schkbd.f

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*> \brief \b SCHKBD
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SCHKBD( NSIZES, MVAL, NVAL, NTYPES, DOTYPE, NRHS,
*                          ISEED, THRESH, A, LDA, BD, BE, S1, S2, X, LDX,
*                          Y, Z, Q, LDQ, PT, LDPT, U, VT, WORK, LWORK,
*                          IWORK, NOUT, INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDPT, LDQ, LDX, LWORK, NOUT, NRHS,
*      $                   NSIZES, NTYPES
*       REAL               THRESH
*       ..
*       .. Array Arguments ..
*       LOGICAL            DOTYPE( * )
*       INTEGER            ISEED( 4 ), IWORK( * ), MVAL( * ), NVAL( * )
*       REAL               A( LDA, * ), BD( * ), BE( * ), PT( LDPT, * ),
*      $                   Q( LDQ, * ), S1( * ), S2( * ), U( LDPT, * ),
*      $                   VT( LDPT, * ), WORK( * ), X( LDX, * ),
*      $                   Y( LDX, * ), Z( LDX, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SCHKBD checks the singular value decomposition (SVD) routines.
*>
*> SGEBRD reduces a real general m by n matrix A to upper or lower
*> bidiagonal form B by an orthogonal transformation:  Q' * A * P = B
*> (or A = Q * B * P').  The matrix B is upper bidiagonal if m >= n
*> and lower bidiagonal if m < n.
*>
*> SORGBR generates the orthogonal matrices Q and P' from SGEBRD.
*> Note that Q and P are not necessarily square.
*>
*> SBDSQR computes the singular value decomposition of the bidiagonal
*> matrix B as B = U S V'.  It is called three times to compute
*>    1)  B = U S1 V', where S1 is the diagonal matrix of singular
*>        values and the columns of the matrices U and V are the left
*>        and right singular vectors, respectively, of B.
*>    2)  Same as 1), but the singular values are stored in S2 and the
*>        singular vectors are not computed.
*>    3)  A = (UQ) S (P'V'), the SVD of the original matrix A.
*> In addition, SBDSQR has an option to apply the left orthogonal matrix
*> U to a matrix X, useful in least squares applications.
*>
*> SBDSDC computes the singular value decomposition of the bidiagonal
*> matrix B as B = U S V' using divide-and-conquer. It is called twice
*> to compute
*>    1) B = U S1 V', where S1 is the diagonal matrix of singular
*>        values and the columns of the matrices U and V are the left
*>        and right singular vectors, respectively, of B.
*>    2) Same as 1), but the singular values are stored in S2 and the
*>        singular vectors are not computed.
*>
*> For each pair of matrix dimensions (M,N) and each selected matrix
*> type, an M by N matrix A and an M by NRHS matrix X are generated.
*> The problem dimensions are as follows
*>    A:          M x N
*>    Q:          M x min(M,N) (but M x M if NRHS > 0)
*>    P:          min(M,N) x N
*>    B:          min(M,N) x min(M,N)
*>    U, V:       min(M,N) x min(M,N)
*>    S1, S2      diagonal, order min(M,N)
*>    X:          M x NRHS
*>
*> For each generated matrix, 14 tests are performed:
*>
*> Test SGEBRD and SORGBR
*>
*> (1)   | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'
*>
*> (2)   | I - Q' Q | / ( M ulp )
*>
*> (3)   | I - PT PT' | / ( N ulp )
*>
*> Test SBDSQR on bidiagonal matrix B
*>
*> (4)   | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
*>
*> (5)   | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X
*>                                                  and   Z = U' Y.
*> (6)   | I - U' U | / ( min(M,N) ulp )
*>
*> (7)   | I - VT VT' | / ( min(M,N) ulp )
*>
*> (8)   S1 contains min(M,N) nonnegative values in decreasing order.
*>       (Return 0 if true, 1/ULP if false.)
*>
*> (9)   | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
*>                                   computing U and V.
*>
*> (10)  0 if the true singular values of B are within THRESH of
*>       those in S1.  2*THRESH if they are not.  (Tested using
*>       SSVDCH)
*>
*> Test SBDSQR on matrix A
*>
*> (11)  | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp )
*>
*> (12)  | X - (QU) Z | / ( |X| max(M,k) ulp )
*>
*> (13)  | I - (QU)'(QU) | / ( M ulp )
*>
*> (14)  | I - (VT PT) (PT'VT') | / ( N ulp )
*>
*> Test SBDSDC on bidiagonal matrix B
*>
*> (15)  | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
*>
*> (16)  | I - U' U | / ( min(M,N) ulp )
*>
*> (17)  | I - VT VT' | / ( min(M,N) ulp )
*>
*> (18)  S1 contains min(M,N) nonnegative values in decreasing order.
*>       (Return 0 if true, 1/ULP if false.)
*>
*> (19)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
*>                                   computing U and V.
*> The possible matrix types are
*>
*> (1)  The zero matrix.
*> (2)  The identity matrix.
*>
*> (3)  A diagonal matrix with evenly spaced entries
*>      1, ..., ULP  and random signs.
*>      (ULP = (first number larger than 1) - 1 )
*> (4)  A diagonal matrix with geometrically spaced entries
*>      1, ..., ULP  and random signs.
*> (5)  A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
*>      and random signs.
*>
*> (6)  Same as (3), but multiplied by SQRT( overflow threshold )
*> (7)  Same as (3), but multiplied by SQRT( underflow threshold )
*>
*> (8)  A matrix of the form  U D V, where U and V are orthogonal and
*>      D has evenly spaced entries 1, ..., ULP with random signs
*>      on the diagonal.
*>
*> (9)  A matrix of the form  U D V, where U and V are orthogonal and
*>      D has geometrically spaced entries 1, ..., ULP with random
*>      signs on the diagonal.
*>
*> (10) A matrix of the form  U D V, where U and V are orthogonal and
*>      D has "clustered" entries 1, ULP,..., ULP with random
*>      signs on the diagonal.
*>
*> (11) Same as (8), but multiplied by SQRT( overflow threshold )
*> (12) Same as (8), but multiplied by SQRT( underflow threshold )
*>
*> (13) Rectangular matrix with random entries chosen from (-1,1).
*> (14) Same as (13), but multiplied by SQRT( overflow threshold )
*> (15) Same as (13), but multiplied by SQRT( underflow threshold )
*>
*> Special case:
*> (16) A bidiagonal matrix with random entries chosen from a
*>      logarithmic distribution on [ulp^2,ulp^(-2)]  (I.e., each
*>      entry is  e^x, where x is chosen uniformly on
*>      [ 2 log(ulp), -2 log(ulp) ] .)  For *this* type:
*>      (a) SGEBRD is not called to reduce it to bidiagonal form.
*>      (b) the bidiagonal is  min(M,N) x min(M,N); if M<N, the
*>          matrix will be lower bidiagonal, otherwise upper.
*>      (c) only tests 5--8 and 14 are performed.
*>
*> A subset of the full set of matrix types may be selected through
*> the logical array DOTYPE.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NSIZES
*> \verbatim
*>          NSIZES is INTEGER
*>          The number of values of M and N contained in the vectors
*>          MVAL and NVAL.  The matrix sizes are used in pairs (M,N).
*> \endverbatim
*>
*> \param[in] MVAL
*> \verbatim
*>          MVAL is INTEGER array, dimension (NM)
*>          The values of the matrix row dimension M.
*> \endverbatim
*>
*> \param[in] NVAL
*> \verbatim
*>          NVAL is INTEGER array, dimension (NM)
*>          The values of the matrix column dimension N.
*> \endverbatim
*>
*> \param[in] NTYPES
*> \verbatim
*>          NTYPES is INTEGER
*>          The number of elements in DOTYPE.   If it is zero, SCHKBD
*>          does nothing.  It must be at least zero.  If it is MAXTYP+1
*>          and NSIZES is 1, then an additional type, MAXTYP+1 is
*>          defined, which is to use whatever matrices are in A and B.
*>          This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
*>          DOTYPE(MAXTYP+1) is .TRUE. .
*> \endverbatim
*>
*> \param[in] DOTYPE
*> \verbatim
*>          DOTYPE is LOGICAL array, dimension (NTYPES)
*>          If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix
*>          of type j will be generated.  If NTYPES is smaller than the
*>          maximum number of types defined (PARAMETER MAXTYP), then
*>          types NTYPES+1 through MAXTYP will not be generated.  If
*>          NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through
*>          DOTYPE(NTYPES) will be ignored.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of columns in the "right-hand side" matrices X, Y,
*>          and Z, used in testing SBDSQR.  If NRHS = 0, then the
*>          operations on the right-hand side will not be tested.
*>          NRHS must be at least 0.
*> \endverbatim
*>
*> \param[in,out] ISEED
*> \verbatim
*>          ISEED is INTEGER array, dimension (4)
*>          On entry ISEED specifies the seed of the random number
*>          generator. The array elements should be between 0 and 4095;
*>          if not they will be reduced mod 4096.  Also, ISEED(4) must
*>          be odd.  The values of ISEED are changed on exit, and can be
*>          used in the next call to SCHKBD to continue the same random
*>          number sequence.
*> \endverbatim
*>
*> \param[in] THRESH
*> \verbatim
*>          THRESH is REAL
*>          The threshold value for the test ratios.  A result is
*>          included in the output file if RESULT >= THRESH.  To have
*>          every test ratio printed, use THRESH = 0.  Note that the
*>          expected value of the test ratios is O(1), so THRESH should
*>          be a reasonably small multiple of 1, e.g., 10 or 100.
*> \endverbatim
*>
*> \param[out] A
*> \verbatim
*>          A is REAL array, dimension (LDA,NMAX)
*>          where NMAX is the maximum value of N in NVAL.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,MMAX),
*>          where MMAX is the maximum value of M in MVAL.
*> \endverbatim
*>
*> \param[out] BD
*> \verbatim
*>          BD is REAL array, dimension
*>                      (max(min(MVAL(j),NVAL(j))))
*> \endverbatim
*>
*> \param[out] BE
*> \verbatim
*>          BE is REAL array, dimension
*>                      (max(min(MVAL(j),NVAL(j))))
*> \endverbatim
*>
*> \param[out] S1
*> \verbatim
*>          S1 is REAL array, dimension
*>                      (max(min(MVAL(j),NVAL(j))))
*> \endverbatim
*>
*> \param[out] S2
*> \verbatim
*>          S2 is REAL array, dimension
*>                      (max(min(MVAL(j),NVAL(j))))
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*>          X is REAL array, dimension (LDX,NRHS)
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the arrays X, Y, and Z.
*>          LDX >= max(1,MMAX)
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*>          Y is REAL array, dimension (LDX,NRHS)
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is REAL array, dimension (LDX,NRHS)
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is REAL array, dimension (LDQ,MMAX)
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q.  LDQ >= max(1,MMAX).
*> \endverbatim
*>
*> \param[out] PT
*> \verbatim
*>          PT is REAL array, dimension (LDPT,NMAX)
*> \endverbatim
*>
*> \param[in] LDPT
*> \verbatim
*>          LDPT is INTEGER
*>          The leading dimension of the arrays PT, U, and V.
*>          LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is REAL array, dimension
*>                      (LDPT,max(min(MVAL(j),NVAL(j))))
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is REAL array, dimension
*>                      (LDPT,max(min(MVAL(j),NVAL(j))))
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The number of entries in WORK.  This must be at least
*>          3(M+N) and  M(M + max(M,N,k) + 1) + N*min(M,N)  for all
*>          pairs  (M,N)=(MM(j),NN(j))
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension at least 8*min(M,N)
*> \endverbatim
*>
*> \param[in] NOUT
*> \verbatim
*>          NOUT is INTEGER
*>          The FORTRAN unit number for printing out error messages
*>          (e.g., if a routine returns IINFO not equal to 0.)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          If 0, then everything ran OK.
*>           -1: NSIZES < 0
*>           -2: Some MM(j) < 0
*>           -3: Some NN(j) < 0
*>           -4: NTYPES < 0
*>           -6: NRHS  < 0
*>           -8: THRESH < 0
*>          -11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ).
*>          -17: LDB < 1 or LDB < MMAX.
*>          -21: LDQ < 1 or LDQ < MMAX.
*>          -23: LDPT< 1 or LDPT< MNMAX.
*>          -27: LWORK too small.
*>          If  SLATMR, SLATMS, SGEBRD, SORGBR, or SBDSQR,
*>              returns an error code, the
*>              absolute value of it is returned.
*>
*>-----------------------------------------------------------------------
*>
*>     Some Local Variables and Parameters:
*>     ---- ----- --------- --- ----------
*>
*>     ZERO, ONE       Real 0 and 1.
*>     MAXTYP          The number of types defined.
*>     NTEST           The number of tests performed, or which can
*>                     be performed so far, for the current matrix.
*>     MMAX            Largest value in NN.
*>     NMAX            Largest value in NN.
*>     MNMIN           min(MM(j), NN(j)) (the dimension of the bidiagonal
*>                     matrix.)
*>     MNMAX           The maximum value of MNMIN for j=1,...,NSIZES.
*>     NFAIL           The number of tests which have exceeded THRESH
*>     COND, IMODE     Values to be passed to the matrix generators.
*>     ANORM           Norm of A; passed to matrix generators.
*>
*>     OVFL, UNFL      Overflow and underflow thresholds.
*>     RTOVFL, RTUNFL  Square roots of the previous 2 values.
*>     ULP, ULPINV     Finest relative precision and its inverse.
*>
*>             The following four arrays decode JTYPE:
*>     KTYPE(j)        The general type (1-10) for type "j".
*>     KMODE(j)        The MODE value to be passed to the matrix
*>                     generator for type "j".
*>     KMAGN(j)        The order of magnitude ( O(1),
*>                     O(overflow^(1/2) ), O(underflow^(1/2) )
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup single_eig
*
*  =====================================================================
      SUBROUTINE schkbd( NSIZES, MVAL, NVAL, NTYPES, DOTYPE, NRHS,
     $                   iseed, thresh, a, lda, bd, be, s1, s2, x, ldx,
     $                   y, z, q, ldq, pt, ldpt, u, vt, work, lwork,
     $                   iwork, nout, info )
*
*  -- LAPACK test routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      INTEGER            info, lda, ldpt, ldq, ldx, lwork, nout, nrhs,
     $                   nsizes, ntypes
      REAL               thresh
*     ..
*     .. Array Arguments ..
      LOGICAL            dotype( * )
      INTEGER            iseed( 4 ), iwork( * ), mval( * ), nval( * )
      REAL               a( lda, * ), bd( * ), be( * ), pt( ldpt, * ),
     $                   q( ldq, * ), s1( * ), s2( * ), u( ldpt, * ),
     $                   vt( ldpt, * ), work( * ), x( ldx, * ),
     $                   y( ldx, * ), z( ldx, * )
*     ..
*
* ======================================================================
*
*     .. Parameters ..
      REAL               zero, one, two, half
      parameter( zero = 0.0e0, one = 1.0e0, two = 2.0e0,
     $                   half = 0.5e0 )
      INTEGER            maxtyp
      parameter( maxtyp = 16 )
*     ..
*     .. Local Scalars ..
      LOGICAL            badmm, badnn, bidiag
      CHARACTER          uplo
      CHARACTER*3        path
      INTEGER            i, iinfo, imode, itype, j, jcol, jsize, jtype,
     $                   log2ui, m, minwrk, mmax, mnmax, mnmin, mq,
     $                   mtypes, n, nfail, nmax, ntest
      REAL               amninv, anorm, cond, ovfl, rtovfl, rtunfl,
     $                   temp1, temp2, ulp, ulpinv, unfl
*     ..
*     .. Local Arrays ..
      INTEGER            idum( 1 ), ioldsd( 4 ), kmagn( maxtyp ),
     $                   kmode( maxtyp ), ktype( maxtyp )
      REAL               dum( 1 ), dumma( 1 ), result( 19 )
*     ..
*     .. External Functions ..
      REAL               slamch, slarnd
      EXTERNAL           slamch, slarnd
*     ..
*     .. External Subroutines ..
      EXTERNAL           alasum, sbdsdc, sbdsqr, sbdt01, sbdt02, sbdt03,
     $                   scopy, sgebrd, sgemm, slabad, slacpy, slahd2,
     $                   slaset, slatmr, slatms, sorgbr, sort01, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, exp, int, log, max, min, sqrt
*     ..
*     .. Scalars in Common ..
      LOGICAL            lerr, ok
      CHARACTER*32       srnamt
      INTEGER            infot, nunit
*     ..
*     .. Common blocks ..
      common             / infoc / infot, nunit, ok, lerr
      common             / srnamc / srnamt
*     ..
*     .. Data statements ..
      DATA               ktype / 1, 2, 5*4, 5*6, 3*9, 10 /
      DATA               kmagn / 2*1, 3*1, 2, 3, 3*1, 2, 3, 1, 2, 3, 0 /
      DATA               kmode / 2*0, 4, 3, 1, 4, 4, 4, 3, 1, 4, 4, 0,
     $                   0, 0, 0 /
*     ..
*     .. Executable Statements ..
*
*     Check for errors
*
      info = 0
*
      badmm = .false.
      badnn = .false.
      mmax = 1
      nmax = 1
      mnmax = 1
      minwrk = 1
      DO 10 j = 1, nsizes
         mmax = max( mmax, mval( j ) )
         IF( mval( j ).LT.0 )
     $      badmm = .true.
         nmax = max( nmax, nval( j ) )
         IF( nval( j ).LT.0 )
     $      badnn = .true.
         mnmax = max( mnmax, min( mval( j ), nval( j ) ) )
         minwrk = max( minwrk, 3*( mval( j )+nval( j ) ),
     $            mval( j )*( mval( j )+max( mval( j ), nval( j ),
     $            nrhs )+1 )+nval( j )*min( nval( j ), mval( j ) ) )
   10 continue
*
*     Check for errors
*
      IF( nsizes.LT.0 ) THEN
         info = -1
      ELSE IF( badmm ) THEN
         info = -2
      ELSE IF( badnn ) THEN
         info = -3
      ELSE IF( ntypes.LT.0 ) THEN
         info = -4
      ELSE IF( nrhs.LT.0 ) THEN
         info = -6
      ELSE IF( lda.LT.mmax ) THEN
         info = -11
      ELSE IF( ldx.LT.mmax ) THEN
         info = -17
      ELSE IF( ldq.LT.mmax ) THEN
         info = -21
      ELSE IF( ldpt.LT.mnmax ) THEN
         info = -23
      ELSE IF( minwrk.GT.lwork ) THEN
         info = -27
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SCHKBD', -info )
         return
      END IF
*
*     Initialize constants
*
      path( 1: 1 ) = 'Single precision'
      path( 2: 3 ) = 'BD'
      nfail = 0
      ntest = 0
      unfl = slamch( 'Safe minimum' )
      ovfl = slamch( 'Overflow' )
      CALL slabad( unfl, ovfl )
      ulp = slamch( 'Precision' )
      ulpinv = one / ulp
      log2ui = int( log( ulpinv ) / log( two ) )
      rtunfl = sqrt( unfl )
      rtovfl = sqrt( ovfl )
      infot = 0
*
*     Loop over sizes, types
*
      DO 200 jsize = 1, nsizes
         m = mval( jsize )
         n = nval( jsize )
         mnmin = min( m, n )
         amninv = one / max( m, n, 1 )
*
         IF( nsizes.NE.1 ) THEN
            mtypes = min( maxtyp, ntypes )
         ELSE
            mtypes = min( maxtyp+1, ntypes )
         END IF
*
         DO 190 jtype = 1, mtypes
            IF( .NOT.dotype( jtype ) )
     $         go to 190
*
            DO 20 j = 1, 4
               ioldsd( j ) = iseed( j )
   20       continue
*
            DO 30 j = 1, 14
               result( j ) = -one
   30       continue
*
            uplo = ' '
*
*           Compute "A"
*
*           Control parameters:
*
*           KMAGN  KMODE        KTYPE
*       =1  O(1)   clustered 1  zero
*       =2  large  clustered 2  identity
*       =3  small  exponential  (none)
*       =4         arithmetic   diagonal, (w/ eigenvalues)
*       =5         random       symmetric, w/ eigenvalues
*       =6                      nonsymmetric, w/ singular values
*       =7                      random diagonal
*       =8                      random symmetric
*       =9                      random nonsymmetric
*       =10                     random bidiagonal (log. distrib.)
*
            IF( mtypes.GT.maxtyp )
     $         go to 100
*
            itype = ktype( jtype )
            imode = kmode( jtype )
*
*           Compute norm
*
            go to( 40, 50, 60 )kmagn( jtype )
*
   40       continue
            anorm = one
            go to 70
*
   50       continue
            anorm = ( rtovfl*ulp )*amninv
            go to 70
*
   60       continue
            anorm = rtunfl*max( m, n )*ulpinv
            go to 70
*
   70       continue
*
            CALL slaset( 'Full', lda, n, zero, zero, a, lda )
            iinfo = 0
            cond = ulpinv
*
            bidiag = .false.
            IF( itype.EQ.1 ) THEN
*
*              Zero matrix
*
               iinfo = 0
*
            ELSE IF( itype.EQ.2 ) THEN
*
*              Identity
*
               DO 80 jcol = 1, mnmin
                  a( jcol, jcol ) = anorm
   80          continue
*
            ELSE IF( itype.EQ.4 ) THEN
*
*              Diagonal Matrix, [Eigen]values Specified
*
               CALL slatms( mnmin, mnmin, 'S', iseed, 'N', work, imode,
     $                      cond, anorm, 0, 0, 'N', a, lda,
     $                      work( mnmin+1 ), iinfo )
*
            ELSE IF( itype.EQ.5 ) THEN
*
*              Symmetric, eigenvalues specified
*
               CALL slatms( mnmin, mnmin, 'S', iseed, 'S', work, imode,
     $                      cond, anorm, m, n, 'N', a, lda,
     $                      work( mnmin+1 ), iinfo )
*
            ELSE IF( itype.EQ.6 ) THEN
*
*              Nonsymmetric, singular values specified
*
               CALL slatms( m, n, 'S', iseed, 'N', work, imode, cond,
     $                      anorm, m, n, 'N', a, lda, work( mnmin+1 ),
     $                      iinfo )
*
            ELSE IF( itype.EQ.7 ) THEN
*
*              Diagonal, random entries
*
               CALL slatmr( mnmin, mnmin, 'S', iseed, 'N', work, 6, one,
     $                      one, 'T', 'N', work( mnmin+1 ), 1, one,
     $                      work( 2*mnmin+1 ), 1, one, 'N', iwork, 0, 0,
     $                      zero, anorm, 'NO', a, lda, iwork, iinfo )
*
            ELSE IF( itype.EQ.8 ) THEN
*
*              Symmetric, random entries
*
               CALL slatmr( mnmin, mnmin, 'S', iseed, 'S', work, 6, one,
     $                      one, 'T', 'N', work( mnmin+1 ), 1, one,
     $                      work( m+mnmin+1 ), 1, one, 'N', iwork, m, n,
     $                      zero, anorm, 'NO', a, lda, iwork, iinfo )
*
            ELSE IF( itype.EQ.9 ) THEN
*
*              Nonsymmetric, random entries
*
               CALL slatmr( m, n, 'S', iseed, 'N', work, 6, one, one,
     $                      'T', 'N', work( mnmin+1 ), 1, one,
     $                      work( m+mnmin+1 ), 1, one, 'N', iwork, m, n,
     $                      zero, anorm, 'NO', a, lda, iwork, iinfo )
*
            ELSE IF( itype.EQ.10 ) THEN
*
*              Bidiagonal, random entries
*
               temp1 = -two*log( ulp )
               DO 90 j = 1, mnmin
                  bd( j ) = exp( temp1*slarnd( 2, iseed ) )
                  IF( j.LT.mnmin )
     $               be( j ) = exp( temp1*slarnd( 2, iseed ) )
   90          continue
*
               iinfo = 0
               bidiag = .true.
               IF( m.GE.n ) THEN
                  uplo = 'U'
               ELSE
                  uplo = 'L'
               END IF
            ELSE
               iinfo = 1
            END IF
*
            IF( iinfo.EQ.0 ) THEN
*
*              Generate Right-Hand Side
*
               IF( bidiag ) THEN
                  CALL slatmr( mnmin, nrhs, 'S', iseed, 'N', work, 6,
     $                         one, one, 'T', 'N', work( mnmin+1 ), 1,
     $                         one, work( 2*mnmin+1 ), 1, one, 'N',
     $                         iwork, mnmin, nrhs, zero, one, 'NO', y,
     $                         ldx, iwork, iinfo )
               ELSE
                  CALL slatmr( m, nrhs, 'S', iseed, 'N', work, 6, one,
     $                         one, 'T', 'N', work( m+1 ), 1, one,
     $                         work( 2*m+1 ), 1, one, 'N', iwork, m,
     $                         nrhs, zero, one, 'NO', x, ldx, iwork,
     $                         iinfo )
               END IF
            END IF
*
*           Error Exit
*
            IF( iinfo.NE.0 ) THEN
               WRITE( nout, fmt = 9998 )'Generator', iinfo, m, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               return
            END IF
*
  100       continue
*
*           Call SGEBRD and SORGBR to compute B, Q, and P, do tests.
*
            IF( .NOT.bidiag ) THEN
*
*              Compute transformations to reduce A to bidiagonal form:
*              B := Q' * A * P.
*
               CALL slacpy( ' ', m, n, a, lda, q, ldq )
               CALL sgebrd( m, n, q, ldq, bd, be, work, work( mnmin+1 ),
     $                      work( 2*mnmin+1 ), lwork-2*mnmin, iinfo )
*
*              Check error code from SGEBRD.
*
               IF( iinfo.NE.0 ) THEN
                  WRITE( nout, fmt = 9998 )'SGEBRD', iinfo, m, n,
     $               jtype, ioldsd
                  info = abs( iinfo )
                  return
               END IF
*
               CALL slacpy( ' ', m, n, q, ldq, pt, ldpt )
               IF( m.GE.n ) THEN
                  uplo = 'U'
               ELSE
                  uplo = 'L'
               END IF
*
*              Generate Q
*
               mq = m
               IF( nrhs.LE.0 )
     $            mq = mnmin
               CALL sorgbr( 'Q', m, mq, n, q, ldq, work,
     $                      work( 2*mnmin+1 ), lwork-2*mnmin, iinfo )
*
*              Check error code from SORGBR.
*
               IF( iinfo.NE.0 ) THEN
                  WRITE( nout, fmt = 9998 )'SORGBR(Q)', iinfo, m, n,
     $               jtype, ioldsd
                  info = abs( iinfo )
                  return
               END IF
*
*              Generate P'
*
               CALL sorgbr( 'P', mnmin, n, m, pt, ldpt, work( mnmin+1 ),
     $                      work( 2*mnmin+1 ), lwork-2*mnmin, iinfo )
*
*              Check error code from SORGBR.
*
               IF( iinfo.NE.0 ) THEN
                  WRITE( nout, fmt = 9998 )'SORGBR(P)', iinfo, m, n,
     $               jtype, ioldsd
                  info = abs( iinfo )
                  return
               END IF
*
*              Apply Q' to an M by NRHS matrix X:  Y := Q' * X.
*
               CALL sgemm( 'Transpose', 'No transpose', m, nrhs, m, one,
     $                     q, ldq, x, ldx, zero, y, ldx )
*
*              Test 1:  Check the decomposition A := Q * B * PT
*                   2:  Check the orthogonality of Q
*                   3:  Check the orthogonality of PT
*
               CALL sbdt01( m, n, 1, a, lda, q, ldq, bd, be, pt, ldpt,
     $                      work, result( 1 ) )
               CALL sort01( 'Columns', m, mq, q, ldq, work, lwork,
     $                      result( 2 ) )
               CALL sort01( 'Rows', mnmin, n, pt, ldpt, work, lwork,
     $                      result( 3 ) )
            END IF
*
*           Use SBDSQR to form the SVD of the bidiagonal matrix B:
*           B := U * S1 * VT, and compute Z = U' * Y.
*
            CALL scopy( mnmin, bd, 1, s1, 1 )
            IF( mnmin.GT.0 )
     $         CALL scopy( mnmin-1, be, 1, work, 1 )
            CALL slacpy( ' ', m, nrhs, y, ldx, z, ldx )
            CALL slaset( 'Full', mnmin, mnmin, zero, one, u, ldpt )
            CALL slaset( 'Full', mnmin, mnmin, zero, one, vt, ldpt )
*
            CALL sbdsqr( uplo, mnmin, mnmin, mnmin, nrhs, s1, work, vt,
     $                   ldpt, u, ldpt, z, ldx, work( mnmin+1 ), iinfo )
*
*           Check error code from SBDSQR.
*
            IF( iinfo.NE.0 ) THEN
               WRITE( nout, fmt = 9998 )'SBDSQR(vects)', iinfo, m, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               IF( iinfo.LT.0 ) THEN
                  return
               ELSE
                  result( 4 ) = ulpinv
                  go to 170
               END IF
            END IF
*
*           Use SBDSQR to compute only the singular values of the
*           bidiagonal matrix B;  U, VT, and Z should not be modified.
*
            CALL scopy( mnmin, bd, 1, s2, 1 )
            IF( mnmin.GT.0 )
     $         CALL scopy( mnmin-1, be, 1, work, 1 )
*
            CALL sbdsqr( uplo, mnmin, 0, 0, 0, s2, work, vt, ldpt, u,
     $                   ldpt, z, ldx, work( mnmin+1 ), iinfo )
*
*           Check error code from SBDSQR.
*
            IF( iinfo.NE.0 ) THEN
               WRITE( nout, fmt = 9998 )'SBDSQR(values)', iinfo, m, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               IF( iinfo.LT.0 ) THEN
                  return
               ELSE
                  result( 9 ) = ulpinv
                  go to 170
               END IF
            END IF
*
*           Test 4:  Check the decomposition B := U * S1 * VT
*                5:  Check the computation Z := U' * Y
*                6:  Check the orthogonality of U
*                7:  Check the orthogonality of VT
*
            CALL sbdt03( uplo, mnmin, 1, bd, be, u, ldpt, s1, vt, ldpt,
     $                   work, result( 4 ) )
            CALL sbdt02( mnmin, nrhs, y, ldx, z, ldx, u, ldpt, work,
     $                   result( 5 ) )
            CALL sort01( 'Columns', mnmin, mnmin, u, ldpt, work, lwork,
     $                   result( 6 ) )
            CALL sort01( 'Rows', mnmin, mnmin, vt, ldpt, work, lwork,
     $                   result( 7 ) )
*
*           Test 8:  Check that the singular values are sorted in
*                    non-increasing order and are non-negative
*
            result( 8 ) = zero
            DO 110 i = 1, mnmin - 1
               IF( s1( i ).LT.s1( i+1 ) )
     $            result( 8 ) = ulpinv
               IF( s1( i ).LT.zero )
     $            result( 8 ) = ulpinv
  110       continue
            IF( mnmin.GE.1 ) THEN
               IF( s1( mnmin ).LT.zero )
     $            result( 8 ) = ulpinv
            END IF
*
*           Test 9:  Compare SBDSQR with and without singular vectors
*
            temp2 = zero
*
            DO 120 j = 1, mnmin
               temp1 = abs( s1( j )-s2( j ) ) /
     $                 max( sqrt( unfl )*max( s1( 1 ), one ),
     $                 ulp*max( abs( s1( j ) ), abs( s2( j ) ) ) )
               temp2 = max( temp1, temp2 )
  120       continue
*
            result( 9 ) = temp2
*
*           Test 10:  Sturm sequence test of singular values
*                     Go up by factors of two until it succeeds
*
            temp1 = thresh*( half-ulp )
*
            DO 130 j = 0, log2ui
*               CALL SSVDCH( MNMIN, BD, BE, S1, TEMP1, IINFO )
               IF( iinfo.EQ.0 )
     $            go to 140
               temp1 = temp1*two
  130       continue
*
  140       continue
            result( 10 ) = temp1
*
*           Use SBDSQR to form the decomposition A := (QU) S (VT PT)
*           from the bidiagonal form A := Q B PT.
*
            IF( .NOT.bidiag ) THEN
               CALL scopy( mnmin, bd, 1, s2, 1 )
               IF( mnmin.GT.0 )
     $            CALL scopy( mnmin-1, be, 1, work, 1 )
*
               CALL sbdsqr( uplo, mnmin, n, m, nrhs, s2, work, pt, ldpt,
     $                      q, ldq, y, ldx, work( mnmin+1 ), iinfo )
*
*              Test 11:  Check the decomposition A := Q*U * S2 * VT*PT
*                   12:  Check the computation Z := U' * Q' * X
*                   13:  Check the orthogonality of Q*U
*                   14:  Check the orthogonality of VT*PT
*
               CALL sbdt01( m, n, 0, a, lda, q, ldq, s2, dumma, pt,
     $                      ldpt, work, result( 11 ) )
               CALL sbdt02( m, nrhs, x, ldx, y, ldx, q, ldq, work,
     $                      result( 12 ) )
               CALL sort01( 'Columns', m, mq, q, ldq, work, lwork,
     $                      result( 13 ) )
               CALL sort01( 'Rows', mnmin, n, pt, ldpt, work, lwork,
     $                      result( 14 ) )
            END IF
*
*           Use SBDSDC to form the SVD of the bidiagonal matrix B:
*           B := U * S1 * VT
*
            CALL scopy( mnmin, bd, 1, s1, 1 )
            IF( mnmin.GT.0 )
     $         CALL scopy( mnmin-1, be, 1, work, 1 )
            CALL slaset( 'Full', mnmin, mnmin, zero, one, u, ldpt )
            CALL slaset( 'Full', mnmin, mnmin, zero, one, vt, ldpt )
*
            CALL sbdsdc( uplo, 'I', mnmin, s1, work, u, ldpt, vt, ldpt,
     $                   dum, idum, work( mnmin+1 ), iwork, iinfo )
*
*           Check error code from SBDSDC.
*
            IF( iinfo.NE.0 ) THEN
               WRITE( nout, fmt = 9998 )'SBDSDC(vects)', iinfo, m, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               IF( iinfo.LT.0 ) THEN
                  return
               ELSE
                  result( 15 ) = ulpinv
                  go to 170
               END IF
            END IF
*
*           Use SBDSDC to compute only the singular values of the
*           bidiagonal matrix B;  U and VT should not be modified.
*
            CALL scopy( mnmin, bd, 1, s2, 1 )
            IF( mnmin.GT.0 )
     $         CALL scopy( mnmin-1, be, 1, work, 1 )
*
            CALL sbdsdc( uplo, 'N', mnmin, s2, work, dum, 1, dum, 1,
     $                   dum, idum, work( mnmin+1 ), iwork, iinfo )
*
*           Check error code from SBDSDC.
*
            IF( iinfo.NE.0 ) THEN
               WRITE( nout, fmt = 9998 )'SBDSDC(values)', iinfo, m, n,
     $            jtype, ioldsd
               info = abs( iinfo )
               IF( iinfo.LT.0 ) THEN
                  return
               ELSE
                  result( 18 ) = ulpinv
                  go to 170
               END IF
            END IF
*
*           Test 15:  Check the decomposition B := U * S1 * VT
*                16:  Check the orthogonality of U
*                17:  Check the orthogonality of VT
*
            CALL sbdt03( uplo, mnmin, 1, bd, be, u, ldpt, s1, vt, ldpt,
     $                   work, result( 15 ) )
            CALL sort01( 'Columns', mnmin, mnmin, u, ldpt, work, lwork,
     $                   result( 16 ) )
            CALL sort01( 'Rows', mnmin, mnmin, vt, ldpt, work, lwork,
     $                   result( 17 ) )
*
*           Test 18:  Check that the singular values are sorted in
*                     non-increasing order and are non-negative
*
            result( 18 ) = zero
            DO 150 i = 1, mnmin - 1
               IF( s1( i ).LT.s1( i+1 ) )
     $            result( 18 ) = ulpinv
               IF( s1( i ).LT.zero )
     $            result( 18 ) = ulpinv
  150       continue
            IF( mnmin.GE.1 ) THEN
               IF( s1( mnmin ).LT.zero )
     $            result( 18 ) = ulpinv
            END IF
*
*           Test 19:  Compare SBDSQR with and without singular vectors
*
            temp2 = zero
*
            DO 160 j = 1, mnmin
               temp1 = abs( s1( j )-s2( j ) ) /
     $                 max( sqrt( unfl )*max( s1( 1 ), one ),
     $                 ulp*max( abs( s1( 1 ) ), abs( s2( 1 ) ) ) )
               temp2 = max( temp1, temp2 )
  160       continue
*
            result( 19 ) = temp2
*
*           End of Loop -- Check for RESULT(j) > THRESH
*
  170       continue
            DO 180 j = 1, 19
               IF( result( j ).GE.thresh ) THEN
                  IF( nfail.EQ.0 )
     $               CALL slahd2( nout, path )
                  WRITE( nout, fmt = 9999 )m, n, jtype, ioldsd, j,
     $               result( j )
                  nfail = nfail + 1
               END IF
  180       continue
            IF( .NOT.bidiag ) THEN
               ntest = ntest + 19
            ELSE
               ntest = ntest + 5
            END IF
*
  190    continue
  200 continue
*
*     Summary
*
      CALL alasum( path, nout, nfail, ntest, 0 )
*
      return
*
*     End of SCHKBD
*
 9999 format( ' M=', i5, ', N=', i5, ', type ', i2, ', seed=',
     $      4( i4, ',' ), ' test(', i2, ')=', g11.4 )
 9998 format( ' SCHKBD: ', a, ' returned INFO=', i6, '.', / 9x, 'M=',
     $      i6, ', N=', i6, ', JTYPE=', i6, ', ISEED=(', 3( i5, ',' ),
     $      i5, ')' )
*
      END