1: *> \brief \b ZLA_HERPVGRW
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLA_HERPVGRW + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zla_herpvgrw.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zla_herpvgrw.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zla_herpvgrw.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * DOUBLE PRECISION FUNCTION ZLA_HERPVGRW( 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: * INTEGER IPIV( * )
30: * COMPLEX*16 A( LDA, * ), AF( LDAF, * )
31: * DOUBLE PRECISION WORK( * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *>
41: *> ZLA_HERPVGRW 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 ZHETRF, .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 ZHETRF.
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 ZHETRF.
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 June 2016
119: *
120: *> \ingroup complex16HEcomputational
121: *
122: * =====================================================================
123: DOUBLE PRECISION FUNCTION ZLA_HERPVGRW( 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: * June 2016
130: *
131: * .. Scalar Arguments ..
132: CHARACTER*1 UPLO
133: INTEGER N, INFO, LDA, LDAF
134: * ..
135: * .. Array Arguments ..
136: INTEGER IPIV( * )
137: COMPLEX*16 A( LDA, * ), AF( LDAF, * )
138: DOUBLE PRECISION WORK( * )
139: * ..
140: *
141: * =====================================================================
142: *
143: * .. Local Scalars ..
144: INTEGER NCOLS, I, J, K, KP
145: DOUBLE PRECISION AMAX, UMAX, RPVGRW, TMP
146: LOGICAL UPPER, LSAME
147: COMPLEX*16 ZDUM
148: * ..
149: * .. External Functions ..
150: EXTERNAL LSAME
151: * ..
152: * .. Intrinsic Functions ..
153: INTRINSIC ABS, REAL, DIMAG, MAX, MIN
154: * ..
155: * .. Statement Functions ..
156: DOUBLE PRECISION CABS1
157: * ..
158: * .. Statement Function Definitions ..
159: CABS1( ZDUM ) = ABS( DBLE ( ZDUM ) ) + ABS( DIMAG ( ZDUM ) )
160: * ..
161: * .. Executable Statements ..
162: *
163: UPPER = LSAME( 'Upper', UPLO )
164: IF ( INFO.EQ.0 ) THEN
165: IF (UPPER) THEN
166: NCOLS = 1
167: ELSE
168: NCOLS = N
169: END IF
170: ELSE
171: NCOLS = INFO
172: END IF
173:
174: RPVGRW = 1.0D+0
175: DO I = 1, 2*N
176: WORK( I ) = 0.0D+0
177: END DO
178: *
179: * Find the max magnitude entry of each column of A. Compute the max
180: * for all N columns so we can apply the pivot permutation while
181: * looping below. Assume a full factorization is the common case.
182: *
183: IF ( UPPER ) THEN
184: DO J = 1, N
185: DO I = 1, J
186: WORK( N+I ) = MAX( CABS1( A( I,J ) ), WORK( N+I ) )
187: WORK( N+J ) = MAX( CABS1( A( I,J ) ), WORK( N+J ) )
188: END DO
189: END DO
190: ELSE
191: DO J = 1, N
192: DO I = J, N
193: WORK( N+I ) = MAX( CABS1( A( I, J ) ), WORK( N+I ) )
194: WORK( N+J ) = MAX( CABS1( A( I, J ) ), WORK( N+J ) )
195: END DO
196: END DO
197: END IF
198: *
199: * Now find the max magnitude entry of each column of U or L. Also
200: * permute the magnitudes of A above so they're in the same order as
201: * the factor.
202: *
203: * The iteration orders and permutations were copied from zsytrs.
204: * Calls to SSWAP would be severe overkill.
205: *
206: IF ( UPPER ) THEN
207: K = N
208: DO WHILE ( K .LT. NCOLS .AND. K.GT.0 )
209: IF ( IPIV( K ).GT.0 ) THEN
210: ! 1x1 pivot
211: KP = IPIV( K )
212: IF ( KP .NE. K ) THEN
213: TMP = WORK( N+K )
214: WORK( N+K ) = WORK( N+KP )
215: WORK( N+KP ) = TMP
216: END IF
217: DO I = 1, K
218: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
219: END DO
220: K = K - 1
221: ELSE
222: ! 2x2 pivot
223: KP = -IPIV( K )
224: TMP = WORK( N+K-1 )
225: WORK( N+K-1 ) = WORK( N+KP )
226: WORK( N+KP ) = TMP
227: DO I = 1, K-1
228: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
229: WORK( K-1 ) =
230: $ MAX( CABS1( AF( I, K-1 ) ), WORK( K-1 ) )
231: END DO
232: WORK( K ) = MAX( CABS1( AF( K, K ) ), WORK( K ) )
233: K = K - 2
234: END IF
235: END DO
236: K = NCOLS
237: DO WHILE ( K .LE. N )
238: IF ( IPIV( K ).GT.0 ) THEN
239: KP = IPIV( K )
240: IF ( KP .NE. K ) THEN
241: TMP = WORK( N+K )
242: WORK( N+K ) = WORK( N+KP )
243: WORK( N+KP ) = TMP
244: END IF
245: K = K + 1
246: ELSE
247: KP = -IPIV( K )
248: TMP = WORK( N+K )
249: WORK( N+K ) = WORK( N+KP )
250: WORK( N+KP ) = TMP
251: K = K + 2
252: END IF
253: END DO
254: ELSE
255: K = 1
256: DO WHILE ( K .LE. NCOLS )
257: IF ( IPIV( K ).GT.0 ) THEN
258: ! 1x1 pivot
259: KP = IPIV( K )
260: IF ( KP .NE. K ) THEN
261: TMP = WORK( N+K )
262: WORK( N+K ) = WORK( N+KP )
263: WORK( N+KP ) = TMP
264: END IF
265: DO I = K, N
266: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
267: END DO
268: K = K + 1
269: ELSE
270: ! 2x2 pivot
271: KP = -IPIV( K )
272: TMP = WORK( N+K+1 )
273: WORK( N+K+1 ) = WORK( N+KP )
274: WORK( N+KP ) = TMP
275: DO I = K+1, N
276: WORK( K ) = MAX( CABS1( AF( I, K ) ), WORK( K ) )
277: WORK( K+1 ) =
278: $ MAX( CABS1( AF( I, K+1 ) ) , WORK( K+1 ) )
279: END DO
280: WORK(K) = MAX( CABS1( AF( K, K ) ), WORK( K ) )
281: K = K + 2
282: END IF
283: END DO
284: K = NCOLS
285: DO WHILE ( K .GE. 1 )
286: IF ( IPIV( K ).GT.0 ) THEN
287: KP = IPIV( K )
288: IF ( KP .NE. K ) THEN
289: TMP = WORK( N+K )
290: WORK( N+K ) = WORK( N+KP )
291: WORK( N+KP ) = TMP
292: END IF
293: K = K - 1
294: ELSE
295: KP = -IPIV( K )
296: TMP = WORK( N+K )
297: WORK( N+K ) = WORK( N+KP )
298: WORK( N+KP ) = TMP
299: K = K - 2
300: ENDIF
301: END DO
302: END IF
303: *
304: * Compute the *inverse* of the max element growth factor. Dividing
305: * by zero would imply the largest entry of the factor's column is
306: * zero. Than can happen when either the column of A is zero or
307: * massive pivots made the factor underflow to zero. Neither counts
308: * as growth in itself, so simply ignore terms with zero
309: * denominators.
310: *
311: IF ( UPPER ) THEN
312: DO I = NCOLS, N
313: UMAX = WORK( I )
314: AMAX = WORK( N+I )
315: IF ( UMAX /= 0.0D+0 ) THEN
316: RPVGRW = MIN( AMAX / UMAX, RPVGRW )
317: END IF
318: END DO
319: ELSE
320: DO I = 1, NCOLS
321: UMAX = WORK( I )
322: AMAX = WORK( N+I )
323: IF ( UMAX /= 0.0D+0 ) THEN
324: RPVGRW = MIN( AMAX / UMAX, RPVGRW )
325: END IF
326: END DO
327: END IF
328:
329: ZLA_HERPVGRW = RPVGRW
330: END
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