA( N ), RWORK( LRWORK )
* INTEGER IWORK( * )
* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
*> matrix [A], where M >= N. The SVD of [A] is written as
*>
*> [A] = [U] * [SIGMA] * [V]^*,
*>
*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
*> diagonal elements, [U] is an M-by-N (or 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]. The columns of [U] and [V] are the left and
*> the right singular vectors of [A], respectively. The matrices [U] and [V]
*> are computed and stored in the arrays U and V, respectively. The diagonal
*> of [SIGMA] is computed and stored in the array SVA.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBA
*> \verbatim
*> JOBA is CHARACTER*1
*> Specifies the level of accuracy:
*> = 'C': This option works well (high relative accuracy) if A = B * D,
*> with well-conditioned B and arbitrary diagonal matrix D.
*> The accuracy cannot be spoiled by COLUMN scaling. The
*> accuracy of the computed output depends on the condition of
*> B, and the procedure aims at the best theoretical accuracy.
*> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
*> bounded by f(M,N)*epsilon* cond(B), independent of D.
*> The input matrix is preprocessed with the QRF with column
*> pivoting. This initial preprocessing and preconditioning by
*> a rank revealing QR factorization is common for all values of
*> JOBA. Additional actions are specified as follows:
*> = 'E': Computation as with 'C' with an additional estimate of the
*> condition number of B. It provides a realistic error bound.
*> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
*> D1, D2, and well-conditioned matrix C, this option gives
*> higher accuracy than the 'C' option. If the structure of the
*> input matrix is not known, and relative accuracy is
*> desirable, then this option is advisable. The input matrix A
*> is preprocessed with QR factorization with FULL (row and
*> column) pivoting.
*> = 'G': Computation as with 'F' with an additional estimate of the
*> condition number of B, where A=B*D. If A has heavily weighted
*> rows, then using this condition number gives too pessimistic
*> error bound.
*> = 'A': Small singular values are not well determined by the data
*> and are considered as noisy; the matrix is treated as
*> numerically rank deficient. The error in the computed
*> singular values is bounded by f(m,n)*epsilon*||A||.
*> The computed SVD A = U * S * V^* restores A up to
*> f(m,n)*epsilon*||A||.
*> This gives the procedure the licence to discard (set to zero)
*> all singular values below N*epsilon*||A||.
*> = 'R': Similar as in 'A'. Rank revealing property of the initial
*> QR factorization is used do reveal (using triangular factor)
*> a gap sigma_{r+1} < epsilon * sigma_r in which case the
*> numerical RANK is declared to be r. The SVD is computed with
*> absolute error bounds, but more accurately than with 'A'.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> Specifies whether to compute the columns of U:
*> = 'U': N columns of U are returned in the array U.
*> = 'F': full set of M left sing. vectors is returned in the array U.
*> = 'W': U may be used as workspace of length M*N. See the description
*> of U.
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether to compute the matrix V:
*> = 'V': N columns of V are returned in the array V; Jacobi rotations
*> are not explicitly accumulated.
*> = 'J': N columns of V are returned in the array V, but they are
*> computed as the product of Jacobi rotations, if JOBT = 'N'.
*> = 'W': V may be used as workspace of length N*N. See the description
*> of V.
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBR
*> \verbatim
*> JOBR is CHARACTER*1
*> Specifies the RANGE for the singular values. Issues the licence to
*> set to zero small positive singular values if they are outside
*> specified range. If A .NE. 0 is scaled so that the largest singular
*> value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
*> the licence to kill columns of A whose norm in c*A is less than
*> SQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN,
*> where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
*> = 'N': Do not kill small columns of c*A. This option assumes that
*> BLAS and QR factorizations and triangular solvers are
*> implemented to work in that range. If the condition of A
*> is greater than BIG, use CGESVJ.
*> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
*> (roughly, as described above). This option is recommended.
*> ===========================
*> For computing the singular values in the FULL range [SFMIN,BIG]
*> use CGESVJ.
*> \endverbatim
*>
*> \param[in] JOBT
*> \verbatim
*> JOBT is CHARACTER*1
*> If the matrix is square then the procedure may determine to use
*> transposed A if A^* seems to be better with respect to convergence.
*> If the matrix is not square, JOBT is ignored.
*> The decision is based on two values of entropy over the adjoint
*> orbit of A^* * A. See the descriptions of RWORK(6) and RWORK(7).
*> = 'T': transpose if entropy test indicates possibly faster
*> convergence of Jacobi process if A^* is taken as input. If A is
*> replaced with A^*, then the row pivoting is included automatically.
*> = 'N': do not speculate.
*> The option 'T' can be used to compute only the singular values, or
*> the full SVD (U, SIGMA and V). For only one set of singular vectors
*> (U or V), the caller should provide both U and V, as one of the
*> matrices is used as workspace if the matrix A is transposed.
*> The implementer can easily remove this constraint and make the
*> code more complicated. See the descriptions of U and V.
*> In general, this option is considered experimental, and 'N'; should
*> be preferred. This is subject to changes in the future.
*> \endverbatim
*>
*> \param[in] JOBP
*> \verbatim
*> JOBP is CHARACTER*1
*> Issues the licence to introduce structured perturbations to drown
*> denormalized numbers. This licence should be active if the
*> denormals are poorly implemented, causing slow computation,
*> especially in cases of fast convergence (!). For details see [1,2].
*> For the sake of simplicity, this perturbations are included only
*> when the full SVD or only the singular values are requested. The
*> implementer/user can easily add the perturbation for the cases of
*> computing one set of singular vectors.
*> = 'P': introduce perturbation
*> = 'N': do not perturb
*> \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. M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> On entry, the M-by-N matrix A.
*> \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 REAL array, dimension (N)
*> On exit,
*> - For RWORK(1)/RWORK(2) = ONE: The singular values of A. During
*> the computation SVA contains Euclidean column norms of the
*> iterated matrices in the array A.
*> - For RWORK(1) .NE. RWORK(2): The singular values of A are
*> (RWORK(1)/RWORK(2)) * SVA(1:N). This factored form is used if
*> sigma_max(A) overflows or if small singular values have been
*> saved from underflow by scaling the input matrix A.
*> - If JOBR='R' then some of the singular values may be returned
*> as exact zeros obtained by "set to zero" because they are
*> below the numerical rank threshold or are denormalized numbers.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is COMPLEX array, dimension ( LDU, N ) or ( LDU, M )
*> If JOBU = 'U', then U contains on exit the M-by-N matrix of
*> the left singular vectors.
*> If JOBU = 'F', then U contains on exit the M-by-M matrix of
*> the left singular vectors, including an ONB
*> of the orthogonal complement of the Range(A).
*> If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N),
*> then U is used as workspace if the procedure
*> replaces A with A^*. In that case, [V] is computed
*> in U as left singular vectors of A^* and then
*> copied back to the V array. This 'W' option is just
*> a reminder to the caller that in this case U is
*> reserved as workspace of length N*N.
*> If JOBU = 'N' U is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U, LDU >= 1.
*> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is COMPLEX array, dimension ( LDV, N )
*> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
*> the right singular vectors;
*> If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N),
*> then V is used as workspace if the procedure
*> replaces A with A^*. In that case, [U] is computed
*> in V as right singular vectors of A^* and then
*> copied back to the U array. This 'W' option is just
*> a reminder to the caller that in this case V is
*> reserved as workspace of length N*N.
*> If JOBV = 'N' V is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V, LDV >= 1.
*> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
*> \endverbatim
*>
*> \param[out] CWORK
*> \verbatim
*> CWORK is COMPLEX array, dimension (MAX(2,LWORK))
*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
*> LRWORK=-1), then on exit CWORK(1) contains the required length of
*> CWORK for the job parameters used in the call.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> Length of CWORK to confirm proper allocation of workspace.
*> LWORK depends on the job:
*>
*> 1. If only SIGMA is needed ( JOBU = 'N', JOBV = 'N' ) and
*> 1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
*> LWORK >= 2*N+1. This is the minimal requirement.
*> ->> For optimal performance (blocked code) the optimal value
*> is LWORK >= N + (N+1)*NB. Here NB is the optimal
*> block size for CGEQP3 and CGEQRF.
*> In general, optimal LWORK is computed as
*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ)).
*> 1.2. .. an estimate of the scaled condition number of A is
*> required (JOBA='E', or 'G'). In this case, LWORK the minimal
*> requirement is LWORK >= N*N + 2*N.
*> ->> For optimal performance (blocked code) the optimal value
*> is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
*> In general, the optimal length LWORK is computed as
*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ),
*> N*N+LWORK(CPOCON)).
*> 2. If SIGMA and the right singular vectors are needed (JOBV = 'V'),
*> (JOBU = 'N')
*> 2.1 .. no scaled condition estimate requested (JOBE = 'N'):
*> -> the minimal requirement is LWORK >= 3*N.
*> -> For optimal performance,
*> LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
*> where NB is the optimal block size for CGEQP3, CGEQRF, CGELQF,
*> CUNMLQ. In general, the optimal length LWORK is computed as
*> LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CGESVJ),
*> N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
*> 2.2 .. an estimate of the scaled condition number of A is
*> required (JOBA='E', or 'G').
*> -> the minimal requirement is LWORK >= 3*N.
*> -> For optimal performance,
*> LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
*> where NB is the optimal block size for CGEQP3, CGEQRF, CGELQF,
*> CUNMLQ. In general, the optimal length LWORK is computed as
*> LWORK >= max(N+LWORK(CGEQP3), LWORK(CPOCON), N+LWORK(CGESVJ),
*> N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
*> 3. If SIGMA and the left singular vectors are needed
*> 3.1 .. no scaled condition estimate requested (JOBE = 'N'):
*> -> the minimal requirement is LWORK >= 3*N.
*> -> For optimal performance:
*> if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
*> where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
*> In general, the optimal length LWORK is computed as
*> LWORK >= max(N+LWORK(CGEQP3), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
*> 3.2 .. an estimate of the scaled condition number of A is
*> required (JOBA='E', or 'G').
*> -> the minimal requirement is LWORK >= 3*N.
*> -> For optimal performance:
*> if JOBU = 'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
*> where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
*> In general, the optimal length LWORK is computed as
*> LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CPOCON),
*> 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
*>
*> 4. If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
*> 4.1. if JOBV = 'V'
*> the minimal requirement is LWORK >= 5*N+2*N*N.
*> 4.2. if JOBV = 'J' the minimal requirement is
*> LWORK >= 4*N+N*N.
*> In both cases, the allocated CWORK can accommodate blocked runs
*> of CGEQP3, CGEQRF, CGELQF, CUNMQR, CUNMLQ.
*>
*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
*> LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
*> minimal length of CWORK for the job parameters used in the call.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (MAX(7,LRWORK))
*> On exit,
*> RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
*> such that SCALE*SVA(1:N) are the computed singular values
*> of A. (See the description of SVA().)
*> RWORK(2) = See the description of RWORK(1).
*> RWORK(3) = SCONDA is an estimate for the condition number of
*> column equilibrated A. (If JOBA = 'E' or 'G')
*> SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
*> It is computed using CPOCON. It holds
*> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
*> where R is the triangular factor from the QRF of A.
*> However, if R is truncated and the numerical rank is
*> determined to be strictly smaller than N, SCONDA is
*> returned as -1, thus indicating that the smallest
*> singular values might be lost.
*>
*> If full SVD is needed, the following two condition numbers are
*> useful for the analysis of the algorithm. They are provided for
*> a developer/implementer who is familiar with the details of
*> the method.
*>
*> RWORK(4) = an estimate of the scaled condition number of the
*> triangular factor in the first QR factorization.
*> RWORK(5) = an estimate of the scaled condition number of the
*> triangular factor in the second QR factorization.
*> The following two parameters are computed if JOBT = 'T'.
*> They are provided for a developer/implementer who is familiar
*> with the details of the method.
*> RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
*> of diag(A^* * A) / Trace(A^* * A) taken as point in the
*> probability simplex.
*> RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
*> If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
*> LRWORK=-1), then on exit RWORK(1) contains the required length of
*> RWORK for the job parameters used in the call.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*> LRWORK is INTEGER
*> Length of RWORK to confirm proper allocation of workspace.
*> LRWORK depends on the job:
*>
*> 1. If only the singular values are requested i.e. if
*> LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
*> then:
*> 1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*> then: LRWORK = max( 7, 2 * M ).
*> 1.2. Otherwise, LRWORK = max( 7, N ).
*> 2. If singular values with the right singular vectors are requested
*> i.e. if
*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
*> .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
*> then:
*> 2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*> then LRWORK = max( 7, 2 * M ).
*> 2.2. Otherwise, LRWORK = max( 7, N ).
*> 3. If singular values with the left singular vectors are requested, i.e. if
*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
*> .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
*> then:
*> 3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*> then LRWORK = max( 7, 2 * M ).
*> 3.2. Otherwise, LRWORK = max( 7, N ).
*> 4. If singular values with both the left and the right singular vectors
*> are requested, i.e. if
*> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
*> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
*> then:
*> 4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
*> then LRWORK = max( 7, 2 * M ).
*> 4.2. Otherwise, LRWORK = max( 7, N ).
*>
*> If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and
*> the length of RWORK is returned in RWORK(1).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, of dimension at least 4, that further depends
*> on the job:
*>
*> 1. If only the singular values are requested then:
*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
*> 2. If the singular values and the right singular vectors are requested then:
*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
*> 3. If the singular values and the left singular vectors are requested then:
*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
*> 4. If the singular values with both the left and the right singular vectors
*> are requested, then:
*> 4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
*> then the length of IWORK is N+M; otherwise the length of IWORK is N.
*> 4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
*> If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
*> then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*N.
*>
*> On exit,
*> IWORK(1) = the numerical rank determined after the initial
*> QR factorization with pivoting. See the descriptions
*> of JOBA and JOBR.
*> IWORK(2) = the number of the computed nonzero singular values
*> IWORK(3) = if nonzero, a warning message:
*> If IWORK(3) = 1 then some of the column norms of A
*> were denormalized floats. The requested high accuracy
*> is not warranted by the data.
*> IWORK(4) = 1 or -1. If IWORK(4) = 1, then the procedure used A^* to
*> do the job as specified by the JOB parameters.
*> If the call to CGEJSV is a workspace query (indicated by LWORK = -1 and
*> LRWORK = -1), then on exit IWORK(1) contains the required length of
*> IWORK for the job parameters used in the call.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> < 0: if INFO = -i, then the i-th argument had an illegal value.
*> = 0: successful exit;
*> > 0: CGEJSV did not converge in the maximal allowed number
*> of sweeps. The computed values may be inaccurate.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexGEsing
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> CGEJSV implements a preconditioned Jacobi SVD algorithm. It uses CGEQP3,
*> CGEQRF, and CGELQF as preprocessors and preconditioners. Optionally, an
*> additional row pivoting can be used as a preprocessor, which in some
*> cases results in much higher accuracy. An example is matrix A with the
*> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
*> diagonal matrices and C is well-conditioned matrix. In that case, complete
*> pivoting in the first QR factorizations provides accuracy dependent on the
*> condition number of C, and independent of D1, D2. Such higher accuracy is
*> not completely understood theoretically, but it works well in practice.
*> Further, if A can be written as A = B*D, with well-conditioned B and some
*> diagonal D, then the high accuracy is guaranteed, both theoretically and
*> in software, independent of D. For more details see [1], [2].
*> The computational range for the singular values can be the full range
*> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
*> & LAPACK routines called by CGEJSV are implemented to work in that range.
*> If that is not the case, then the restriction for safe computation with
*> the singular values in the range of normalized IEEE numbers is that the
*> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
*> overflow. This code (CGEJSV) is best used in this restricted range,
*> meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
*> returned as zeros. See JOBR for details on this.
*> Further, this implementation is somewhat slower than the one described
*> in [1,2] due to replacement of some non-LAPACK components, and because
*> the choice of some tuning parameters in the iterative part (CGESVJ) is
*> left to the implementer on a particular machine.
*> The rank revealing QR factorization (in this code: CGEQP3) should be
*> implemented as in [3]. We have a new version of CGEQP3 under development
*> that is more robust than the current one in LAPACK, with a cleaner cut in
*> rank deficient cases. It will be available in the SIGMA library [4].
*> If M is much larger than N, it is obvious that the initial QRF with
*> column pivoting can be preprocessed by the QRF without pivoting. That
*> well known trick is not used in CGEJSV because in some cases heavy row
*> weighting can be treated with complete pivoting. The overhead in cases
*> M much larger than N is then only due to pivoting, but the benefits in
*> terms of accuracy have prevailed. The implementer/user can incorporate
*> this extra QRF step easily. The implementer can also improve data movement
*> (matrix transpose, matrix copy, matrix transposed copy) - this
*> implementation of CGEJSV uses only the simplest, naive data movement.
*> \endverbatim
*
*> \par Contributor:
* ==================
*>
*> Zlatko Drmac (Zagreb, Croatia)
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] 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.
*> [2] 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.
*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
*> factorization software - a case study.
*> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
*> LAPACK Working note 176.
*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*> QSVD, (H,K)-SVD computations.
*> Department of Mathematics, University of Zagreb, 2008, 2016.
*> \endverbatim
*
*> \par Bugs, examples and comments:
* =================================
*>
*> Please report all bugs and send interesting examples and/or comments to
*> [email protected]. Thank you.
*>
* =====================================================================
SUBROUTINE CGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
$ M, N, A, LDA, SVA, U, LDU, V, LDV,
$ CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
IMPLICIT NONE
INTEGER INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
REAL SVA( N ), RWORK( LRWORK )
INTEGER IWORK( * )
CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
* ..
*
* ===========================================================================
*
* .. Local Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ), CONE = ( 1.0E0, 0.0E0 ) )
* ..
* .. Local Scalars ..
COMPLEX CTEMP
REAL AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
$ CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN, MAXPRJ, SCALEM,
$ SCONDA, SFMIN, SMALL, TEMP1, USCAL1, USCAL2, XSC
INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LQUERY,
$ LSVEC, L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN, NOSCAL,
$ ROWPIV, RSVEC, TRANSP
*
INTEGER OPTWRK, MINWRK, MINRWRK, MINIWRK
INTEGER LWCON, LWLQF, LWQP3, LWQRF, LWUNMLQ, LWUNMQR, LWUNMQRM,
$ LWSVDJ, LWSVDJV, LRWQP3, LRWCON, LRWSVDJ, IWOFF
INTEGER LWRK_CGELQF, LWRK_CGEQP3, LWRK_CGEQP3N, LWRK_CGEQRF,
$ LWRK_CGESVJ, LWRK_CGESVJV, LWRK_CGESVJU, LWRK_CUNMLQ,
$ LWRK_CUNMQR, LWRK_CUNMQRM
* ..
* .. Local Arrays
COMPLEX CDUMMY(1)
REAL RDUMMY(1)
*
* .. Intrinsic Functions ..
INTRINSIC ABS, CMPLX, CONJG, ALOG, MAX, MIN, REAL, NINT, SQRT
* ..
* .. External Functions ..
REAL SLAMCH, SCNRM2
INTEGER ISAMAX, ICAMAX
LOGICAL LSAME
EXTERNAL ISAMAX, ICAMAX, LSAME, SLAMCH, SCNRM2
* ..
* .. External Subroutines ..
EXTERNAL SLASSQ, CCOPY, CGELQF, CGEQP3, CGEQRF, CLACPY, CLAPMR,
$ CLASCL, SLASCL, CLASET, CLASSQ, CLASWP, CUNGQR, CUNMLQ,
$ CUNMQR, CPOCON, SSCAL, CSSCAL, CSWAP, CTRSM, CLACGV,
$ XERBLA
*
EXTERNAL CGESVJ
* ..
*
* Test the input arguments
*
LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
JRACC = LSAME( JOBV, 'J' )
RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
L2RANK = LSAME( JOBA, 'R' )
L2ABER = LSAME( JOBA, 'A' )
ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
L2TRAN = LSAME( JOBT, 'T' ) .AND. ( M .EQ. N )
L2KILL = LSAME( JOBR, 'R' )
DEFR = LSAME( JOBR, 'N' )
L2PERT = LSAME( JOBP, 'P' )
*
LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )
*
IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
$ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
INFO = - 1
ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
$ ( LSAME( JOBU, 'W' ) .AND. RSVEC .AND. L2TRAN ) ) ) THEN
INFO = - 2
ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
$ ( LSAME( JOBV, 'W' ) .AND. LSVEC .AND. L2TRAN ) ) ) THEN
INFO = - 3
ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
INFO = - 4
ELSE IF ( .NOT. ( LSAME(JOBT,'T') .OR. LSAME(JOBT,'N') ) ) THEN
INFO = - 5
ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
INFO = - 6
ELSE IF ( M .LT. 0 ) THEN
INFO = - 7
ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
INFO = - 8
ELSE IF ( LDA .LT. M ) THEN
INFO = - 10
ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
INFO = - 13
ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
INFO = - 15
ELSE
* #:)
INFO = 0
END IF
*
IF ( INFO .EQ. 0 ) THEN
* .. compute the minimal and the optimal workspace lengths
* [[The expressions for computing the minimal and the optimal
* values of LCWORK, LRWORK are written with a lot of redundancy and
* can be simplified. However, this verbose form is useful for
* maintenance and modifications of the code.]]
*
* .. minimal workspace length for CGEQP3 of an M x N matrix,
* CGEQRF of an N x N matrix, CGELQF of an N x N matrix,
* CUNMLQ for computing N x N matrix, CUNMQR for computing N x N
* matrix, CUNMQR for computing M x N matrix, respectively.
LWQP3 = N+1
LWQRF = MAX( 1, N )
LWLQF = MAX( 1, N )
LWUNMLQ = MAX( 1, N )
LWUNMQR = MAX( 1, N )
LWUNMQRM = MAX( 1, M )
* .. minimal workspace length for CPOCON of an N x N matrix
LWCON = 2 * N
* .. minimal workspace length for CGESVJ of an N x N matrix,
* without and with explicit accumulation of Jacobi rotations
LWSVDJ = MAX( 2 * N, 1 )
LWSVDJV = MAX( 2 * N, 1 )
* .. minimal REAL workspace length for CGEQP3, CPOCON, CGESVJ
LRWQP3 = 2 * N
LRWCON = N
LRWSVDJ = N
IF ( LQUERY ) THEN
CALL CGEQP3( M, N, A, LDA, IWORK, CDUMMY, CDUMMY, -1,
$ RDUMMY, IERR )
LWRK_CGEQP3 = INT( CDUMMY(1) )
CALL CGEQRF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
LWRK_CGEQRF = INT( CDUMMY(1) )
CALL CGELQF( N, N, A, LDA, CDUMMY, CDUMMY,-1, IERR )
LWRK_CGELQF = INT( CDUMMY(1) )
END IF
MINWRK = 2
OPTWRK = 2
MINIWRK = N
IF ( .NOT. (LSVEC .OR. RSVEC ) ) THEN
* .. minimal and optimal sizes of the complex workspace if
* only the singular values are requested
IF ( ERREST ) THEN
MINWRK = MAX( N+LWQP3, N**2+LWCON, N+LWQRF, LWSVDJ )
ELSE
MINWRK = MAX( N+LWQP3, N+LWQRF, LWSVDJ )
END IF
IF ( LQUERY ) THEN
CALL CGESVJ( 'L', 'N', 'N', N, N, A, LDA, SVA, N, V,
$ LDV, CDUMMY, -1, RDUMMY, -1, IERR )
LWRK_CGESVJ = INT( CDUMMY(1) )
IF ( ERREST ) THEN
OPTWRK = MAX( N+LWRK_CGEQP3, N**2+LWCON,
$ N+LWRK_CGEQRF, LWRK_CGESVJ )
ELSE
OPTWRK = MAX( N+LWRK_CGEQP3, N+LWRK_CGEQRF,
$ LWRK_CGESVJ )
END IF
END IF
IF ( L2TRAN .OR. ROWPIV ) THEN
IF ( ERREST ) THEN
MINRWRK = MAX( 7, 2*M, LRWQP3, LRWCON, LRWSVDJ )
ELSE
MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
END IF
ELSE
IF ( ERREST ) THEN
MINRWRK = MAX( 7, LRWQP3, LRWCON, LRWSVDJ )
ELSE
MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
END IF
END IF
IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
ELSE IF ( RSVEC .AND. (.NOT.LSVEC) ) THEN
* .. minimal and optimal sizes of the complex workspace if the
* singular values and the right singular vectors are requested
IF ( ERREST ) THEN
MINWRK = MAX( N+LWQP3, LWCON, LWSVDJ, N+LWLQF,
$ 2*N+LWQRF, N+LWSVDJ, N+LWUNMLQ )
ELSE
MINWRK = MAX( N+LWQP3, LWSVDJ, N+LWLQF, 2*N+LWQRF,
$ N+LWSVDJ, N+LWUNMLQ )
END IF
IF ( LQUERY ) THEN
CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
$ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
LWRK_CGESVJ = INT( CDUMMY(1) )
CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
$ V, LDV, CDUMMY, -1, IERR )
LWRK_CUNMLQ = INT( CDUMMY(1) )
IF ( ERREST ) THEN
OPTWRK = MAX( N+LWRK_CGEQP3, LWCON, LWRK_CGESVJ,
$ N+LWRK_CGELQF, 2*N+LWRK_CGEQRF,
$ N+LWRK_CGESVJ, N+LWRK_CUNMLQ )
ELSE
OPTWRK = MAX( N+LWRK_CGEQP3, LWRK_CGESVJ,N+LWRK_CGELQF,
$ 2*N+LWRK_CGEQRF, N+LWRK_CGESVJ,
$ N+LWRK_CUNMLQ )
END IF
END IF
IF ( L2TRAN .OR. ROWPIV ) THEN
IF ( ERREST ) THEN
MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
ELSE
MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
END IF
ELSE
IF ( ERREST ) THEN
MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
ELSE
MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
END IF
END IF
IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
ELSE IF ( LSVEC .AND. (.NOT.RSVEC) ) THEN
* .. minimal and optimal sizes of the complex workspace if the
* singular values and the left singular vectors are requested
IF ( ERREST ) THEN
MINWRK = N + MAX( LWQP3,LWCON,N+LWQRF,LWSVDJ,LWUNMQRM )
ELSE
MINWRK = N + MAX( LWQP3, N+LWQRF, LWSVDJ, LWUNMQRM )
END IF
IF ( LQUERY ) THEN
CALL CGESVJ( 'L', 'U', 'N', N,N, U, LDU, SVA, N, A,
$ LDA, CDUMMY, -1, RDUMMY, -1, IERR )
LWRK_CGESVJ = INT( CDUMMY(1) )
CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
$ LDU, CDUMMY, -1, IERR )
LWRK_CUNMQRM = INT( CDUMMY(1) )
IF ( ERREST ) THEN
OPTWRK = N + MAX( LWRK_CGEQP3, LWCON, N+LWRK_CGEQRF,
$ LWRK_CGESVJ, LWRK_CUNMQRM )
ELSE
OPTWRK = N + MAX( LWRK_CGEQP3, N+LWRK_CGEQRF,
$ LWRK_CGESVJ, LWRK_CUNMQRM )
END IF
END IF
IF ( L2TRAN .OR. ROWPIV ) THEN
IF ( ERREST ) THEN
MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
ELSE
MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ )
END IF
ELSE
IF ( ERREST ) THEN
MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
ELSE
MINRWRK = MAX( 7, LRWQP3, LRWSVDJ )
END IF
END IF
IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
ELSE
* .. minimal and optimal sizes of the complex workspace if the
* full SVD is requested
IF ( .NOT. JRACC ) THEN
IF ( ERREST ) THEN
MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+N**2+LWCON,
$ 2*N+LWQRF, 2*N+LWQP3,
$ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
$ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
$ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
$ N+N**2+LWSVDJ, N+LWUNMQRM )
ELSE
MINWRK = MAX( N+LWQP3, 2*N+N**2+LWCON,
$ 2*N+LWQRF, 2*N+LWQP3,
$ 2*N+N**2+N+LWLQF, 2*N+N**2+N+N**2+LWCON,
$ 2*N+N**2+N+LWSVDJ, 2*N+N**2+N+LWSVDJV,
$ 2*N+N**2+N+LWUNMQR,2*N+N**2+N+LWUNMLQ,
$ N+N**2+LWSVDJ, N+LWUNMQRM )
END IF
MINIWRK = MINIWRK + N
IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
ELSE
IF ( ERREST ) THEN
MINWRK = MAX( N+LWQP3, N+LWCON, 2*N+LWQRF,
$ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
$ N+LWUNMQRM )
ELSE
MINWRK = MAX( N+LWQP3, 2*N+LWQRF,
$ 2*N+N**2+LWSVDJV, 2*N+N**2+N+LWUNMQR,
$ N+LWUNMQRM )
END IF
IF ( ROWPIV .OR. L2TRAN ) MINIWRK = MINIWRK + M
END IF
IF ( LQUERY ) THEN
CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
$ LDU, CDUMMY, -1, IERR )
LWRK_CUNMQRM = INT( CDUMMY(1) )
CALL CUNMQR( 'L', 'N', N, N, N, A, LDA, CDUMMY, U,
$ LDU, CDUMMY, -1, IERR )
LWRK_CUNMQR = INT( CDUMMY(1) )
IF ( .NOT. JRACC ) THEN
CALL CGEQP3( N,N, A, LDA, IWORK, CDUMMY,CDUMMY, -1,
$ RDUMMY, IERR )
LWRK_CGEQP3N = INT( CDUMMY(1) )
CALL CGESVJ( 'L', 'U', 'N', N, N, U, LDU, SVA,
$ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
LWRK_CGESVJ = INT( CDUMMY(1) )
CALL CGESVJ( 'U', 'U', 'N', N, N, U, LDU, SVA,
$ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
LWRK_CGESVJU = INT( CDUMMY(1) )
CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
$ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
LWRK_CGESVJV = INT( CDUMMY(1) )
CALL CUNMLQ( 'L', 'C', N, N, N, A, LDA, CDUMMY,
$ V, LDV, CDUMMY, -1, IERR )
LWRK_CUNMLQ = INT( CDUMMY(1) )
IF ( ERREST ) THEN
OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON,
$ 2*N+N**2+LWCON, 2*N+LWRK_CGEQRF,
$ 2*N+LWRK_CGEQP3N,
$ 2*N+N**2+N+LWRK_CGELQF,
$ 2*N+N**2+N+N**2+LWCON,
$ 2*N+N**2+N+LWRK_CGESVJ,
$ 2*N+N**2+N+LWRK_CGESVJV,
$ 2*N+N**2+N+LWRK_CUNMQR,
$ 2*N+N**2+N+LWRK_CUNMLQ,
$ N+N**2+LWRK_CGESVJU,
$ N+LWRK_CUNMQRM )
ELSE
OPTWRK = MAX( N+LWRK_CGEQP3,
$ 2*N+N**2+LWCON, 2*N+LWRK_CGEQRF,
$ 2*N+LWRK_CGEQP3N,
$ 2*N+N**2+N+LWRK_CGELQF,
$ 2*N+N**2+N+N**2+LWCON,
$ 2*N+N**2+N+LWRK_CGESVJ,
$ 2*N+N**2+N+LWRK_CGESVJV,
$ 2*N+N**2+N+LWRK_CUNMQR,
$ 2*N+N**2+N+LWRK_CUNMLQ,
$ N+N**2+LWRK_CGESVJU,
$ N+LWRK_CUNMQRM )
END IF
ELSE
CALL CGESVJ( 'L', 'U', 'V', N, N, U, LDU, SVA,
$ N, V, LDV, CDUMMY, -1, RDUMMY, -1, IERR )
LWRK_CGESVJV = INT( CDUMMY(1) )
CALL CUNMQR( 'L', 'N', N, N, N, CDUMMY, N, CDUMMY,
$ V, LDV, CDUMMY, -1, IERR )
LWRK_CUNMQR = INT( CDUMMY(1) )
CALL CUNMQR( 'L', 'N', M, N, N, A, LDA, CDUMMY, U,
$ LDU, CDUMMY, -1, IERR )
LWRK_CUNMQRM = INT( CDUMMY(1) )
IF ( ERREST ) THEN
OPTWRK = MAX( N+LWRK_CGEQP3, N+LWCON,
$ 2*N+LWRK_CGEQRF, 2*N+N**2,
$ 2*N+N**2+LWRK_CGESVJV,
$ 2*N+N**2+N+LWRK_CUNMQR,N+LWRK_CUNMQRM )
ELSE
OPTWRK = MAX( N+LWRK_CGEQP3, 2*N+LWRK_CGEQRF,
$ 2*N+N**2, 2*N+N**2+LWRK_CGESVJV,
$ 2*N+N**2+N+LWRK_CUNMQR,
$ N+LWRK_CUNMQRM )
END IF
END IF
END IF
IF ( L2TRAN .OR. ROWPIV ) THEN
MINRWRK = MAX( 7, 2*M, LRWQP3, LRWSVDJ, LRWCON )
ELSE
MINRWRK = MAX( 7, LRWQP3, LRWSVDJ, LRWCON )
END IF
END IF
MINWRK = MAX( 2, MINWRK )
OPTWRK = MAX( OPTWRK, MINWRK )
IF ( LWORK .LT. MINWRK .AND. (.NOT.LQUERY) ) INFO = - 17
IF ( LRWORK .LT. MINRWRK .AND. (.NOT.LQUERY) ) INFO = - 19
END IF
*
IF ( INFO .NE. 0 ) THEN
* #:(
CALL XERBLA( 'CGEJSV', - INFO )
RETURN
ELSE IF ( LQUERY ) THEN
CWORK(1) = OPTWRK
CWORK(2) = MINWRK
RWORK(1) = MINRWRK
IWORK(1) = MAX( 4, MINIWRK )
RETURN
END IF
*
* Quick return for void matrix (Y3K safe)
* #:)
IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
IWORK(1:4) = 0
RWORK(1:7) = 0
RETURN
ENDIF
*
* Determine whether the matrix U should be M x N or M x M
*
IF ( LSVEC ) THEN
N1 = N
IF ( LSAME( JOBU, 'F' ) ) N1 = M
END IF
*
* Set numerical parameters
*
*! NOTE: Make sure SLAMCH() does not fail on the target architecture.
*
EPSLN = SLAMCH('Epsilon')
SFMIN = SLAMCH('SafeMinimum')
SMALL = SFMIN / EPSLN
BIG = SLAMCH('O')
* BIG = ONE / SFMIN
*
* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
*
*(!) If necessary, scale SVA() to protect the largest norm from
* overflow. It is possible that this scaling pushes the smallest
* column norm left from the underflow threshold (extreme case).
*
SCALEM = ONE / SQRT(REAL(M)*REAL(N))
NOSCAL = .TRUE.
GOSCAL = .TRUE.
DO 1874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL CLASSQ( M, A(1,p), 1, AAPP, AAQQ )
IF ( AAPP .GT. BIG ) THEN
INFO = - 9
CALL XERBLA( 'CGEJSV', -INFO )
RETURN
END IF
AAQQ = SQRT(AAQQ)
IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN