1SLAR1V(1)           LAPACK auxiliary routine (version 3.1)           SLAR1V(1)
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NAME

6       SLAR1V  -  the (scaled) r-th column of the inverse of the sumbmatrix in
7       rows B1 through BN of the tridiagonal matrix L D L^T - sigma I
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SYNOPSIS

10       SUBROUTINE SLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z,
11                          WANTNC,  NEGCNT,  ZTZ,  MINGMA,  R,  ISUPPZ, NRMINV,
12                          RESID, RQCORR, WORK )
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14           LOGICAL        WANTNC
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16           INTEGER        B1, BN, N, NEGCNT, R
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18           REAL           GAPTOL,  LAMBDA,  MINGMA,  NRMINV,  PIVMIN,   RESID,
19                          RQCORR, ZTZ
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21           INTEGER        ISUPPZ( * )
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23           REAL           D( * ), L( * ), LD( * ), LLD( * ), WORK( * )
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25           REAL           Z( * )
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PURPOSE

28       SLAR1V  computes the (scaled) r-th column of the inverse of the sumbma‐
29       trix in rows B1 through BN of the tridiagonal matrix L D L^T - sigma I.
30       When  sigma  is close to an eigenvalue, the computed vector is an accu‐
31       rate eigenvector. Usually, r corresponds to the index where the  eigen‐
32       vector  is  largest  in magnitude.  The following steps accomplish this
33       computation :
34       (a) Stationary qd transform,  L D L^T - sigma I = L(+) D(+) L(+)^T, (b)
35       Progressive  qd  transform,  L  D L^T - sigma I = U(-) D(-) U(-)^T, (c)
36       Computation of the diagonal elements of the inverse of
37           L D L^T - sigma I by combining the above transforms, and choosing
38           r as the index where the diagonal of the inverse is (one of the)
39           largest in magnitude.
40       (d) Computation of the (scaled) r-th column of the inverse using the
41           twisted factorization obtained by combining the top part of the
42           the stationary and the bottom part of the progressive transform.
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ARGUMENTS

46       N        (input) INTEGER
47                The order of the matrix L D L^T.
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49       B1       (input) INTEGER
50                First index of the submatrix of L D L^T.
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52       BN       (input) INTEGER
53                Last index of the submatrix of L D L^T.
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55       LAMBDA    (input) REAL
56                 The shift. In  order  to  compute  an  accurate  eigenvector,
57                 LAMBDA should be a good approximation to an eigenvalue of L D
58                 L^T.
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60       L        (input) REAL             array, dimension (N-1)
61                The (n-1) subdiagonal elements of the unit  bidiagonal  matrix
62                L, in elements 1 to N-1.
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64       D        (input) REAL             array, dimension (N)
65                The n diagonal elements of the diagonal matrix D.
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67       LD       (input) REAL             array, dimension (N-1)
68                The n-1 elements L(i)*D(i).
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70       LLD      (input) REAL             array, dimension (N-1)
71                The n-1 elements L(i)*L(i)*D(i).
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73       PIVMIN   (input) REAL
74                The minimum pivot in the Sturm sequence.
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76       GAPTOL   (input) REAL
77                Tolerance that indicates when eigenvector entries are negligi‐
78                ble w.r.t. their contribution to the residual.
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80       Z        (input/output) REAL             array, dimension (N)
81                On input, all entries of Z must be set to  0.   On  output,  Z
82                contains  the (scaled) r-th column of the inverse. The scaling
83                is such that Z(R) equals 1.
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85       WANTNC   (input) LOGICAL
86                Specifies whether NEGCNT has to be computed.
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88       NEGCNT   (output) INTEGER
89                If WANTNC is .TRUE. then NEGCNT  =  the  number  of  pivots  <
90                pivmin  in  the  matrix factorization L D L^T, and NEGCNT = -1
91                otherwise.
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93       ZTZ      (output) REAL
94                The square of the 2-norm of Z.
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96       MINGMA   (output) REAL
97                The reciprocal of the largest (in magnitude) diagonal  element
98                of the inverse of L D L^T - sigma I.
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100       R        (input/output) INTEGER
101                The  twist index for the twisted factorization used to compute
102                Z.  On input, 0 <= R <= N. If R is input as 0, R is set to the
103                index  where (L D L^T - sigma I)^{-1} is largest in magnitude.
104                If 1 <= R <= N, R is unchanged.  On  output,  R  contains  the
105                twist  index  used  to  compute  Z.  Ideally, R designates the
106                position of the maximum entry in the eigenvector.
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108       ISUPPZ   (output) INTEGER array, dimension (2)
109                The support of the vector in Z, i.e., the vector Z is  nonzero
110                only in elements ISUPPZ(1) through ISUPPZ( 2 ).
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112       NRMINV   (output) REAL
113                NRMINV = 1/SQRT( ZTZ )
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115       RESID    (output) REAL
116                The  residual  of  the FP vector.  RESID = ABS( MINGMA )/SQRT(
117                ZTZ )
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119       RQCORR   (output) REAL
120                The  Rayleigh  Quotient  correction  to  LAMBDA.    RQCORR   =
121                MINGMA*TMP
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123       WORK     (workspace) REAL             array, dimension (4*N)
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FURTHER DETAILS

126       Based on contributions by
127          Beresford Parlett, University of California, Berkeley, USA
128          Jim Demmel, University of California, Berkeley, USA
129          Inderjit Dhillon, University of Texas, Austin, USA
130          Osni Marques, LBNL/NERSC, USA
131          Christof Voemel, University of California, Berkeley, USA
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136 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       SLAR1V(1)