TESTING/EIG/dchkbd.f(3)

NAME
TESTING/EIG/dchkbd.f

SYNOPSIS

  Functions/Subroutines
subroutine dchkbd (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)
DCHKBD

Function/Subroutine Documentation

  subroutine dchkbd (integer nsizes, integer, dimension( * ) mval, integer, dimension( * ) nval, integer ntypes, logical, dimension( * ) dotype, integer nrhs, integer, dimension( 4 ) iseed, double precision thresh, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) bd, double precision, dimension( * ) be, double precision, dimension( * ) s1, double precision, dimension( * ) s2, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( ldx, * ) y, double precision, dimension( ldx, * ) z, double precision, dimension( ldq, * ) q, integer ldq, double precision, dimension( ldpt, * ) pt, integer ldpt, double precision, dimension( ldpt, * ) u, double precision, dimension( ldpt, * ) vt, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer nout, integer info)

DCHKBD

Purpose:

     DCHKBD checks the singular value decomposition (SVD) routines.
     DGEBRD 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.
     DORGBR generates the orthogonal matrices Q and P' from DGEBRD.
     Note that Q and P are not necessarily square.
     DBDSQR 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, DBDSQR has an option to apply the left orthogonal matrix
     U to a matrix X, useful in least squares applications.
     DBDSDC 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.
      DBDSVDX computes the singular value decomposition of the bidiagonal
      matrix B as B = U S V' using bisection and inverse iteration. It is
      called six times to compute
         1) B = U S1 V', RANGE='A', 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) B = U S1 V', RANGE='I', with 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
         4) Same as 3), but the singular values are stored in S2 and the
             singular vectors are not computed.
         5) B = U S1 V', RANGE='V', with 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
         6) Same as 5), 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 DGEBRD and DORGBR
     (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 DBDSQR 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
           DSVDCH)
     Test DBDSQR 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 DBDSDC 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.
      Test DBDSVDX on bidiagonal matrix B
      (20)  | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
      (21)  | I - U' U | / ( min(M,N) ulp )
      (22)  | I - VT VT' | / ( min(M,N) ulp )
      (23)  S1 contains min(M,N) nonnegative values in decreasing order.
            (Return 0 if true, 1/ULP if false.)
      (24)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
                                        computing U and V.
      (25)  | S1 - U' B VT' | / ( |S| n ulp )    DBDSVDX('V', 'I')
      (26)  | I - U' U | / ( min(M,N) ulp )
      (27)  | I - VT VT' | / ( min(M,N) ulp )
      (28)  S1 contains min(M,N) nonnegative values in decreasing order.
            (Return 0 if true, 1/ULP if false.)
      (29)  | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
                                        computing U and V.
      (30)  | S1 - U' B VT' | / ( |S1| n ulp )   DBDSVDX('V', 'V')
      (31)  | I - U' U | / ( min(M,N) ulp )
      (32)  | I - VT VT' | / ( min(M,N) ulp )
      (33)  S1 contains min(M,N) nonnegative values in decreasing order.
            (Return 0 if true, 1/ULP if false.)
      (34)  | 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) DGEBRD 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.

Parameters
    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).

    MVAL

              MVAL is INTEGER array, dimension (NM)
              The values of the matrix row dimension M.

    NVAL

              NVAL is INTEGER array, dimension (NM)
              The values of the matrix column dimension N.

    NTYPES

              NTYPES is INTEGER
              The number of elements in DOTYPE.   If it is zero, DCHKBD
              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. .

    DOTYPE

              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.

    NRHS

              NRHS is INTEGER
              The number of columns in the 'right-hand side' matrices X, Y,
              and Z, used in testing DBDSQR.  If NRHS = 0, then the
              operations on the right-hand side will not be tested.
              NRHS must be at least 0.

    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 values of ISEED are changed on exit, and can be
              used in the next call to DCHKBD to continue the same random
              number sequence.

    THRESH

              THRESH is DOUBLE PRECISION
              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.

    A

              A is DOUBLE PRECISION array, dimension (LDA,NMAX)
              where NMAX is the maximum value of N in NVAL.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,MMAX),
              where MMAX is the maximum value of M in MVAL.

    BD

              BD is DOUBLE PRECISION array, dimension
                          (max(min(MVAL(j),NVAL(j))))

    BE

              BE is DOUBLE PRECISION array, dimension
                          (max(min(MVAL(j),NVAL(j))))

    S1

              S1 is DOUBLE PRECISION array, dimension
                          (max(min(MVAL(j),NVAL(j))))

    S2

              S2 is DOUBLE PRECISION array, dimension
                          (max(min(MVAL(j),NVAL(j))))

    X

              X is DOUBLE PRECISION array, dimension (LDX,NRHS)

    LDX

              LDX is INTEGER
              The leading dimension of the arrays X, Y, and Z.
              LDX >= max(1,MMAX)

    Y

              Y is DOUBLE PRECISION array, dimension (LDX,NRHS)

    Z

              Z is DOUBLE PRECISION array, dimension (LDX,NRHS)

    Q

              Q is DOUBLE PRECISION array, dimension (LDQ,MMAX)

    LDQ

              LDQ is INTEGER
              The leading dimension of the array Q.  LDQ >= max(1,MMAX).

    PT

              PT is DOUBLE PRECISION array, dimension (LDPT,NMAX)

    LDPT

              LDPT is INTEGER
              The leading dimension of the arrays PT, U, and V.
              LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).

    U

              U is DOUBLE PRECISION array, dimension
                          (LDPT,max(min(MVAL(j),NVAL(j))))

    VT

              VT is DOUBLE PRECISION array, dimension
                          (LDPT,max(min(MVAL(j),NVAL(j))))

    WORK

              WORK is DOUBLE PRECISION array, dimension (LWORK)

    LWORK

              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))

    IWORK

              IWORK is INTEGER array, dimension at least 8*min(M,N)

    NOUT

              NOUT is INTEGER
              The FORTRAN unit number for printing out error messages
              (e.g., if a routine returns IINFO not equal to 0.)

    INFO

              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  DLATMR, SLATMS, DGEBRD, DORGBR, or DBDSQR,
                  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) )

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Definition at line 489 of file dchkbd.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.