SBDSVDX computes the singular value decomposition (SVD) of a real


  N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT,


  where S is a diagonal matrix with non-negative diagonal elements


  (the singular values of B), and U and VT are orthogonal matrices


  of left and right singular vectors, respectively.


  Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ]


  and superdiagonal E = [ e_1 e_2 ... e_N-1 ], SBDSVDX computes the


  singular value decompositon of B through the eigenvalues and


  eigenvectors of the N*2-by-N*2 tridiagonal matrix


        |  0  d_1                |


        | d_1  0  e_1            |


  TGK = |     e_1  0  d_2        |


        |         d_2  .   .     |


        |              .   .   . |


  If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then


  (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) /


  sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and


  P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ].


  Given a TGK matrix, one can either a) compute -s,-v and change signs


  so that the singular values (and corresponding vectors) are already in


  descending order (as in SGESVD/SGESDD) or b) compute s,v and reorder


  the values (and corresponding vectors). SBDSVDX implements a) by


  calling SSTEVX (bisection plus inverse iteration, to be replaced


  with a version of the Multiple Relative Robust Representation


  algorithm. (See P. Willems and B. Lang, A framework for the MR^3


  algorithm: theory and implementation, SIAM J. Sci. Comput.,


  35:740-766, 2013.)

UPLO

          UPLO is CHARACTER*1


          = 'U':  B is upper bidiagonal;


          = 'L':  B is lower bidiagonal.

JOBZ

          JOBZ is CHARACTER*1


          = 'N':  Compute singular values only;


          = 'V':  Compute singular values and singular vectors.

RANGE

          RANGE is CHARACTER*1


          = 'A': all singular values will be found.


          = 'V': all singular values in the half-open interval [VL,VU)


                 will be found.


          = 'I': the IL-th through IU-th singular values will be found.

N

          N is INTEGER


          The order of the bidiagonal matrix.  N >= 0.

D

          D is REAL array, dimension (N)


          The n diagonal elements of the bidiagonal matrix B.

E

          E is REAL array, dimension (max(1,N-1))


          The (n-1) superdiagonal elements of the bidiagonal matrix


          B in elements 1 to N-1.

VL

         VL is REAL


          If RANGE='V', the lower bound of the interval to


          be searched for singular values. VU > VL.


          Not referenced if RANGE = 'A' or 'I'.

VU

         VU is REAL


          If RANGE='V', the upper bound of the interval to


          be searched for singular values. VU > VL.


          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER


          If RANGE='I', the index of the


          smallest singular value to be returned.


          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.


          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER


          If RANGE='I', the index of the


          largest singular value to be returned.


          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.


          Not referenced if RANGE = 'A' or 'V'.

NS

          NS is INTEGER


          The total number of singular values found.  0 <= NS <= N.


          If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.

S

          S is REAL array, dimension (N)


          The first NS elements contain the selected singular values in


          ascending order.

Z

          Z is REAL array, dimension (2*N,K) )


          If JOBZ = 'V', then if INFO = 0 the first NS columns of Z


          contain the singular vectors of the matrix B corresponding to


          the selected singular values, with U in rows 1 to N and V


          in rows N+1 to N*2, i.e.


          Z = [ U ]


              [ V ]


          If JOBZ = 'N', then Z is not referenced.


          Note: The user must ensure that at least K = NS+1 columns are


          supplied in the array Z; if RANGE = 'V', the exact value of


          NS is not known in advance and an upper bound must be used.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z. LDZ >= 1, and if


          JOBZ = 'V', LDZ >= max(2,N*2).

WORK

          WORK is REAL array, dimension (14*N)

IWORK

          IWORK is INTEGER array, dimension (12*N)


          If JOBZ = 'V', then if INFO = 0, the first NS elements of


          IWORK are zero. If INFO > 0, then IWORK contains the indices


          of the eigenvectors that failed to converge in DSTEVX.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value


          > 0:  if INFO = i, then i eigenvectors failed to converge


                   in SSTEVX. The indices of the eigenvectors


                   (as returned by SSTEVX) are stored in the


                   array IWORK.


                if INFO = N*2 + 1, an internal error occurred.

Univ. of Tennessee

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NAG Ltd.

June 2016