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[out] 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: *> \ingroup complex16SYcomputational
119: *
120: * =====================================================================
121: DOUBLE PRECISION FUNCTION ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
122: $ LDAF, IPIV, WORK )
123: *
124: * -- LAPACK computational routine --
125: * -- LAPACK is a software package provided by Univ. of Tennessee, --
126: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
127: *
128: * .. Scalar Arguments ..
129: CHARACTER*1 UPLO
130: INTEGER N, INFO, LDA, LDAF
131: * ..
132: * .. Array Arguments ..
133: COMPLEX*16 A( LDA, * ), AF( LDAF, * )
134: DOUBLE PRECISION WORK( * )
135: INTEGER IPIV( * )
136: * ..
137: *
138: * =====================================================================
139: *
140: * .. Local Scalars ..
141: INTEGER NCOLS, I, J, K, KP
142: DOUBLE PRECISION AMAX, UMAX, RPVGRW, TMP
143: LOGICAL UPPER
144: COMPLEX*16 ZDUM
145: * ..
146: * .. Intrinsic Functions ..
147: INTRINSIC ABS, REAL, DIMAG, MAX, MIN
148: * ..
149: * .. External Subroutines ..
150: EXTERNAL LSAME
151: LOGICAL LSAME
152: * ..
153: * .. Statement Functions ..
154: DOUBLE PRECISION CABS1
155: * ..
156: * .. Statement Function Definitions ..
157: CABS1( ZDUM ) = ABS( DBLE ( ZDUM ) ) + ABS( DIMAG ( ZDUM ) )
158: * ..
159: * .. Executable Statements ..
160: *
161: UPPER = LSAME( 'Upper', UPLO )
162: IF ( INFO.EQ.0 ) THEN
163: IF ( UPPER ) THEN
164: NCOLS = 1
165: ELSE
166: NCOLS = N
167: END IF
168: ELSE
169: NCOLS = INFO
170: END IF
171:
172: RPVGRW = 1.0D+0
173: DO I = 1, 2*N
174: WORK( I ) = 0.0D+0
175: END DO
176: *
177: * Find the max magnitude entry of each column of A. Compute the max
178: * for all N columns so we can apply the pivot permutation while
179: * looping below. Assume a full factorization is the common case.
180: *
181: IF ( UPPER ) THEN
182: DO J = 1, N
183: DO I = 1, J
184: WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) )
185: WORK( N+J ) = MAX( CABS1( A( I, J ) ), WORK( N+J ) )
186: END DO
187: END DO
188: ELSE
189: DO J = 1, N
190: DO I = J, N
191: WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) )
192: WORK( N+J ) = MAX( CABS1( A( I, J ) ), WORK( N+J ) )
193: END DO
194: END DO
195: END IF
196: *
197: * Now find the max magnitude entry of each column of U or L. Also
198: * permute the magnitudes of A above so they're in the same order as
199: * the factor.
200: *
201: * The iteration orders and permutations were copied from zsytrs.
202: * Calls to SSWAP would be severe overkill.
203: *
204: IF ( UPPER ) THEN
205: K = N
206: DO WHILE ( K .LT. NCOLS .AND. K.GT.0 )
207: IF ( IPIV( K ).GT.0 ) THEN
208: ! 1x1 pivot
209: KP = IPIV( K )
210: IF ( KP .NE. K ) THEN
211: TMP = WORK( N+K )
212: WORK( N+K ) = WORK( N+KP )
213: WORK( N+KP ) = TMP
214: END IF
215: DO I = 1, K
216: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
217: END DO
218: K = K - 1
219: ELSE
220: ! 2x2 pivot
221: KP = -IPIV( K )
222: TMP = WORK( N+K-1 )
223: WORK( N+K-1 ) = WORK( N+KP )
224: WORK( N+KP ) = TMP
225: DO I = 1, K-1
226: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
227: WORK( K-1 ) =
228: $ MAX( CABS1( AF( I, K-1 ) ), WORK( K-1 ) )
229: END DO
230: WORK( K ) = MAX( CABS1( AF( K, K ) ), WORK( K ) )
231: K = K - 2
232: END IF
233: END DO
234: K = NCOLS
235: DO WHILE ( K .LE. N )
236: IF ( IPIV( K ).GT.0 ) THEN
237: KP = IPIV( K )
238: IF ( KP .NE. K ) THEN
239: TMP = WORK( N+K )
240: WORK( N+K ) = WORK( N+KP )
241: WORK( N+KP ) = TMP
242: END IF
243: K = K + 1
244: ELSE
245: KP = -IPIV( K )
246: TMP = WORK( N+K )
247: WORK( N+K ) = WORK( N+KP )
248: WORK( N+KP ) = TMP
249: K = K + 2
250: END IF
251: END DO
252: ELSE
253: K = 1
254: DO WHILE ( K .LE. NCOLS )
255: IF ( IPIV( K ).GT.0 ) THEN
256: ! 1x1 pivot
257: KP = IPIV( K )
258: IF ( KP .NE. K ) THEN
259: TMP = WORK( N+K )
260: WORK( N+K ) = WORK( N+KP )
261: WORK( N+KP ) = TMP
262: END IF
263: DO I = K, N
264: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
265: END DO
266: K = K + 1
267: ELSE
268: ! 2x2 pivot
269: KP = -IPIV( K )
270: TMP = WORK( N+K+1 )
271: WORK( N+K+1 ) = WORK( N+KP )
272: WORK( N+KP ) = TMP
273: DO I = K+1, N
274: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
275: WORK( K+1 ) =
276: $ MAX( CABS1( AF( I, K+1 ) ), WORK( K+1 ) )
277: END DO
278: WORK( K ) = MAX( CABS1( AF( K, K ) ), WORK( K ) )
279: K = K + 2
280: END IF
281: END DO
282: K = NCOLS
283: DO WHILE ( K .GE. 1 )
284: IF ( IPIV( K ).GT.0 ) THEN
285: KP = IPIV( K )
286: IF ( KP .NE. K ) THEN
287: TMP = WORK( N+K )
288: WORK( N+K ) = WORK( N+KP )
289: WORK( N+KP ) = TMP
290: END IF
291: K = K - 1
292: ELSE
293: KP = -IPIV( K )
294: TMP = WORK( N+K )
295: WORK( N+K ) = WORK( N+KP )
296: WORK( N+KP ) = TMP
297: K = K - 2
298: ENDIF
299: END DO
300: END IF
301: *
302: * Compute the *inverse* of the max element growth factor. Dividing
303: * by zero would imply the largest entry of the factor's column is
304: * zero. Than can happen when either the column of A is zero or
305: * massive pivots made the factor underflow to zero. Neither counts
306: * as growth in itself, so simply ignore terms with zero
307: * denominators.
308: *
309: IF ( UPPER ) THEN
310: DO I = NCOLS, N
311: UMAX = WORK( I )
312: AMAX = WORK( N+I )
313: IF ( UMAX /= 0.0D+0 ) THEN
314: RPVGRW = MIN( AMAX / UMAX, RPVGRW )
315: END IF
316: END DO
317: ELSE
318: DO I = 1, NCOLS
319: UMAX = WORK( I )
320: AMAX = WORK( N+I )
321: IF ( UMAX /= 0.0D+0 ) THEN
322: RPVGRW = MIN( AMAX / UMAX, RPVGRW )
323: END IF
324: END DO
325: END IF
326:
327: ZLA_SYRPVGRW = RPVGRW
328: *
329: * End of ZLA_SYRPVGRW
330: *
331: END
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