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NAME
     sbdsqr - compute the singular value decomposition (SVD) of a
     real N-by-N (upper or lower) bidiagonal matrix B

SYNOPSIS
     SUBROUTINE SBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT,  LDVT,
               U, LDU, C, LDC, WORK, INFO )

     CHARACTER UPLO

     INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU

     REAL C( LDC, * ), D( * ), E( * ), U( LDU, * ), VT(  LDVT,  *
               ), WORK( * )



     #include <sunperf.h>

     void sbdsqr(char uplo, int n, int ncvt, int  nru,  int  ncc,
               float  *d,  float  *e, float *svt, int ldvt, float
               *su, int ldu, float *sc, int ldc, int *info) ;

PURPOSE
     SBDSQR computes the singular value decomposition (SVD) of  a
     real N-by-N (upper or lower) bidiagonal matrix B:  B = Q * S
     * P' (P' denotes the transpose of P), where S is a  diagonal
     matrix  with  non-negative  diagonal  elements (the singular
     values of B), and Q and P are orthogonal matrices.

     The routine computes S, and optionally computes U * Q, P'  *
     VT, or Q' * C, for given real input matrices U, VT, and C.

     See "Computing  Small Singular Values of Bidiagonal Matrices
     With Guaranteed High Relative Accuracy," by J. Demmel and W.
     Kahan, LAPACK Working Note #3 (or SIAM J. Sci. Statist. Com-
     put. vol. 11, no. 5, pp. 873-912, Sept 1990) and
     "Accurate singular values and differential  qd  algorithms,"
     by  B.  Parlett  and V. Fernando, Technical Report CPAM-554,
     Mathematics Department, University of California  at  Berke-
     ley, July 1992 for a detailed description of the algorithm.


ARGUMENTS
     UPLO      (input) CHARACTER*1
               = 'U':  B is upper bidiagonal;
               = 'L':  B is lower bidiagonal.

     N         (input) INTEGER
               The order of the matrix B.  N >= 0.

     NCVT      (input) INTEGER
               The number of columns of the matrix VT. NCVT >= 0.

     NRU       (input) INTEGER
               The number of rows of the matrix U. NRU >= 0.

     NCC       (input) INTEGER
               The number of columns of the matrix C. NCC >= 0.

     D         (input/output) REAL array, dimension (N)
               On entry, the n diagonal elements of the  bidiago-
               nal  matrix  B.   On exit, if INFO=0, the singular
               values of B in decreasing order.

     E         (input/output) REAL array, dimension (N)
               On entry, the elements of E contain the  offdiago-
               nal elements of the bidiagonal matrix whose SVD is
               desired. On normal exit (INFO  =  0),  E  is  des-
               troyed.   If the algorithm does not converge (INFO
               > 0), D and E will contain the diagonal and super-
               diagonal  elements of a bidiagonal matrix orthogo-
               nally equivalent to the one given as  input.  E(N)
               is used for workspace.

     VT        (input/output) REAL array, dimension (LDVT, NCVT)
               On entry, an N-by-NCVT matrix VT.  On exit, VT  is
               overwritten  by  P' * VT.  VT is not referenced if
               NCVT = 0.

     LDVT      (input) INTEGER
               The leading dimension of the array  VT.   LDVT  >=
               max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

     U         (input/output) REAL array, dimension (LDU, N)
               On entry, an NRU-by-N matrix U.   On  exit,  U  is
               overwritten  by U * Q.  U is not referenced if NRU
               = 0.

     LDU       (input) INTEGER
               The leading dimension of  the  array  U.   LDU  >=
               max(1,NRU).

     C         (input/output) REAL array, dimension (LDC, NCC)
               On entry, an N-by-NCC matrix C.   On  exit,  C  is
               overwritten by Q' * C.  C is not referenced if NCC
               = 0.

     LDC       (input) INTEGER
               The leading dimension of  the  array  C.   LDC  >=
               max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

     WORK      (workspace) REAL array, dimension
               2*N  if only singular values wanted (NCVT = NRU  =
               NCC = 0) max( 1, 4*N-4 ) otherwise

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  If INFO = -i, the i-th argument had an ille-
               gal value
               > 0:  the algorithm did not converge; D and E con-
               tain  the elements of a bidiagonal matrix which is
               orthogonally similar to the input  matrix  B;   if
               INFO  =  i,  i elements of E have not converged to
               zero.

PARAMETERS
     TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))  TOLMUL
               controls the convergence criterion of the QR loop.
               If it is positive, TOLMUL*EPS is the desired rela-
               tive  precision  in  the computed singular values.
               If it is  negative,  abs(TOLMUL*EPS*sigma_max)  is
               the  desired  absolute  accuracy  in  the computed
               singular values (corresponds to relative  accuracy
               abs(TOLMUL*EPS)  in  the  largest  singular value.
               abs(TOLMUL) should be between  1  and  1/EPS,  and
               preferably  between  10 (for fast convergence) and
               .1/EPS (for there  to  be  some  accuracy  in  the
               results).  Default is to lose at either one eighth
               or 2 of the available decimal digits in each  com-
               puted singular value (whichever is smaller).

     MAXITR  INTEGER, default = 6  MAXITR  controls  the  maximum
               number  of  passes  of  the  algorithm through its
               inner loop. The algorithms stops (and so fails  to
               converge)  if  the  number  of  passes through the
               inner loop exceeds MAXITR*N**2.

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