1SLASD3(1)           LAPACK auxiliary routine (version 3.1)           SLASD3(1)
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NAME

6       SLASD3  - 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 SLASD3( 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           REAL           D(  *  ), DSIGMA( * ), Q( LDQ, * ), U( LDU, * ), U2(
18                          LDU2, * ), VT( LDVT, * ), VT2( LDVT2, * ), Z( * )
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PURPOSE

21       SLASD3 finds all the square roots of the roots of the secular equation,
22       as defined by the values in D and Z.  It makes the appropriate calls to
23       SLASD4 and then updates the singular vectors by matrix multiplication.
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25       This code makes very mild assumptions about floating point  arithmetic.
26       It  will  work  on  machines  with a guard digit in add/subtract, or on
27       those binary machines without guard digits which subtract like the Cray
28       XMP,  Cray  YMP,  Cray  C  90, or Cray 2.  It could conceivably fail on
29       hexadecimal or decimal machines without guard digits, but  we  know  of
30       none.
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32       SLASD3 is called from SLASD1.
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ARGUMENTS

36       NL     (input) INTEGER
37              The row dimension of the upper block.  NL >= 1.
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39       NR     (input) INTEGER
40              The row dimension of the lower block.  NR >= 1.
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42       SQRE   (input) INTEGER
43              = 0: the lower block is an NR-by-NR square matrix.
44              = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
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46              The  bidiagonal matrix has N = NL + NR + 1 rows and M = N + SQRE
47              >= N columns.
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49       K      (input) INTEGER
50              The size of the secular equation, 1 =< K = < N.
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52       D      (output) REAL array, dimension(K)
53              On exit the square roots of the roots of the  secular  equation,
54              in ascending order.
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56       Q      (workspace) REAL array,
57              dimension at least (LDQ,K).
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59       LDQ    (input) INTEGER
60              The leading dimension of the array Q.  LDQ >= K.
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62              DSIGMA  (input/output) REAL array, dimension(K) The first K ele‐
63              ments of this array contain the old roots of the deflated updat‐
64              ing problem.  These are the poles of the secular equation.
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66       U      (output) REAL array, dimension (LDU, N)
67              The  last N - K columns of this matrix contain the deflated left
68              singular vectors.
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70       LDU    (input) INTEGER
71              The leading dimension of the array U.  LDU >= N.
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73       U2     (input) REAL array, dimension (LDU2, N)
74              The first K columns of this matrix contain the non-deflated left
75              singular vectors for the split problem.
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77       LDU2   (input) INTEGER
78              The leading dimension of the array U2.  LDU2 >= N.
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80       VT     (output) REAL array, dimension (LDVT, M)
81              The  last M - K columns of VT' contain the deflated right singu‐
82              lar vectors.
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84       LDVT   (input) INTEGER
85              The leading dimension of the array VT.  LDVT >= N.
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87       VT2    (input/output) REAL array, dimension (LDVT2, N)
88              The first K columns of VT2' contain the non-deflated right  sin‐
89              gular vectors for the split problem.
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91       LDVT2  (input) INTEGER
92              The leading dimension of the array VT2.  LDVT2 >= N.
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94       IDXC   (input) INTEGER array, dimension (N)
95              The  permutation  used  to arrange the columns of U (and rows of
96              VT) into  three  groups:   the  first  group  contains  non-zero
97              entries only at and above (or before) NL +1; the second contains
98              non-zero entries only at and below  (or  after)  NL+2;  and  the
99              third  is  dense.  The  first  column of U and the row of VT are
100              treated separately, however.
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102              The rows of the singular vectors found by SLASD4 must  be  like‐
103              wise permuted before the matrix multiplies can take place.
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105       CTOT   (input) INTEGER array, dimension (4)
106              A count of the total number of the various types of columns in U
107              (or rows in VT), as described in IDXC. The fourth column type is
108              any column which has been deflated.
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110       Z      (input/output) REAL array, dimension (K)
111              The first K elements of this array contain the components of the
112              deflation-adjusted updating row vector.
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114       INFO   (output) INTEGER
115              = 0:  successful exit.
116              < 0:  if INFO = -i, the i-th argument had an illegal value.
117              > 0:  if INFO = 1, an singular value did not converge
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FURTHER DETAILS

120       Based on contributions by
121          Ming Gu and Huan Ren, Computer Science Division, University of
122          California at Berkeley, USA
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127 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       SLASD3(1)