.TH "TESTING/EIG/dchkbd.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
TESTING/EIG/dchkbd.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBdchkbd\fP (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)"
.br
.RI "\fBDCHKBD\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

.PP
\fBDCHKBD\fP 
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 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\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fINSIZES\fP 
.PP
.nf
          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)\&.
.fi
.PP
.br
\fIMVAL\fP 
.PP
.nf
          MVAL is INTEGER array, dimension (NM)
          The values of the matrix row dimension M\&.
.fi
.PP
.br
\fINVAL\fP 
.PP
.nf
          NVAL is INTEGER array, dimension (NM)
          The values of the matrix column dimension N\&.
.fi
.PP
.br
\fINTYPES\fP 
.PP
.nf
          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\&. \&.
.fi
.PP
.br
\fIDOTYPE\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fINRHS\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIISEED\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fITHRESH\fP 
.PP
.nf
          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\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA,NMAX)
          where NMAX is the maximum value of N in NVAL\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,MMAX),
          where MMAX is the maximum value of M in MVAL\&.
.fi
.PP
.br
\fIBD\fP 
.PP
.nf
          BD is DOUBLE PRECISION array, dimension
                      (max(min(MVAL(j),NVAL(j))))
.fi
.PP
.br
\fIBE\fP 
.PP
.nf
          BE is DOUBLE PRECISION array, dimension
                      (max(min(MVAL(j),NVAL(j))))
.fi
.PP
.br
\fIS1\fP 
.PP
.nf
          S1 is DOUBLE PRECISION array, dimension
                      (max(min(MVAL(j),NVAL(j))))
.fi
.PP
.br
\fIS2\fP 
.PP
.nf
          S2 is DOUBLE PRECISION array, dimension
                      (max(min(MVAL(j),NVAL(j))))
.fi
.PP
.br
\fIX\fP 
.PP
.nf
          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
.fi
.PP
.br
\fILDX\fP 
.PP
.nf
          LDX is INTEGER
          The leading dimension of the arrays X, Y, and Z\&.
          LDX >= max(1,MMAX)
.fi
.PP
.br
\fIY\fP 
.PP
.nf
          Y is DOUBLE PRECISION array, dimension (LDX,NRHS)
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is DOUBLE PRECISION array, dimension (LDX,NRHS)
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is DOUBLE PRECISION array, dimension (LDQ,MMAX)
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q\&.  LDQ >= max(1,MMAX)\&.
.fi
.PP
.br
\fIPT\fP 
.PP
.nf
          PT is DOUBLE PRECISION array, dimension (LDPT,NMAX)
.fi
.PP
.br
\fILDPT\fP 
.PP
.nf
          LDPT is INTEGER
          The leading dimension of the arrays PT, U, and V\&.
          LDPT >= max(1, max(min(MVAL(j),NVAL(j))))\&.
.fi
.PP
.br
\fIU\fP 
.PP
.nf
          U is DOUBLE PRECISION array, dimension
                      (LDPT,max(min(MVAL(j),NVAL(j))))
.fi
.PP
.br
\fIVT\fP 
.PP
.nf
          VT is DOUBLE PRECISION array, dimension
                      (LDPT,max(min(MVAL(j),NVAL(j))))
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (LWORK)
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          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))
.fi
.PP
.br
\fIIWORK\fP 
.PP
.nf
          IWORK is INTEGER array, dimension at least 8*min(M,N)
.fi
.PP
.br
\fINOUT\fP 
.PP
.nf
          NOUT is INTEGER
          The FORTRAN unit number for printing out error messages
          (e\&.g\&., if a routine returns IINFO not equal to 0\&.)
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          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) )
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP

.PP
Definition at line \fB489\fP of file \fBdchkbd\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.