DLASD3 finds all the square roots of the roots of the secular


 equation, as defined by the values in D and Z.  It makes the


 appropriate calls to DLASD4 and then updates the singular


 vectors by matrix multiplication.


 This code makes very mild assumptions about floating point


 arithmetic. It will work on machines with a guard digit in


 add/subtract, or on those binary machines without guard digits


 which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.


 It could conceivably fail on hexadecimal or decimal machines


 without guard digits, but we know of none.


 DLASD3 is called from DLASD1.

NL

          NL is INTEGER


         The row dimension of the upper block.  NL >= 1.

NR

          NR is INTEGER


         The row dimension of the lower block.  NR >= 1.

SQRE

          SQRE is INTEGER


         = 0: the lower block is an NR-by-NR square matrix.


         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.


         The bidiagonal matrix has N = NL + NR + 1 rows and


         M = N + SQRE >= N columns.

K

          K is INTEGER


         The size of the secular equation, 1 =< K = < N.

D

          D is DOUBLE PRECISION array, dimension(K)


         On exit the square roots of the roots of the secular equation,


         in ascending order.

Q

          Q is DOUBLE PRECISION array,


                     dimension at least (LDQ,K).

LDQ

          LDQ is INTEGER


         The leading dimension of the array Q.  LDQ >= K.

DSIGMA

          DSIGMA is DOUBLE PRECISION array, dimension(K)


         The first K elements of this array contain the old roots


         of the deflated updating problem.  These are the poles


         of the secular equation.

U

          U is DOUBLE PRECISION array, dimension (LDU, N)


         The last N - K columns of this matrix contain the deflated


         left singular vectors.

LDU

          LDU is INTEGER


         The leading dimension of the array U.  LDU >= N.

U2

          U2 is DOUBLE PRECISION array, dimension (LDU2, N)


         The first K columns of this matrix contain the non-deflated


         left singular vectors for the split problem.

LDU2

          LDU2 is INTEGER


         The leading dimension of the array U2.  LDU2 >= N.

VT

          VT is DOUBLE PRECISION array, dimension (LDVT, M)


         The last M - K columns of VT**T contain the deflated


         right singular vectors.

LDVT

          LDVT is INTEGER


         The leading dimension of the array VT.  LDVT >= N.

VT2

          VT2 is DOUBLE PRECISION array, dimension (LDVT2, N)


         The first K columns of VT2**T contain the non-deflated


         right singular vectors for the split problem.

LDVT2

          LDVT2 is INTEGER


         The leading dimension of the array VT2.  LDVT2 >= N.

IDXC

          IDXC is INTEGER array, dimension ( N )


         The permutation used to arrange the columns of U (and rows of


         VT) into three groups:  the first group contains non-zero


         entries only at and above (or before) NL +1; the second


         contains non-zero entries only at and below (or after) NL+2;


         and the third is dense. The first column of U and the row of


         VT are treated separately, however.


         The rows of the singular vectors found by DLASD4


         must be likewise permuted before the matrix multiplies can


         take place.

CTOT

          CTOT is INTEGER array, dimension ( 4 )


         A count of the total number of the various types of columns


         in U (or rows in VT), as described in IDXC. The fourth column


         type is any column which has been deflated.

Z

          Z is DOUBLE PRECISION array, dimension (K)


         The first K elements of this array contain the components


         of the deflation-adjusted updating row vector.

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.

September 2012

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