Using a divide and conquer approach, SLASDA computes the singular


 value decomposition (SVD) of a real upper bidiagonal N-by-M matrix


 B with diagonal D and offdiagonal E, where M = N + SQRE. The


 algorithm computes the singular values in the SVD B = U * S * VT.


 The orthogonal matrices U and VT are optionally computed in


 compact form.


 A related subroutine, SLASD0, computes the singular values and


 the singular vectors in explicit form.

ICOMPQ

          ICOMPQ is INTEGER


         Specifies whether singular vectors are to be computed


         in compact form, as follows


         = 0: Compute singular values only.


         = 1: Compute singular vectors of upper bidiagonal


              matrix in compact form.

SMLSIZ

          SMLSIZ is INTEGER


         The maximum size of the subproblems at the bottom of the


         computation tree.

N

          N is INTEGER


         The row dimension of the upper bidiagonal matrix. This is


         also the dimension of the main diagonal array D.

SQRE

          SQRE is INTEGER


         Specifies the column dimension of the bidiagonal matrix.


         = 0: The bidiagonal matrix has column dimension M = N;


         = 1: The bidiagonal matrix has column dimension M = N + 1.

D

          D is REAL array, dimension ( N )


         On entry D contains the main diagonal of the bidiagonal


         matrix. On exit D, if INFO = 0, contains its singular values.

E

          E is REAL array, dimension ( M-1 )


         Contains the subdiagonal entries of the bidiagonal matrix.


         On exit, E has been destroyed.

U

          U is REAL array,


         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced


         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left


         singular vector matrices of all subproblems at the bottom


         level.

LDU

          LDU is INTEGER, LDU = > N.


         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,


         GIVNUM, and Z.

VT

          VT is REAL array,


         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced


         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right


         singular vector matrices of all subproblems at the bottom


         level.

K

          K is INTEGER array, dimension ( N )


         if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.


         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th


         secular equation on the computation tree.

DIFL

          DIFL is REAL array, dimension ( LDU, NLVL ),


         where NLVL = floor(log_2 (N/SMLSIZ))).

DIFR

          DIFR is REAL array,


                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and


                  dimension ( N ) if ICOMPQ = 0.


         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)


         record distances between singular values on the I-th


         level and singular values on the (I -1)-th level, and


         DIFR(1:N, 2 * I ) contains the normalizing factors for


         the right singular vector matrix. See SLASD8 for details.

Z

          Z is REAL array,


                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and


                  dimension ( N ) if ICOMPQ = 0.


         The first K elements of Z(1, I) contain the components of


         the deflation-adjusted updating row vector for subproblems


         on the I-th level.

POLES

          POLES is REAL array,


         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced


         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and


         POLES(1, 2*I) contain  the new and old singular values


         involved in the secular equations on the I-th level.

GIVPTR

          GIVPTR is INTEGER array,


         dimension ( N ) if ICOMPQ = 1, and not referenced if


         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records


         the number of Givens rotations performed on the I-th


         problem on the computation tree.

GIVCOL

          GIVCOL is INTEGER array,


         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not


         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,


         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations


         of Givens rotations performed on the I-th level on the


         computation tree.

LDGCOL

          LDGCOL is INTEGER, LDGCOL = > N.


         The leading dimension of arrays GIVCOL and PERM.

PERM

          PERM is INTEGER array, dimension ( LDGCOL, NLVL )


         if ICOMPQ = 1, and not referenced


         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records


         permutations done on the I-th level of the computation tree.

GIVNUM

          GIVNUM is REAL array,


         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not


         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,


         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-


         values of Givens rotations performed on the I-th level on


         the computation tree.

C

          C is REAL array,


         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.


         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,


         C( I ) contains the C-value of a Givens rotation related to


         the right null space of the I-th subproblem.

S

          S is REAL array, dimension ( N ) if


         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1


         and the I-th subproblem is not square, on exit, S( I )


         contains the S-value of a Givens rotation related to


         the right null space of the I-th subproblem.

WORK

          WORK is REAL array, dimension


         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).

IWORK

          IWORK is INTEGER array, dimension (7*N).

INFO

          INFO is INTEGER


          = 0:  successful exit.


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


          > 0:  if INFO = 1, a singular value did not converge

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA