sbdsqr.f - Man Page

subroutine sbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, work, info)
SBDSQR

SBDSQR  

Purpose:

 SBDSQR computes the singular values and, optionally, the right and/or
 left singular vectors from the singular value decomposition (SVD) of
 a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
 zero-shift QR algorithm.  The SVD of B has the form

    B = Q * S * P**T

 where S is the diagonal matrix of singular values, Q is an orthogonal
 matrix of left singular vectors, and P is an orthogonal matrix of
 right singular vectors.  If left singular vectors are requested, this
 subroutine actually returns U*Q instead of Q, and, if right singular
 vectors are requested, this subroutine returns P**T*VT instead of
 P**T, for given real input matrices U and VT.  When U and VT are the
 orthogonal matrices that reduce a general matrix A to bidiagonal
 form:  A = U*B*VT, as computed by SGEBRD, then

    A = (U*Q) * S * (P**T*VT)

 is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
 for a given real input matrix 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. Comput. 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 Berkeley, July 1992
 for a detailed description of the algorithm.

Parameters

UPLO

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

N

          N is INTEGER
          The order of the matrix B.  N >= 0.

NCVT

          NCVT is INTEGER
          The number of columns of the matrix VT. NCVT >= 0.

NRU

          NRU is INTEGER
          The number of rows of the matrix U. NRU >= 0.

NCC

          NCC is INTEGER
          The number of columns of the matrix C. NCC >= 0.

D

          D is REAL array, dimension (N)
          On entry, the n diagonal elements of the bidiagonal matrix B.
          On exit, if INFO=0, the singular values of B in decreasing
          order.

E

          E is REAL array, dimension (N-1)
          On entry, the N-1 offdiagonal elements of the bidiagonal
          matrix B.
          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
          will contain the diagonal and superdiagonal elements of a
          bidiagonal matrix orthogonally equivalent to the one given
          as input.

VT

          VT is REAL array, dimension (LDVT, NCVT)
          On entry, an N-by-NCVT matrix VT.
          On exit, VT is overwritten by P**T * VT.
          Not referenced if NCVT = 0.

LDVT

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

U

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

LDU

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

C

          C is REAL array, dimension (LDC, NCC)
          On entry, an N-by-NCC matrix C.
          On exit, C is overwritten by Q**T * C.
          Not referenced if NCC = 0.

LDC

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

WORK

          WORK is REAL array, dimension (4*N)

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 0:
             if NCVT = NRU = NCC = 0,
                = 1, a split was marked by a positive value in E
                = 2, current block of Z not diagonalized after 30*N
                     iterations (in inner while loop)
                = 3, termination criterion of outer while loop not met
                     (program created more than N unreduced blocks)
             else NCVT = NRU = NCC = 0,
                   the algorithm did not converge; D and E contain 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.

Internal 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 relative
             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 computed 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.

Note:

  Bug report from Cezary Dendek.
  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
  removed since it can overflow pretty easily (for N larger or equal
  than 18,919). We instead use MAXITDIVN = MAXITR*N.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 238 of file sbdsqr.f.