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dla_porpvgrw.f
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1 *> \brief \b DLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DLA_PORPVGRW + dependencies
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * DOUBLE PRECISION FUNCTION DLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF,
22 * LDAF, WORK )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER*1 UPLO
26 * INTEGER NCOLS, LDA, LDAF
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), WORK( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *>
39 *> DLA_PORPVGRW computes the reciprocal pivot growth factor
40 *> norm(A)/norm(U). The "max absolute element" norm is used. If this is
41 *> much less than 1, the stability of the LU factorization of the
42 *> (equilibrated) matrix A could be poor. This also means that the
43 *> solution X, estimated condition numbers, and error bounds could be
44 *> unreliable.
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] UPLO
51 *> \verbatim
52 *> UPLO is CHARACTER*1
53 *> = 'U': Upper triangle of A is stored;
54 *> = 'L': Lower triangle of A is stored.
55 *> \endverbatim
56 *>
57 *> \param[in] NCOLS
58 *> \verbatim
59 *> NCOLS is INTEGER
60 *> The number of columns of the matrix A. NCOLS >= 0.
61 *> \endverbatim
62 *>
63 *> \param[in] A
64 *> \verbatim
65 *> A is DOUBLE PRECISION array, dimension (LDA,N)
66 *> On entry, the N-by-N matrix A.
67 *> \endverbatim
68 *>
69 *> \param[in] LDA
70 *> \verbatim
71 *> LDA is INTEGER
72 *> The leading dimension of the array A. LDA >= max(1,N).
73 *> \endverbatim
74 *>
75 *> \param[in] AF
76 *> \verbatim
77 *> AF is DOUBLE PRECISION array, dimension (LDAF,N)
78 *> The triangular factor U or L from the Cholesky factorization
79 *> A = U**T*U or A = L*L**T, as computed by DPOTRF.
80 *> \endverbatim
81 *>
82 *> \param[in] LDAF
83 *> \verbatim
84 *> LDAF is INTEGER
85 *> The leading dimension of the array AF. LDAF >= max(1,N).
86 *> \endverbatim
87 *>
88 *> \param[in] WORK
89 *> \verbatim
90 *> WORK is DOUBLE PRECISION array, dimension (2*N)
91 *> \endverbatim
92 *
93 * Authors:
94 * ========
95 *
96 *> \author Univ. of Tennessee
97 *> \author Univ. of California Berkeley
98 *> \author Univ. of Colorado Denver
99 *> \author NAG Ltd.
100 *
101 *> \date September 2012
102 *
103 *> \ingroup doublePOcomputational
104 *
105 * =====================================================================
106  DOUBLE PRECISION FUNCTION dla_porpvgrw( UPLO, NCOLS, A, LDA, AF,
107  $ ldaf, work )
108 *
109 * -- LAPACK computational routine (version 3.4.2) --
110 * -- LAPACK is a software package provided by Univ. of Tennessee, --
111 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
112 * September 2012
113 *
114 * .. Scalar Arguments ..
115  CHARACTER*1 uplo
116  INTEGER ncols, lda, ldaf
117 * ..
118 * .. Array Arguments ..
119  DOUBLE PRECISION a( lda, * ), af( ldaf, * ), work( * )
120 * ..
121 *
122 * =====================================================================
123 *
124 * .. Local Scalars ..
125  INTEGER i, j
126  DOUBLE PRECISION amax, umax, rpvgrw
127  LOGICAL upper
128 * ..
129 * .. Intrinsic Functions ..
130  INTRINSIC abs, max, min
131 * ..
132 * .. External Functions ..
133  EXTERNAL lsame, dlaset
134  LOGICAL lsame
135 * ..
136 * .. Executable Statements ..
137 *
138  upper = lsame( 'Upper', uplo )
139 *
140 * DPOTRF will have factored only the NCOLSxNCOLS leading minor, so
141 * we restrict the growth search to that minor and use only the first
142 * 2*NCOLS workspace entries.
143 *
144  rpvgrw = 1.0d+0
145  DO i = 1, 2*ncols
146  work( i ) = 0.0d+0
147  END DO
148 *
149 * Find the max magnitude entry of each column.
150 *
151  IF ( upper ) THEN
152  DO j = 1, ncols
153  DO i = 1, j
154  work( ncols+j ) =
155  $ max( abs( a( i, j ) ), work( ncols+j ) )
156  END DO
157  END DO
158  ELSE
159  DO j = 1, ncols
160  DO i = j, ncols
161  work( ncols+j ) =
162  $ max( abs( a( i, j ) ), work( ncols+j ) )
163  END DO
164  END DO
165  END IF
166 *
167 * Now find the max magnitude entry of each column of the factor in
168 * AF. No pivoting, so no permutations.
169 *
170  IF ( lsame( 'Upper', uplo ) ) THEN
171  DO j = 1, ncols
172  DO i = 1, j
173  work( j ) = max( abs( af( i, j ) ), work( j ) )
174  END DO
175  END DO
176  ELSE
177  DO j = 1, ncols
178  DO i = j, ncols
179  work( j ) = max( abs( af( i, j ) ), work( j ) )
180  END DO
181  END DO
182  END IF
183 *
184 * Compute the *inverse* of the max element growth factor. Dividing
185 * by zero would imply the largest entry of the factor's column is
186 * zero. Than can happen when either the column of A is zero or
187 * massive pivots made the factor underflow to zero. Neither counts
188 * as growth in itself, so simply ignore terms with zero
189 * denominators.
190 *
191  IF ( lsame( 'Upper', uplo ) ) THEN
192  DO i = 1, ncols
193  umax = work( i )
194  amax = work( ncols+i )
195  IF ( umax /= 0.0d+0 ) THEN
196  rpvgrw = min( amax / umax, rpvgrw )
197  END IF
198  END DO
199  ELSE
200  DO i = 1, ncols
201  umax = work( i )
202  amax = work( ncols+i )
203  IF ( umax /= 0.0d+0 ) THEN
204  rpvgrw = min( amax / umax, rpvgrw )
205  END IF
206  END DO
207  END IF
208 
209  dla_porpvgrw = rpvgrw
210  END