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.

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.

  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 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.

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