TESTING/EIG/schkbd.f(3)

NAME
TESTING/EIG/schkbd.f

SYNOPSIS

  Functions/Subroutines
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)
SCHKBD

Function/Subroutine Documentation

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

SCHKBD

Purpose:

    !>
    !> 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.
    !>
    !>  SBDSVDX 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 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.
    !>  Test SBDSVDX 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 )    SBDSVDX('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 )   SBDSVDX('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  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  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.
    !>

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, 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. .
    !>

    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  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.
    !>

    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 SCHKBD to continue the same random
    !>          number sequence.
    !>

    THRESH

    !>          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.
    !>

    A

    !>          A is REAL 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 REAL array, dimension
    !>                      (max(min(MVAL(j),NVAL(j))))
    !>

    BE

    !>          BE is REAL array, dimension
    !>                      (max(min(MVAL(j),NVAL(j))))
    !>

    S1

    !>          S1 is REAL array, dimension
    !>                      (max(min(MVAL(j),NVAL(j))))
    !>

    S2

    !>          S2 is REAL array, dimension
    !>                      (max(min(MVAL(j),NVAL(j))))
    !>

    X

    !>          X is REAL 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 REAL array, dimension (LDX,NRHS)
    !>

    Z

    !>          Z is REAL array, dimension (LDX,NRHS)
    !>

    Q

    !>          Q is REAL array, dimension (LDQ,MMAX)
    !>

    LDQ

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

    PT

    !>          PT is REAL 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 REAL array, dimension
    !>                      (LDPT,max(min(MVAL(j),NVAL(j))))
    !>

    VT

    !>          VT is REAL array, dimension
    !>                      (LDPT,max(min(MVAL(j),NVAL(j))))
    !>

    WORK

    !>          WORK is REAL 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  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 .
    !>     KMODE(j)        The MODE value to be passed to the matrix
    !>                     generator for type .
    !>     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 schkbd.f.

Author

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