schkbd.f

来自「famous linear algebra library (LAPACK) p」· F 代码 · 共 969 行 · 第 1/3 页

      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 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. 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, * )
*     ..
*
*  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.
*
*  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.
*
*  Arguments
*  ==========
*
*  NSIZES  (input) 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    (input) INTEGER array, dimension (NM)
*          The values of the matrix row dimension M.
*
*  NVAL    (input) INTEGER array, dimension (NM)
*          The values of the matrix column dimension N.
*
*  NTYPES  (input) 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  (input) 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    (input) 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.
*
*  ISEED   (input/output) 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  (input) 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       (workspace) REAL array, dimension (LDA,NMAX)
*          where NMAX is the maximum value of N in NVAL.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,MMAX),
*          where MMAX is the maximum value of M in MVAL.
*
*  BD      (workspace) REAL array, dimension
*                      (max(min(MVAL(j),NVAL(j))))
*
*  BE      (workspace) REAL array, dimension
*                      (max(min(MVAL(j),NVAL(j))))
*
*  S1      (workspace) REAL array, dimension
*                      (max(min(MVAL(j),NVAL(j))))
*
*  S2      (workspace) REAL array, dimension
*                      (max(min(MVAL(j),NVAL(j))))
*
*  X       (workspace) REAL array, dimension (LDX,NRHS)
*
*  LDX     (input) INTEGER
*          The leading dimension of the arrays X, Y, and Z.
*          LDX >= max(1,MMAX)
*
*  Y       (workspace) REAL array, dimension (LDX,NRHS)
*
*  Z       (workspace) REAL array, dimension (LDX,NRHS)
*
*  Q       (workspace) REAL array, dimension (LDQ,MMAX)
*
*  LDQ     (input) INTEGER
*          The leading dimension of the array Q.  LDQ >= max(1,MMAX).
*
*  PT      (workspace) REAL array, dimension (LDPT,NMAX)
*
*  LDPT    (input) INTEGER
*          The leading dimension of the arrays PT, U, and V.
*          LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).
*
*  U       (workspace) REAL array, dimension
*                      (LDPT,max(min(MVAL(j),NVAL(j))))
*
*  V       (workspace) REAL array, dimension
*                      (LDPT,max(min(MVAL(j),NVAL(j))))
*
*  WORK    (workspace) REAL array, dimension (LWORK)
*
*  LWORK   (input) 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   (workspace) INTEGER array, dimension at least 8*min(M,N)
*
*  NOUT    (input) INTEGER
*          The FORTRAN unit number for printing out error messages
*          (e.g., if a routine returns IINFO not equal to 0.)
*
*  INFO    (output) 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) )
*