1DLASD3(1)           LAPACK auxiliary routine (version 3.1)           DLASD3(1)
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NAME

6       DLASD3  - all the square roots of the roots of the secular equation, as
7       defined by the values in D and Z
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SYNOPSIS

10       SUBROUTINE DLASD3( NL, NR, SQRE, K, D, Q,  LDQ,  DSIGMA,  U,  LDU,  U2,
11                          LDU2, VT, LDVT, VT2, LDVT2, IDXC, CTOT, Z, INFO )
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13           INTEGER        INFO, K, LDQ, LDU, LDU2, LDVT, LDVT2, NL, NR, SQRE
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15           INTEGER        CTOT( * ), IDXC( * )
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17           DOUBLE         PRECISION  D( * ), DSIGMA( * ), Q( LDQ, * ), U( LDU,
18                          * ), U2( LDU2, * ), VT( LDVT, * ), VT2( LDVT2, *  ),
19                          Z( * )
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PURPOSE

22       DLASD3 finds all the square roots of the roots of the secular equation,
23       as defined by the values in D and Z.  It makes the appropriate calls to
24       DLASD4 and then updates the singular vectors by matrix multiplication.
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26       This  code makes very mild assumptions about floating point arithmetic.
27       It will work on machines with a guard  digit  in  add/subtract,  or  on
28       those binary machines without guard digits which subtract like the Cray
29       XMP, Cray YMP, Cray C 90, or Cray 2.   It  could  conceivably  fail  on
30       hexadecimal  or  decimal  machines without guard digits, but we know of
31       none.
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33       DLASD3 is called from DLASD1.
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ARGUMENTS

37       NL     (input) INTEGER
38              The row dimension of the upper block.  NL >= 1.
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40       NR     (input) INTEGER
41              The row dimension of the lower block.  NR >= 1.
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43       SQRE   (input) INTEGER
44              = 0: the lower block is an NR-by-NR square matrix.
45              = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
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47              The bidiagonal matrix has N = NL + NR + 1 rows and M = N +  SQRE
48              >= N columns.
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50       K      (input) INTEGER
51              The size of the secular equation, 1 =< K = < N.
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53       D      (output) DOUBLE PRECISION array, dimension(K)
54              On  exit  the square roots of the roots of the secular equation,
55              in ascending order.
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57       Q      (workspace) DOUBLE PRECISION array,
58              dimension at least (LDQ,K).
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60       LDQ    (input) INTEGER
61              The leading dimension of the array Q.  LDQ >= K.
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63              DSIGMA (input) DOUBLE PRECISION array, dimension(K) The first  K
64              elements  of  this  array  contain the old roots of the deflated
65              updating problem.  These are the poles of the secular equation.
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67       U      (output) DOUBLE PRECISION array, dimension (LDU, N)
68              The last N - K columns of this matrix contain the deflated  left
69              singular vectors.
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71       LDU    (input) INTEGER
72              The leading dimension of the array U.  LDU >= N.
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74       U2     (input/output) DOUBLE PRECISION array, dimension (LDU2, N)
75              The first K columns of this matrix contain the non-deflated left
76              singular vectors for the split problem.
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78       LDU2   (input) INTEGER
79              The leading dimension of the array U2.  LDU2 >= N.
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81       VT     (output) DOUBLE PRECISION array, dimension (LDVT, M)
82              The last M - K columns of VT' contain the deflated right  singu‐
83              lar vectors.
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85       LDVT   (input) INTEGER
86              The leading dimension of the array VT.  LDVT >= N.
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88       VT2    (input/output) DOUBLE PRECISION array, dimension (LDVT2, N)
89              The  first K columns of VT2' contain the non-deflated right sin‐
90              gular vectors for the split problem.
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92       LDVT2  (input) INTEGER
93              The leading dimension of the array VT2.  LDVT2 >= N.
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95       IDXC   (input) INTEGER array, dimension ( N )
96              The permutation used to arrange the columns of U  (and  rows  of
97              VT)  into  three  groups:   the  first  group  contains non-zero
98              entries only at and above (or before) NL +1; the second contains
99              non-zero  entries  only  at  and  below (or after) NL+2; and the
100              third is dense. The first column of U and  the  row  of  VT  are
101              treated separately, however.
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103              The  rows  of the singular vectors found by DLASD4 must be like‐
104              wise permuted before the matrix multiplies can take place.
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106       CTOT   (input) INTEGER array, dimension ( 4 )
107              A count of the total number of the various types of columns in U
108              (or rows in VT), as described in IDXC. The fourth column type is
109              any column which has been deflated.
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111       Z      (input) DOUBLE PRECISION array, dimension (K)
112              The first K elements of this array contain the components of the
113              deflation-adjusted updating row vector.
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115       INFO   (output) INTEGER
116              = 0:  successful exit.
117              < 0:  if INFO = -i, the i-th argument had an illegal value.
118              > 0:  if INFO = 1, an singular value did not converge
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FURTHER DETAILS

121       Based on contributions by
122          Ming Gu and Huan Ren, Computer Science Division, University of
123          California at Berkeley, USA
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128 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       DLASD3(1)