1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
|
*> \brief \b CLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLA_PORPVGRW + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cla_porpvgrw.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cla_porpvgrw.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cla_porpvgrw.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* REAL FUNCTION CLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF, LDAF, WORK )
*
* .. Scalar Arguments ..
* CHARACTER*1 UPLO
* INTEGER NCOLS, LDA, LDAF
* ..
* .. Array Arguments ..
* COMPLEX A( LDA, * ), AF( LDAF, * )
* REAL WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*>
*> CLA_PORPVGRW computes the reciprocal pivot growth factor
*> norm(A)/norm(U). The "max absolute element" norm is used. If this is
*> much less than 1, the stability of the LU factorization of the
*> (equilibrated) matrix A could be poor. This also means that the
*> solution X, estimated condition numbers, and error bounds could be
*> unreliable.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] NCOLS
*> \verbatim
*> NCOLS is INTEGER
*> The number of columns of the matrix A. NCOLS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> On entry, the N-by-N matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*> AF is COMPLEX array, dimension (LDAF,N)
*> The triangular factor U or L from the Cholesky factorization
*> A = U**T*U or A = L*L**T, as computed by CPOTRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] WORK
*> \verbatim
*> WORK is REAL array, dimension (2*N)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup complexPOcomputational
*
* =====================================================================
REAL FUNCTION CLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF, LDAF, WORK )
*
* -- LAPACK computational routine (version 3.6.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER*1 UPLO
INTEGER NCOLS, LDA, LDAF
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), AF( LDAF, * )
REAL WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, J
REAL AMAX, UMAX, RPVGRW
LOGICAL UPPER
COMPLEX ZDUM
* ..
* .. External Functions ..
EXTERNAL LSAME
LOGICAL LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, REAL, AIMAG
* ..
* .. Statement Functions ..
REAL CABS1
* ..
* .. Statement Function Definitions ..
CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
* ..
* .. Executable Statements ..
UPPER = LSAME( 'Upper', UPLO )
*
* SPOTRF will have factored only the NCOLSxNCOLS leading minor, so
* we restrict the growth search to that minor and use only the first
* 2*NCOLS workspace entries.
*
RPVGRW = 1.0
DO I = 1, 2*NCOLS
WORK( I ) = 0.0
END DO
*
* Find the max magnitude entry of each column.
*
IF ( UPPER ) THEN
DO J = 1, NCOLS
DO I = 1, J
WORK( NCOLS+J ) =
$ MAX( CABS1( A( I, J ) ), WORK( NCOLS+J ) )
END DO
END DO
ELSE
DO J = 1, NCOLS
DO I = J, NCOLS
WORK( NCOLS+J ) =
$ MAX( CABS1( A( I, J ) ), WORK( NCOLS+J ) )
END DO
END DO
END IF
*
* Now find the max magnitude entry of each column of the factor in
* AF. No pivoting, so no permutations.
*
IF ( LSAME( 'Upper', UPLO ) ) THEN
DO J = 1, NCOLS
DO I = 1, J
WORK( J ) = MAX( CABS1( AF( I, J ) ), WORK( J ) )
END DO
END DO
ELSE
DO J = 1, NCOLS
DO I = J, NCOLS
WORK( J ) = MAX( CABS1( AF( I, J ) ), WORK( J ) )
END DO
END DO
END IF
*
* Compute the *inverse* of the max element growth factor. Dividing
* by zero would imply the largest entry of the factor's column is
* zero. Than can happen when either the column of A is zero or
* massive pivots made the factor underflow to zero. Neither counts
* as growth in itself, so simply ignore terms with zero
* denominators.
*
IF ( LSAME( 'Upper', UPLO ) ) THEN
DO I = 1, NCOLS
UMAX = WORK( I )
AMAX = WORK( NCOLS+I )
IF ( UMAX /= 0.0 ) THEN
RPVGRW = MIN( AMAX / UMAX, RPVGRW )
END IF
END DO
ELSE
DO I = 1, NCOLS
UMAX = WORK( I )
AMAX = WORK( NCOLS+I )
IF ( UMAX /= 0.0 ) THEN
RPVGRW = MIN( AMAX / UMAX, RPVGRW )
END IF
END DO
END IF
CLA_PORPVGRW = RPVGRW
END
|
