1: *> \brief \b ZLA_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
9: *> Download ZLA_SYRPVGRW + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zla_syrpvgrw.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zla_syrpvgrw.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zla_syrpvgrw.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * DOUBLE PRECISION FUNCTION ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
22: * LDAF, IPIV, WORK )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER*1 UPLO
26: * INTEGER N, INFO, LDA, LDAF
27: * ..
28: * .. Array Arguments ..
29: * COMPLEX*16 A( LDA, * ), AF( LDAF, * )
30: * DOUBLE PRECISION WORK( * )
31: * INTEGER IPIV( * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *>
41: *> ZLA_SYRPVGRW computes the reciprocal pivot growth factor
42: *> norm(A)/norm(U). The "max absolute element" norm is used. If this is
43: *> much less than 1, the stability of the LU factorization of the
44: *> (equilibrated) matrix A could be poor. This also means that the
45: *> solution X, estimated condition numbers, and error bounds could be
46: *> unreliable.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] UPLO
53: *> \verbatim
54: *> UPLO is CHARACTER*1
55: *> = 'U': Upper triangle of A is stored;
56: *> = 'L': Lower triangle of A is stored.
57: *> \endverbatim
58: *>
59: *> \param[in] N
60: *> \verbatim
61: *> N is INTEGER
62: *> The number of linear equations, i.e., the order of the
63: *> matrix A. N >= 0.
64: *> \endverbatim
65: *>
66: *> \param[in] INFO
67: *> \verbatim
68: *> INFO is INTEGER
69: *> The value of INFO returned from ZSYTRF, .i.e., the pivot in
70: *> column INFO is exactly 0.
71: *> \endverbatim
72: *>
73: *> \param[in] A
74: *> \verbatim
75: *> A is COMPLEX*16 array, dimension (LDA,N)
76: *> On entry, the N-by-N matrix A.
77: *> \endverbatim
78: *>
79: *> \param[in] LDA
80: *> \verbatim
81: *> LDA is INTEGER
82: *> The leading dimension of the array A. LDA >= max(1,N).
83: *> \endverbatim
84: *>
85: *> \param[in] AF
86: *> \verbatim
87: *> AF is COMPLEX*16 array, dimension (LDAF,N)
88: *> The block diagonal matrix D and the multipliers used to
89: *> obtain the factor U or L as computed by ZSYTRF.
90: *> \endverbatim
91: *>
92: *> \param[in] LDAF
93: *> \verbatim
94: *> LDAF is INTEGER
95: *> The leading dimension of the array AF. LDAF >= max(1,N).
96: *> \endverbatim
97: *>
98: *> \param[in] IPIV
99: *> \verbatim
100: *> IPIV is INTEGER array, dimension (N)
101: *> Details of the interchanges and the block structure of D
102: *> as determined by ZSYTRF.
103: *> \endverbatim
104: *>
105: *> \param[in] WORK
106: *> \verbatim
107: *> WORK is DOUBLE PRECISION array, dimension (2*N)
108: *> \endverbatim
109: *
110: * Authors:
111: * ========
112: *
113: *> \author Univ. of Tennessee
114: *> \author Univ. of California Berkeley
115: *> \author Univ. of Colorado Denver
116: *> \author NAG Ltd.
117: *
118: *> \date December 2016
119: *
120: *> \ingroup complex16SYcomputational
121: *
122: * =====================================================================
123: DOUBLE PRECISION FUNCTION ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
124: $ LDAF, IPIV, WORK )
125: *
126: * -- LAPACK computational routine (version 3.7.0) --
127: * -- LAPACK is a software package provided by Univ. of Tennessee, --
128: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
129: * December 2016
130: *
131: * .. Scalar Arguments ..
132: CHARACTER*1 UPLO
133: INTEGER N, INFO, LDA, LDAF
134: * ..
135: * .. Array Arguments ..
136: COMPLEX*16 A( LDA, * ), AF( LDAF, * )
137: DOUBLE PRECISION WORK( * )
138: INTEGER IPIV( * )
139: * ..
140: *
141: * =====================================================================
142: *
143: * .. Local Scalars ..
144: INTEGER NCOLS, I, J, K, KP
145: DOUBLE PRECISION AMAX, UMAX, RPVGRW, TMP
146: LOGICAL UPPER
147: COMPLEX*16 ZDUM
148: * ..
149: * .. Intrinsic Functions ..
150: INTRINSIC ABS, REAL, DIMAG, MAX, MIN
151: * ..
152: * .. External Subroutines ..
153: EXTERNAL LSAME
154: LOGICAL LSAME
155: * ..
156: * .. Statement Functions ..
157: DOUBLE PRECISION CABS1
158: * ..
159: * .. Statement Function Definitions ..
160: CABS1( ZDUM ) = ABS( DBLE ( ZDUM ) ) + ABS( DIMAG ( ZDUM ) )
161: * ..
162: * .. Executable Statements ..
163: *
164: UPPER = LSAME( 'Upper', UPLO )
165: IF ( INFO.EQ.0 ) THEN
166: IF ( UPPER ) THEN
167: NCOLS = 1
168: ELSE
169: NCOLS = N
170: END IF
171: ELSE
172: NCOLS = INFO
173: END IF
174:
175: RPVGRW = 1.0D+0
176: DO I = 1, 2*N
177: WORK( I ) = 0.0D+0
178: END DO
179: *
180: * Find the max magnitude entry of each column of A. Compute the max
181: * for all N columns so we can apply the pivot permutation while
182: * looping below. Assume a full factorization is the common case.
183: *
184: IF ( UPPER ) THEN
185: DO J = 1, N
186: DO I = 1, J
187: WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) )
188: WORK( N+J ) = MAX( CABS1( A( I, J ) ), WORK( N+J ) )
189: END DO
190: END DO
191: ELSE
192: DO J = 1, N
193: DO I = J, N
194: WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) )
195: WORK( N+J ) = MAX( CABS1( A( I, J ) ), WORK( N+J ) )
196: END DO
197: END DO
198: END IF
199: *
200: * Now find the max magnitude entry of each column of U or L. Also
201: * permute the magnitudes of A above so they're in the same order as
202: * the factor.
203: *
204: * The iteration orders and permutations were copied from zsytrs.
205: * Calls to SSWAP would be severe overkill.
206: *
207: IF ( UPPER ) THEN
208: K = N
209: DO WHILE ( K .LT. NCOLS .AND. K.GT.0 )
210: IF ( IPIV( K ).GT.0 ) THEN
211: ! 1x1 pivot
212: KP = IPIV( K )
213: IF ( KP .NE. K ) THEN
214: TMP = WORK( N+K )
215: WORK( N+K ) = WORK( N+KP )
216: WORK( N+KP ) = TMP
217: END IF
218: DO I = 1, K
219: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
220: END DO
221: K = K - 1
222: ELSE
223: ! 2x2 pivot
224: KP = -IPIV( K )
225: TMP = WORK( N+K-1 )
226: WORK( N+K-1 ) = WORK( N+KP )
227: WORK( N+KP ) = TMP
228: DO I = 1, K-1
229: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
230: WORK( K-1 ) =
231: $ MAX( CABS1( AF( I, K-1 ) ), WORK( K-1 ) )
232: END DO
233: WORK( K ) = MAX( CABS1( AF( K, K ) ), WORK( K ) )
234: K = K - 2
235: END IF
236: END DO
237: K = NCOLS
238: DO WHILE ( K .LE. N )
239: IF ( IPIV( K ).GT.0 ) THEN
240: KP = IPIV( K )
241: IF ( KP .NE. K ) THEN
242: TMP = WORK( N+K )
243: WORK( N+K ) = WORK( N+KP )
244: WORK( N+KP ) = TMP
245: END IF
246: K = K + 1
247: ELSE
248: KP = -IPIV( K )
249: TMP = WORK( N+K )
250: WORK( N+K ) = WORK( N+KP )
251: WORK( N+KP ) = TMP
252: K = K + 2
253: END IF
254: END DO
255: ELSE
256: K = 1
257: DO WHILE ( K .LE. NCOLS )
258: IF ( IPIV( K ).GT.0 ) THEN
259: ! 1x1 pivot
260: KP = IPIV( K )
261: IF ( KP .NE. K ) THEN
262: TMP = WORK( N+K )
263: WORK( N+K ) = WORK( N+KP )
264: WORK( N+KP ) = TMP
265: END IF
266: DO I = K, N
267: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
268: END DO
269: K = K + 1
270: ELSE
271: ! 2x2 pivot
272: KP = -IPIV( K )
273: TMP = WORK( N+K+1 )
274: WORK( N+K+1 ) = WORK( N+KP )
275: WORK( N+KP ) = TMP
276: DO I = K+1, N
277: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
278: WORK( K+1 ) =
279: $ MAX( CABS1( AF( I, K+1 ) ), WORK( K+1 ) )
280: END DO
281: WORK( K ) = MAX( CABS1( AF( K, K ) ), WORK( K ) )
282: K = K + 2
283: END IF
284: END DO
285: K = NCOLS
286: DO WHILE ( K .GE. 1 )
287: IF ( IPIV( K ).GT.0 ) THEN
288: KP = IPIV( K )
289: IF ( KP .NE. K ) THEN
290: TMP = WORK( N+K )
291: WORK( N+K ) = WORK( N+KP )
292: WORK( N+KP ) = TMP
293: END IF
294: K = K - 1
295: ELSE
296: KP = -IPIV( K )
297: TMP = WORK( N+K )
298: WORK( N+K ) = WORK( N+KP )
299: WORK( N+KP ) = TMP
300: K = K - 2
301: ENDIF
302: END DO
303: END IF
304: *
305: * Compute the *inverse* of the max element growth factor. Dividing
306: * by zero would imply the largest entry of the factor's column is
307: * zero. Than can happen when either the column of A is zero or
308: * massive pivots made the factor underflow to zero. Neither counts
309: * as growth in itself, so simply ignore terms with zero
310: * denominators.
311: *
312: IF ( UPPER ) THEN
313: DO I = NCOLS, N
314: UMAX = WORK( I )
315: AMAX = WORK( N+I )
316: IF ( UMAX /= 0.0D+0 ) THEN
317: RPVGRW = MIN( AMAX / UMAX, RPVGRW )
318: END IF
319: END DO
320: ELSE
321: DO I = 1, NCOLS
322: UMAX = WORK( I )
323: AMAX = WORK( N+I )
324: IF ( UMAX /= 0.0D+0 ) THEN
325: RPVGRW = MIN( AMAX / UMAX, RPVGRW )
326: END IF
327: END DO
328: END IF
329:
330: ZLA_SYRPVGRW = RPVGRW
331: END
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