LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ schkbb()

subroutine schkbb ( integer  NSIZES,
integer, dimension( * )  MVAL,
integer, dimension( * )  NVAL,
integer  NWDTHS,
integer, dimension( * )  KK,
integer  NTYPES,
logical, dimension( * )  DOTYPE,
integer  NRHS,
integer, dimension( 4 )  ISEED,
real  THRESH,
integer  NOUNIT,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( ldab, * )  AB,
integer  LDAB,
real, dimension( * )  BD,
real, dimension( * )  BE,
real, dimension( ldq, * )  Q,
integer  LDQ,
real, dimension( ldp, * )  P,
integer  LDP,
real, dimension( ldc, * )  C,
integer  LDC,
real, dimension( ldc, * )  CC,
real, dimension( * )  WORK,
integer  LWORK,
real, dimension( * )  RESULT,
integer  INFO 
)

SCHKBB

Purpose:
 SCHKBB tests the reduction of a general real rectangular band
 matrix to bidiagonal form.

 SGBBRD factors a general band matrix A as  Q B P* , where * means
 transpose, B is upper bidiagonal, and Q and P are orthogonal;
 SGBBRD can also overwrite a given matrix C with Q* C .

 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 C are generated.
 The problem dimensions are as follows
    A:          M x N
    Q:          M x M
    P:          N x N
    B:          min(M,N) x min(M,N)
    C:          M x NRHS

 For each generated matrix, 4 tests are performed:

 (1)   | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'

 (2)   | I - Q' Q | / ( M ulp )

 (3)   | I - PT PT' | / ( N ulp )

 (4)   | Y - Q' C | / ( |Y| max(M,NRHS) ulp ), where Y = Q' C.

 The "types" are specified by a logical array DOTYPE( 1:NTYPES );
 if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
 Currently, the list of possible types is:

 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 )
Parameters
[in]NSIZES
          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).
          If NSIZES is zero, SCHKBB does nothing.  NSIZES must be at
          least zero.
[in]MVAL
          MVAL is INTEGER array, dimension (NSIZES)
          The values of the matrix row dimension M.
[in]NVAL
          NVAL is INTEGER array, dimension (NSIZES)
          The values of the matrix column dimension N.
[in]NWDTHS
          NWDTHS is INTEGER
          The number of bandwidths to use.  If it is zero,
          SCHKBB does nothing.  It must be at least zero.
[in]KK
          KK is INTEGER array, dimension (NWDTHS)
          An array containing the bandwidths to be used for the band
          matrices.  The values must be at least zero.
[in]NTYPES
          NTYPES is INTEGER
          The number of elements in DOTYPE.   If it is zero, SCHKBB
          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 matrix is in A.  This
          is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
          DOTYPE(MAXTYP+1) is .TRUE. .
[in]DOTYPE
          DOTYPE is LOGICAL array, dimension (NTYPES)
          If DOTYPE(j) is .TRUE., then for each size in NN a
          matrix of that size and 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.
[in]NRHS
          NRHS is INTEGER
          The number of columns in the "right-hand side" matrix C.
          If NRHS = 0, then the operations on the right-hand side will
          not be tested. NRHS must be at least 0.
[in,out]ISEED
          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 random number generator uses a linear
          congruential sequence limited to small integers, and so
          should produce machine independent random numbers. The
          values of ISEED are changed on exit, and can be used in the
          next call to SCHKBB to continue the same random number
          sequence.
[in]THRESH
          THRESH is REAL
          A test will count as "failed" if the "error", computed as
          described above, exceeds THRESH.  Note that the error
          is scaled to be O(1), so THRESH should be a reasonably
          small multiple of 1, e.g., 10 or 100.  In particular,
          it should not depend on the precision (single vs. double)
          or the size of the matrix.  It must be at least zero.
[in]NOUNIT
          NOUNIT is INTEGER
          The FORTRAN unit number for printing out error messages
          (e.g., if a routine returns IINFO not equal to 0.)
[in,out]A
          A is REAL array, dimension
                            (LDA, max(NN))
          Used to hold the matrix A.
[in]LDA
          LDA is INTEGER
          The leading dimension of A.  It must be at least 1
          and at least max( NN ).
[out]AB
          AB is REAL array, dimension (LDAB, max(NN))
          Used to hold A in band storage format.
[in]LDAB
          LDAB is INTEGER
          The leading dimension of AB.  It must be at least 2 (not 1!)
          and at least max( KK )+1.
[out]BD
          BD is REAL array, dimension (max(NN))
          Used to hold the diagonal of the bidiagonal matrix computed
          by SGBBRD.
[out]BE
          BE is REAL array, dimension (max(NN))
          Used to hold the off-diagonal of the bidiagonal matrix
          computed by SGBBRD.
[out]Q
          Q is REAL array, dimension (LDQ, max(NN))
          Used to hold the orthogonal matrix Q computed by SGBBRD.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of Q.  It must be at least 1
          and at least max( NN ).
[out]P
          P is REAL array, dimension (LDP, max(NN))
          Used to hold the orthogonal matrix P computed by SGBBRD.
[in]LDP
          LDP is INTEGER
          The leading dimension of P.  It must be at least 1
          and at least max( NN ).
[out]C
          C is REAL array, dimension (LDC, max(NN))
          Used to hold the matrix C updated by SGBBRD.
[in]LDC
          LDC is INTEGER
          The leading dimension of U.  It must be at least 1
          and at least max( NN ).
[out]CC
          CC is REAL array, dimension (LDC, max(NN))
          Used to hold a copy of the matrix C.
[out]WORK
          WORK is REAL array, dimension (LWORK)
[in]LWORK
          LWORK is INTEGER
          The number of entries in WORK.  This must be at least
          max( LDA+1, max(NN)+1 )*max(NN).
[out]RESULT
          RESULT is REAL array, dimension (4)
          The values computed by the tests described above.
          The values are currently limited to 1/ulp, to avoid
          overflow.
[out]INFO
          INFO is INTEGER
          If 0, then everything ran OK.

-----------------------------------------------------------------------

       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.
       NTESTT          The total number of tests performed so far.
       NMAX            Largest value in NN.
       NMATS           The number of matrices generated so far.
       NERRS           The number of tests which have exceeded THRESH
                       so far.
       COND, IMODE     Values to be passed to the matrix generators.
       ANORM           Norm of A; passed to matrix generators.

       OVFL, UNFL      Overflow and underflow thresholds.
       ULP, ULPINV     Finest relative precision and its inverse.
       RTOVFL, RTUNFL  Square roots of the previous 2 values.
               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) )
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 351 of file schkbb.f.

355 *
356 * -- LAPACK test routine (input) --
357 * -- LAPACK is a software package provided by Univ. of Tennessee, --
358 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
359 *
360 * .. Scalar Arguments ..
361  INTEGER INFO, LDA, LDAB, LDC, LDP, LDQ, LWORK, NOUNIT,
362  $ NRHS, NSIZES, NTYPES, NWDTHS
363  REAL THRESH
364 * ..
365 * .. Array Arguments ..
366  LOGICAL DOTYPE( * )
367  INTEGER ISEED( 4 ), KK( * ), MVAL( * ), NVAL( * )
368  REAL A( LDA, * ), AB( LDAB, * ), BD( * ), BE( * ),
369  $ C( LDC, * ), CC( LDC, * ), P( LDP, * ),
370  $ Q( LDQ, * ), RESULT( * ), WORK( * )
371 * ..
372 *
373 * =====================================================================
374 *
375 * .. Parameters ..
376  REAL ZERO, ONE
377  parameter( zero = 0.0e0, one = 1.0e0 )
378  INTEGER MAXTYP
379  parameter( maxtyp = 15 )
380 * ..
381 * .. Local Scalars ..
382  LOGICAL BADMM, BADNN, BADNNB
383  INTEGER I, IINFO, IMODE, ITYPE, J, JCOL, JR, JSIZE,
384  $ JTYPE, JWIDTH, K, KL, KMAX, KU, M, MMAX, MNMAX,
385  $ MNMIN, MTYPES, N, NERRS, NMATS, NMAX, NTEST,
386  $ NTESTT
387  REAL AMNINV, ANORM, COND, OVFL, RTOVFL, RTUNFL, ULP,
388  $ ULPINV, UNFL
389 * ..
390 * .. Local Arrays ..
391  INTEGER IDUMMA( 1 ), IOLDSD( 4 ), KMAGN( MAXTYP ),
392  $ KMODE( MAXTYP ), KTYPE( MAXTYP )
393 * ..
394 * .. External Functions ..
395  REAL SLAMCH
396  EXTERNAL slamch
397 * ..
398 * .. External Subroutines ..
399  EXTERNAL sbdt01, sbdt02, sgbbrd, slacpy, slahd2, slaset,
401 * ..
402 * .. Intrinsic Functions ..
403  INTRINSIC abs, max, min, real, sqrt
404 * ..
405 * .. Data statements ..
406  DATA ktype / 1, 2, 5*4, 5*6, 3*9 /
407  DATA kmagn / 2*1, 3*1, 2, 3, 3*1, 2, 3, 1, 2, 3 /
408  DATA kmode / 2*0, 4, 3, 1, 4, 4, 4, 3, 1, 4, 4, 0,
409  $ 0, 0 /
410 * ..
411 * .. Executable Statements ..
412 *
413 * Check for errors
414 *
415  ntestt = 0
416  info = 0
417 *
418 * Important constants
419 *
420  badmm = .false.
421  badnn = .false.
422  mmax = 1
423  nmax = 1
424  mnmax = 1
425  DO 10 j = 1, nsizes
426  mmax = max( mmax, mval( j ) )
427  IF( mval( j ).LT.0 )
428  $ badmm = .true.
429  nmax = max( nmax, nval( j ) )
430  IF( nval( j ).LT.0 )
431  $ badnn = .true.
432  mnmax = max( mnmax, min( mval( j ), nval( j ) ) )
433  10 CONTINUE
434 *
435  badnnb = .false.
436  kmax = 0
437  DO 20 j = 1, nwdths
438  kmax = max( kmax, kk( j ) )
439  IF( kk( j ).LT.0 )
440  $ badnnb = .true.
441  20 CONTINUE
442 *
443 * Check for errors
444 *
445  IF( nsizes.LT.0 ) THEN
446  info = -1
447  ELSE IF( badmm ) THEN
448  info = -2
449  ELSE IF( badnn ) THEN
450  info = -3
451  ELSE IF( nwdths.LT.0 ) THEN
452  info = -4
453  ELSE IF( badnnb ) THEN
454  info = -5
455  ELSE IF( ntypes.LT.0 ) THEN
456  info = -6
457  ELSE IF( nrhs.LT.0 ) THEN
458  info = -8
459  ELSE IF( lda.LT.nmax ) THEN
460  info = -13
461  ELSE IF( ldab.LT.2*kmax+1 ) THEN
462  info = -15
463  ELSE IF( ldq.LT.nmax ) THEN
464  info = -19
465  ELSE IF( ldp.LT.nmax ) THEN
466  info = -21
467  ELSE IF( ldc.LT.nmax ) THEN
468  info = -23
469  ELSE IF( ( max( lda, nmax )+1 )*nmax.GT.lwork ) THEN
470  info = -26
471  END IF
472 *
473  IF( info.NE.0 ) THEN
474  CALL xerbla( 'SCHKBB', -info )
475  RETURN
476  END IF
477 *
478 * Quick return if possible
479 *
480  IF( nsizes.EQ.0 .OR. ntypes.EQ.0 .OR. nwdths.EQ.0 )
481  $ RETURN
482 *
483 * More Important constants
484 *
485  unfl = slamch( 'Safe minimum' )
486  ovfl = one / unfl
487  ulp = slamch( 'Epsilon' )*slamch( 'Base' )
488  ulpinv = one / ulp
489  rtunfl = sqrt( unfl )
490  rtovfl = sqrt( ovfl )
491 *
492 * Loop over sizes, widths, types
493 *
494  nerrs = 0
495  nmats = 0
496 *
497  DO 160 jsize = 1, nsizes
498  m = mval( jsize )
499  n = nval( jsize )
500  mnmin = min( m, n )
501  amninv = one / real( max( 1, m, n ) )
502 *
503  DO 150 jwidth = 1, nwdths
504  k = kk( jwidth )
505  IF( k.GE.m .AND. k.GE.n )
506  $ GO TO 150
507  kl = max( 0, min( m-1, k ) )
508  ku = max( 0, min( n-1, k ) )
509 *
510  IF( nsizes.NE.1 ) THEN
511  mtypes = min( maxtyp, ntypes )
512  ELSE
513  mtypes = min( maxtyp+1, ntypes )
514  END IF
515 *
516  DO 140 jtype = 1, mtypes
517  IF( .NOT.dotype( jtype ) )
518  $ GO TO 140
519  nmats = nmats + 1
520  ntest = 0
521 *
522  DO 30 j = 1, 4
523  ioldsd( j ) = iseed( j )
524  30 CONTINUE
525 *
526 * Compute "A".
527 *
528 * Control parameters:
529 *
530 * KMAGN KMODE KTYPE
531 * =1 O(1) clustered 1 zero
532 * =2 large clustered 2 identity
533 * =3 small exponential (none)
534 * =4 arithmetic diagonal, (w/ singular values)
535 * =5 random log (none)
536 * =6 random nonhermitian, w/ singular values
537 * =7 (none)
538 * =8 (none)
539 * =9 random nonhermitian
540 *
541  IF( mtypes.GT.maxtyp )
542  $ GO TO 90
543 *
544  itype = ktype( jtype )
545  imode = kmode( jtype )
546 *
547 * Compute norm
548 *
549  GO TO ( 40, 50, 60 )kmagn( jtype )
550 *
551  40 CONTINUE
552  anorm = one
553  GO TO 70
554 *
555  50 CONTINUE
556  anorm = ( rtovfl*ulp )*amninv
557  GO TO 70
558 *
559  60 CONTINUE
560  anorm = rtunfl*max( m, n )*ulpinv
561  GO TO 70
562 *
563  70 CONTINUE
564 *
565  CALL slaset( 'Full', lda, n, zero, zero, a, lda )
566  CALL slaset( 'Full', ldab, n, zero, zero, ab, ldab )
567  iinfo = 0
568  cond = ulpinv
569 *
570 * Special Matrices -- Identity & Jordan block
571 *
572 * Zero
573 *
574  IF( itype.EQ.1 ) THEN
575  iinfo = 0
576 *
577  ELSE IF( itype.EQ.2 ) THEN
578 *
579 * Identity
580 *
581  DO 80 jcol = 1, n
582  a( jcol, jcol ) = anorm
583  80 CONTINUE
584 *
585  ELSE IF( itype.EQ.4 ) THEN
586 *
587 * Diagonal Matrix, singular values specified
588 *
589  CALL slatms( m, n, 'S', iseed, 'N', work, imode, cond,
590  $ anorm, 0, 0, 'N', a, lda, work( m+1 ),
591  $ iinfo )
592 *
593  ELSE IF( itype.EQ.6 ) THEN
594 *
595 * Nonhermitian, singular values specified
596 *
597  CALL slatms( m, n, 'S', iseed, 'N', work, imode, cond,
598  $ anorm, kl, ku, 'N', a, lda, work( m+1 ),
599  $ iinfo )
600 *
601  ELSE IF( itype.EQ.9 ) THEN
602 *
603 * Nonhermitian, random entries
604 *
605  CALL slatmr( m, n, 'S', iseed, 'N', work, 6, one, one,
606  $ 'T', 'N', work( n+1 ), 1, one,
607  $ work( 2*n+1 ), 1, one, 'N', idumma, kl,
608  $ ku, zero, anorm, 'N', a, lda, idumma,
609  $ iinfo )
610 *
611  ELSE
612 *
613  iinfo = 1
614  END IF
615 *
616 * Generate Right-Hand Side
617 *
618  CALL slatmr( m, nrhs, 'S', iseed, 'N', work, 6, one, one,
619  $ 'T', 'N', work( m+1 ), 1, one,
620  $ work( 2*m+1 ), 1, one, 'N', idumma, m, nrhs,
621  $ zero, one, 'NO', c, ldc, idumma, iinfo )
622 *
623  IF( iinfo.NE.0 ) THEN
624  WRITE( nounit, fmt = 9999 )'Generator', iinfo, n,
625  $ jtype, ioldsd
626  info = abs( iinfo )
627  RETURN
628  END IF
629 *
630  90 CONTINUE
631 *
632 * Copy A to band storage.
633 *
634  DO 110 j = 1, n
635  DO 100 i = max( 1, j-ku ), min( m, j+kl )
636  ab( ku+1+i-j, j ) = a( i, j )
637  100 CONTINUE
638  110 CONTINUE
639 *
640 * Copy C
641 *
642  CALL slacpy( 'Full', m, nrhs, c, ldc, cc, ldc )
643 *
644 * Call SGBBRD to compute B, Q and P, and to update C.
645 *
646  CALL sgbbrd( 'B', m, n, nrhs, kl, ku, ab, ldab, bd, be,
647  $ q, ldq, p, ldp, cc, ldc, work, iinfo )
648 *
649  IF( iinfo.NE.0 ) THEN
650  WRITE( nounit, fmt = 9999 )'SGBBRD', iinfo, n, jtype,
651  $ ioldsd
652  info = abs( iinfo )
653  IF( iinfo.LT.0 ) THEN
654  RETURN
655  ELSE
656  result( 1 ) = ulpinv
657  GO TO 120
658  END IF
659  END IF
660 *
661 * Test 1: Check the decomposition A := Q * B * P'
662 * 2: Check the orthogonality of Q
663 * 3: Check the orthogonality of P
664 * 4: Check the computation of Q' * C
665 *
666  CALL sbdt01( m, n, -1, a, lda, q, ldq, bd, be, p, ldp,
667  $ work, result( 1 ) )
668  CALL sort01( 'Columns', m, m, q, ldq, work, lwork,
669  $ result( 2 ) )
670  CALL sort01( 'Rows', n, n, p, ldp, work, lwork,
671  $ result( 3 ) )
672  CALL sbdt02( m, nrhs, c, ldc, cc, ldc, q, ldq, work,
673  $ result( 4 ) )
674 *
675 * End of Loop -- Check for RESULT(j) > THRESH
676 *
677  ntest = 4
678  120 CONTINUE
679  ntestt = ntestt + ntest
680 *
681 * Print out tests which fail.
682 *
683  DO 130 jr = 1, ntest
684  IF( result( jr ).GE.thresh ) THEN
685  IF( nerrs.EQ.0 )
686  $ CALL slahd2( nounit, 'SBB' )
687  nerrs = nerrs + 1
688  WRITE( nounit, fmt = 9998 )m, n, k, ioldsd, jtype,
689  $ jr, result( jr )
690  END IF
691  130 CONTINUE
692 *
693  140 CONTINUE
694  150 CONTINUE
695  160 CONTINUE
696 *
697 * Summary
698 *
699  CALL slasum( 'SBB', nounit, nerrs, ntestt )
700  RETURN
701 *
702  9999 FORMAT( ' SCHKBB: ', a, ' returned INFO=', i5, '.', / 9x, 'M=',
703  $ i5, ' N=', i5, ' K=', i5, ', JTYPE=', i5, ', ISEED=(',
704  $ 3( i5, ',' ), i5, ')' )
705  9998 FORMAT( ' M =', i4, ' N=', i4, ', K=', i3, ', seed=',
706  $ 4( i4, ',' ), ' type ', i2, ', test(', i2, ')=', g10.3 )
707 *
708 * End of SCHKBB
709 *
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slatms(M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, KL, KU, PACK, A, LDA, WORK, INFO)
SLATMS
Definition: slatms.f:321
subroutine slatmr(M, N, DIST, ISEED, SYM, D, MODE, COND, DMAX, RSIGN, GRADE, DL, MODEL, CONDL, DR, MODER, CONDR, PIVTNG, IPIVOT, KL, KU, SPARSE, ANORM, PACK, A, LDA, IWORK, INFO)
SLATMR
Definition: slatmr.f:471
subroutine sgbbrd(VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q, LDQ, PT, LDPT, C, LDC, WORK, INFO)
SGBBRD
Definition: sgbbrd.f:187
subroutine sort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
SORT01
Definition: sort01.f:116
subroutine sbdt02(M, N, B, LDB, C, LDC, U, LDU, WORK, RESID)
SBDT02
Definition: sbdt02.f:112
subroutine sbdt01(M, N, KD, A, LDA, Q, LDQ, D, E, PT, LDPT, WORK, RESID)
SBDT01
Definition: sbdt01.f:141
subroutine slahd2(IOUNIT, PATH)
SLAHD2
Definition: slahd2.f:65
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
subroutine slasum(TYPE, IOUNIT, IE, NRUN)
SLASUM
Definition: slasum.f:41
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