SRC/cbdsqr.f(3) Library Functions Manual SRC/cbdsqr.f(3)


subroutine cbdsqr (uplo, n, ncvt, nru, ncc, d, e, vt, ldvt, u, ldu, c, ldc, rwork, info)



 CBDSQR 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**H
 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**H*VT instead of
 P**H, for given complex input matrices U and VT.  When U and VT are
 the unitary matrices that reduce a general matrix A to bidiagonal
 form: A = U*B*VT, as computed by CGEBRD, then
    A = (U*Q) * S * (P**H*VT)
 is the SVD of A.  Optionally, the subroutine may also compute Q**H*C
 for a given complex 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.


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


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


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


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


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


          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


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


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


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


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


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


          RWORK is REAL array, dimension (4*N)


          INFO is INTEGER
          = 0:  successful exit
          < 0:  If INFO = -i, the i-th argument had an illegal value
          > 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.


  Bug report from Cezary Dendek.
  On November 3rd 2023, 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.


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Definition at line 231 of file cbdsqr.f.

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