LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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sla_syrpvgrw.f
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1*> \brief \b SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* REAL FUNCTION SLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, IPIV,
22* WORK )
23*
24* .. Scalar Arguments ..
25* CHARACTER*1 UPLO
26* INTEGER N, INFO, LDA, LDAF
27* ..
28* .. Array Arguments ..
29* INTEGER IPIV( * )
30* REAL A( LDA, * ), AF( LDAF, * ), WORK( * )
31* ..
32*
33*
34*> \par Purpose:
35* =============
36*>
37*> \verbatim
38*>
39*>
40*> SLA_SYRPVGRW computes the reciprocal pivot growth factor
41*> norm(A)/norm(U). The "max absolute element" norm is used. If this is
42*> much less than 1, the stability of the LU factorization of the
43*> (equilibrated) matrix A could be poor. This also means that the
44*> solution X, estimated condition numbers, and error bounds could be
45*> unreliable.
46*> \endverbatim
47*
48* Arguments:
49* ==========
50*
51*> \param[in] UPLO
52*> \verbatim
53*> UPLO is CHARACTER*1
54*> = 'U': Upper triangle of A is stored;
55*> = 'L': Lower triangle of A is stored.
56*> \endverbatim
57*>
58*> \param[in] N
59*> \verbatim
60*> N is INTEGER
61*> The number of linear equations, i.e., the order of the
62*> matrix A. N >= 0.
63*> \endverbatim
64*>
65*> \param[in] INFO
66*> \verbatim
67*> INFO is INTEGER
68*> The value of INFO returned from SSYTRF, .i.e., the pivot in
69*> column INFO is exactly 0.
70*> \endverbatim
71*>
72*> \param[in] A
73*> \verbatim
74*> A is REAL array, dimension (LDA,N)
75*> On entry, the N-by-N matrix A.
76*> \endverbatim
77*>
78*> \param[in] LDA
79*> \verbatim
80*> LDA is INTEGER
81*> The leading dimension of the array A. LDA >= max(1,N).
82*> \endverbatim
83*>
84*> \param[in] AF
85*> \verbatim
86*> AF is REAL array, dimension (LDAF,N)
87*> The block diagonal matrix D and the multipliers used to
88*> obtain the factor U or L as computed by SSYTRF.
89*> \endverbatim
90*>
91*> \param[in] LDAF
92*> \verbatim
93*> LDAF is INTEGER
94*> The leading dimension of the array AF. LDAF >= max(1,N).
95*> \endverbatim
96*>
97*> \param[in] IPIV
98*> \verbatim
99*> IPIV is INTEGER array, dimension (N)
100*> Details of the interchanges and the block structure of D
101*> as determined by SSYTRF.
102*> \endverbatim
103*>
104*> \param[out] WORK
105*> \verbatim
106*> WORK is REAL array, dimension (2*N)
107*> \endverbatim
108*
109* Authors:
110* ========
111*
112*> \author Univ. of Tennessee
113*> \author Univ. of California Berkeley
114*> \author Univ. of Colorado Denver
115*> \author NAG Ltd.
116*
117*> \ingroup realSYcomputational
118*
119* =====================================================================
120 REAL function sla_syrpvgrw( uplo, n, info, a, lda, af, ldaf, ipiv,
121 \$ work )
122*
123* -- LAPACK computational routine --
124* -- LAPACK is a software package provided by Univ. of Tennessee, --
125* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
126*
127* .. Scalar Arguments ..
128 CHARACTER*1 uplo
129 INTEGER n, info, lda, ldaf
130* ..
131* .. Array Arguments ..
132 INTEGER ipiv( * )
133 REAL a( lda, * ), af( ldaf, * ), work( * )
134* ..
135*
136* =====================================================================
137*
138* .. Local Scalars ..
139 INTEGER ncols, i, j, k, kp
140 REAL amax, umax, rpvgrw, tmp
141 LOGICAL upper
142* ..
143* .. Intrinsic Functions ..
144 INTRINSIC abs, max, min
145* ..
146* .. External Functions ..
147 EXTERNAL lsame
148 LOGICAL lsame
149* ..
150* .. Executable Statements ..
151*
152 upper = lsame( 'Upper', uplo )
153 IF ( info.EQ.0 ) THEN
154 IF ( upper ) THEN
155 ncols = 1
156 ELSE
157 ncols = n
158 END IF
159 ELSE
160 ncols = info
161 END IF
162
163 rpvgrw = 1.0
164 DO i = 1, 2*n
165 work( i ) = 0.0
166 END DO
167*
168* Find the max magnitude entry of each column of A. Compute the max
169* for all N columns so we can apply the pivot permutation while
170* looping below. Assume a full factorization is the common case.
171*
172 IF ( upper ) THEN
173 DO j = 1, n
174 DO i = 1, j
175 work( n+i ) = max( abs( a( i, j ) ), work( n+i ) )
176 work( n+j ) = max( abs( a( i, j ) ), work( n+j ) )
177 END DO
178 END DO
179 ELSE
180 DO j = 1, n
181 DO i = j, n
182 work( n+i ) = max( abs( a( i, j ) ), work( n+i ) )
183 work( n+j ) = max( abs( a( i, j ) ), work( n+j ) )
184 END DO
185 END DO
186 END IF
187*
188* Now find the max magnitude entry of each column of U or L. Also
189* permute the magnitudes of A above so they're in the same order as
190* the factor.
191*
192* The iteration orders and permutations were copied from ssytrs.
193* Calls to SSWAP would be severe overkill.
194*
195 IF ( upper ) THEN
196 k = n
197 DO WHILE ( k .LT. ncols .AND. k.GT.0 )
198 IF ( ipiv( k ).GT.0 ) THEN
199! 1x1 pivot
200 kp = ipiv( k )
201 IF ( kp .NE. k ) THEN
202 tmp = work( n+k )
203 work( n+k ) = work( n+kp )
204 work( n+kp ) = tmp
205 END IF
206 DO i = 1, k
207 work( k ) = max( abs( af( i, k ) ), work( k ) )
208 END DO
209 k = k - 1
210 ELSE
211! 2x2 pivot
212 kp = -ipiv( k )
213 tmp = work( n+k-1 )
214 work( n+k-1 ) = work( n+kp )
215 work( n+kp ) = tmp
216 DO i = 1, k-1
217 work( k ) = max( abs( af( i, k ) ), work( k ) )
218 work( k-1 ) = max( abs( af( i, k-1 ) ), work( k-1 ) )
219 END DO
220 work( k ) = max( abs( af( k, k ) ), work( k ) )
221 k = k - 2
222 END IF
223 END DO
224 k = ncols
225 DO WHILE ( k .LE. n )
226 IF ( ipiv( k ).GT.0 ) THEN
227 kp = ipiv( k )
228 IF ( kp .NE. k ) THEN
229 tmp = work( n+k )
230 work( n+k ) = work( n+kp )
231 work( n+kp ) = tmp
232 END IF
233 k = k + 1
234 ELSE
235 kp = -ipiv( k )
236 tmp = work( n+k )
237 work( n+k ) = work( n+kp )
238 work( n+kp ) = tmp
239 k = k + 2
240 END IF
241 END DO
242 ELSE
243 k = 1
244 DO WHILE ( k .LE. ncols )
245 IF ( ipiv( k ).GT.0 ) THEN
246! 1x1 pivot
247 kp = ipiv( k )
248 IF ( kp .NE. k ) THEN
249 tmp = work( n+k )
250 work( n+k ) = work( n+kp )
251 work( n+kp ) = tmp
252 END IF
253 DO i = k, n
254 work( k ) = max( abs( af( i, k ) ), work( k ) )
255 END DO
256 k = k + 1
257 ELSE
258! 2x2 pivot
259 kp = -ipiv( k )
260 tmp = work( n+k+1 )
261 work( n+k+1 ) = work( n+kp )
262 work( n+kp ) = tmp
263 DO i = k+1, n
264 work( k ) = max( abs( af( i, k ) ), work( k ) )
265 work( k+1 ) = max( abs( af(i, k+1 ) ), work( k+1 ) )
266 END DO
267 work( k ) = max( abs( af( k, k ) ), work( k ) )
268 k = k + 2
269 END IF
270 END DO
271 k = ncols
272 DO WHILE ( k .GE. 1 )
273 IF ( ipiv( k ).GT.0 ) THEN
274 kp = ipiv( k )
275 IF ( kp .NE. k ) THEN
276 tmp = work( n+k )
277 work( n+k ) = work( n+kp )
278 work( n+kp ) = tmp
279 END IF
280 k = k - 1
281 ELSE
282 kp = -ipiv( k )
283 tmp = work( n+k )
284 work( n+k ) = work( n+kp )
285 work( n+kp ) = tmp
286 k = k - 2
287 ENDIF
288 END DO
289 END IF
290*
291* Compute the *inverse* of the max element growth factor. Dividing
292* by zero would imply the largest entry of the factor's column is
293* zero. Than can happen when either the column of A is zero or
294* massive pivots made the factor underflow to zero. Neither counts
295* as growth in itself, so simply ignore terms with zero
296* denominators.
297*
298 IF ( upper ) THEN
299 DO i = ncols, n
300 umax = work( i )
301 amax = work( n+i )
302 IF ( umax /= 0.0 ) THEN
303 rpvgrw = min( amax / umax, rpvgrw )
304 END IF
305 END DO
306 ELSE
307 DO i = 1, ncols
308 umax = work( i )
309 amax = work( n+i )
310 IF ( umax /= 0.0 ) THEN
311 rpvgrw = min( amax / umax, rpvgrw )
312 END IF
313 END DO
314 END IF
315
316 sla_syrpvgrw = rpvgrw
317*
318* End of SLA_SYRPVGRW
319*
320 END
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function sla_syrpvgrw(UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, WORK)
SLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite m...
Definition: sla_syrpvgrw.f:122