linalg: add functions needed for MSCG

*> \brief <b> DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices</b>

*

*  =========== DOCUMENTATION ===========

*

* Online html documentation available at

*            http://www.netlib.org/lapack/explore-html/

*

*> \htmlonly

*> Download DGELSD + dependencies

*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgelsd.f">

*> [TGZ]</a>

*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgelsd.f">

*> [ZIP]</a>

*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgelsd.f">

*> [TXT]</a>

*> \endhtmlonly

*

*  Definition:

*  ===========

*

*       SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,

*                          WORK, LWORK, IWORK, INFO )

*

*       .. Scalar Arguments ..

*       INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

*       DOUBLE PRECISION   RCOND

*       ..

*       .. Array Arguments ..

*       INTEGER            IWORK( * )

*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )

*       ..

*

*

*> \par Purpose:

*  =============

*>

*> \verbatim

*>

*> DGELSD computes the minimum-norm solution to a real linear least

*> squares problem:

*>     minimize 2-norm(| b - A*x |)

*> using the singular value decomposition (SVD) of A. A is an M-by-N

*> matrix which may be rank-deficient.

*>

*> Several right hand side vectors b and solution vectors x can be

*> handled in a single call; they are stored as the columns of the

*> M-by-NRHS right hand side matrix B and the N-by-NRHS solution

*> matrix X.

*>

*> The problem is solved in three steps:

*> (1) Reduce the coefficient matrix A to bidiagonal form with

*>     Householder transformations, reducing the original problem

*>     into a "bidiagonal least squares problem" (BLS)

*> (2) Solve the BLS using a divide and conquer approach.

*> (3) Apply back all the Householder transformations to solve

*>     the original least squares problem.

*>

*> The effective rank of A is determined by treating as zero those

*> singular values which are less than RCOND times the largest singular

*> value.

*>

*> 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] M

*> \verbatim

*>          M is INTEGER

*>          The number of rows of A. M >= 0.

*> \endverbatim

*>

*> \param[in] N

*> \verbatim

*>          N is INTEGER

*>          The number of columns of A. N >= 0.

*> \endverbatim

*>

*> \param[in] NRHS

*> \verbatim

*>          NRHS is INTEGER

*>          The number of right hand sides, i.e., the number of columns

*>          of the matrices B and X. NRHS >= 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, A has been destroyed.

*> \endverbatim

*>

*> \param[in] LDA

*> \verbatim

*>          LDA is INTEGER

*>          The leading dimension of the array A.  LDA >= max(1,M).

*> \endverbatim

*>

*> \param[in,out] B

*> \verbatim

*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)

*>          On entry, the M-by-NRHS right hand side matrix B.

*>          On exit, B is overwritten by the N-by-NRHS solution

*>          matrix X.  If m >= n and RANK = n, the residual

*>          sum-of-squares for the solution in the i-th column is given

*>          by the sum of squares of elements n+1:m in that column.

*> \endverbatim

*>

*> \param[in] LDB

*> \verbatim

*>          LDB is INTEGER

*>          The leading dimension of the array B. LDB >= max(1,max(M,N)).

*> \endverbatim

*>

*> \param[out] S

*> \verbatim

*>          S is DOUBLE PRECISION array, dimension (min(M,N))

*>          The singular values of A in decreasing order.

*>          The condition number of A in the 2-norm = S(1)/S(min(m,n)).

*> \endverbatim

*>

*> \param[in] RCOND

*> \verbatim

*>          RCOND is DOUBLE PRECISION

*>          RCOND is used to determine the effective rank of A.

*>          Singular values S(i) <= RCOND*S(1) are treated as zero.

*>          If RCOND < 0, machine precision is used instead.

*> \endverbatim

*>

*> \param[out] RANK

*> \verbatim

*>          RANK is INTEGER

*>          The effective rank of A, i.e., the number of singular values

*>          which are greater than RCOND*S(1).

*> \endverbatim

*>

*> \param[out] WORK

*> \verbatim

*>          WORK is DOUBLE PRECISION 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 must be at least 1.

*>          The exact minimum amount of workspace needed depends on M,

*>          N and NRHS. As long as LWORK is at least

*>              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,

*>          if M is greater than or equal to N or

*>              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,

*>          if M is less than N, the code will execute correctly.

*>          SMLSIZ is returned by ILAENV and is equal to the maximum

*>          size of the subproblems at the bottom of the computation

*>          tree (usually about 25), and

*>             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )

*>          For good performance, LWORK should generally be larger.

*>

*>          If LWORK = -1, then a workspace query is assumed; the routine

*>          only calculates the optimal size of the WORK array, returns

*>          this value as the first entry of the WORK array, and no error

*>          message related to LWORK is issued by XERBLA.

*> \endverbatim

*>

*> \param[out] IWORK

*> \verbatim

*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))

*>          LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN),

*>          where MINMN = MIN( M,N ).

*>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

*> \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 algorithm for computing the SVD failed to converge;

*>                if INFO = i, i off-diagonal elements of an intermediate

*>                bidiagonal form did not converge to zero.

*> \endverbatim

*

*  Authors:

*  ========

*

*> \author Univ. of Tennessee

*> \author Univ. of California Berkeley

*> \author Univ. of Colorado Denver

*> \author NAG Ltd.

*

*> \date June 2017

*

*> \ingroup doubleGEsolve

*

*> \par Contributors:

*  ==================

*>

*>     Ming Gu and Ren-Cang Li, Computer Science Division, University of

*>       California at Berkeley, USA \n

*>     Osni Marques, LBNL/NERSC, USA \n

*

*  =====================================================================

      SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,

     $                   WORK, LWORK, IWORK, INFO )

*

*  -- LAPACK driver routine (version 3.7.1) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     June 2017

*

*     .. Scalar Arguments ..

      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

      DOUBLE PRECISION   RCOND

*     ..

*     .. Array Arguments ..

      INTEGER            IWORK( * )

      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )

*     ..

*

*  =====================================================================

*

*     .. Parameters ..

      DOUBLE PRECISION   ZERO, ONE, TWO

      PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )

*     ..

*     .. Local Scalars ..

      LOGICAL            LQUERY

      INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,

     $                   LDWORK, LIWORK, MAXMN, MAXWRK, MINMN, MINWRK,

     $                   MM, MNTHR, NLVL, NWORK, SMLSIZ, WLALSD

      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM

*     ..

*     .. External Subroutines ..

      EXTERNAL           DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,

     $                   DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA

*     ..

*     .. External Functions ..

      INTEGER            ILAENV

      DOUBLE PRECISION   DLAMCH, DLANGE

      EXTERNAL           ILAENV, DLAMCH, DLANGE

*     ..

*     .. Intrinsic Functions ..

      INTRINSIC          DBLE, INT, LOG, MAX, MIN

*     ..

*     .. Executable Statements ..

*

*     Test the input arguments.

*

      INFO = 0

      MINMN = MIN( M, N )

      MAXMN = MAX( M, N )

      MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )

      LQUERY = ( LWORK.EQ.-1 )

      IF( M.LT.0 ) THEN

         INFO = -1

      ELSE IF( N.LT.0 ) THEN

         INFO = -2

      ELSE IF( NRHS.LT.0 ) THEN

         INFO = -3

      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN

         INFO = -5

      ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN

         INFO = -7

      END IF

*

      SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )

*

*     Compute workspace.

*     (Note: Comments in the code beginning "Workspace:" describe the

*     minimal amount of workspace needed at that point in the code,

*     as well as the preferred amount for good performance.

*     NB refers to the optimal block size for the immediately

*     following subroutine, as returned by ILAENV.)

*

      MINWRK = 1

      LIWORK = 1

      MINMN = MAX( 1, MINMN )

      NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /

     $       LOG( TWO ) ) + 1, 0 )

*

      IF( INFO.EQ.0 ) THEN

         MAXWRK = 0

         LIWORK = 3*MINMN*NLVL + 11*MINMN

         MM = M

         IF( M.GE.N .AND. M.GE.MNTHR ) THEN

*

*           Path 1a - overdetermined, with many more rows than columns.

*

            MM = N

            MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,

     $               -1, -1 ) )

            MAXWRK = MAX( MAXWRK, N+NRHS*

     $               ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )

         END IF

         IF( M.GE.N ) THEN

*

*           Path 1 - overdetermined or exactly determined.

*

            MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*

     $               ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )

            MAXWRK = MAX( MAXWRK, 3*N+NRHS*

     $               ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )

            MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*

     $               ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )

            WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2

            MAXWRK = MAX( MAXWRK, 3*N+WLALSD )

            MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )

         END IF

         IF( N.GT.M ) THEN

            WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2

            IF( N.GE.MNTHR ) THEN

*

*              Path 2a - underdetermined, with many more columns

*              than rows.

*

               MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )

               MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*

     $                  ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )

               MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*

     $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )

               MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*

     $                  ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )

               IF( NRHS.GT.1 ) THEN

                  MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )

               ELSE

                  MAXWRK = MAX( MAXWRK, M*M+2*M )

               END IF

               MAXWRK = MAX( MAXWRK, M+NRHS*

     $                  ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )

               MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )

!     XXX: Ensure the Path 2a case below is triggered.  The workspace

!     calculation should use queries for all routines eventually.

               MAXWRK = MAX( MAXWRK,

     $              4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )

            ELSE

*

*              Path 2 - remaining underdetermined cases.

*

               MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,

     $                  -1, -1 )

               MAXWRK = MAX( MAXWRK, 3*M+NRHS*

     $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )

               MAXWRK = MAX( MAXWRK, 3*M+M*

     $                  ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )

               MAXWRK = MAX( MAXWRK, 3*M+WLALSD )

            END IF

            MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )

         END IF

         MINWRK = MIN( MINWRK, MAXWRK )

         WORK( 1 ) = MAXWRK

         IWORK( 1 ) = LIWORK

         IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN

            INFO = -12

         END IF

      END IF

*

      IF( INFO.NE.0 ) THEN

         CALL XERBLA( 'DGELSD', -INFO )

         RETURN

      ELSE IF( LQUERY ) THEN

         GO TO 10

      END IF

*

*     Quick return if possible.

*

      IF( M.EQ.0 .OR. N.EQ.0 ) THEN

         RANK = 0

         RETURN

      END IF

*

*     Get machine parameters.

*

      EPS = DLAMCH( 'P' )

      SFMIN = DLAMCH( 'S' )

      SMLNUM = SFMIN / EPS

      BIGNUM = ONE / SMLNUM

      CALL DLABAD( SMLNUM, BIGNUM )

*

*     Scale A if max entry outside range [SMLNUM,BIGNUM].

*

      ANRM = DLANGE( 'M', M, N, A, LDA, WORK )

      IASCL = 0

      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN

*

*        Scale matrix norm up to SMLNUM.

*

         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )

         IASCL = 1

      ELSE IF( ANRM.GT.BIGNUM ) THEN

*

*        Scale matrix norm down to BIGNUM.

*

         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )

         IASCL = 2

      ELSE IF( ANRM.EQ.ZERO ) THEN

*

*        Matrix all zero. Return zero solution.

*

         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )

         CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )

         RANK = 0

         GO TO 10

      END IF