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Mon Jan 27 09:28:34 2014 UTC (10 years, 3 months ago) by
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1: *> \brief \b ZHEEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHEEQUB + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheequb.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheequb.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheequb.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
22: *
23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, N
25: * DOUBLE PRECISION AMAX, SCOND
26: * CHARACTER UPLO
27: * ..
28: * .. Array Arguments ..
29: * COMPLEX*16 A( LDA, * ), WORK( * )
30: * DOUBLE PRECISION S( * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZHEEQUB computes row and column scalings intended to equilibrate a
40: *> Hermitian matrix A and reduce its condition number
41: *> (with respect to the two-norm). S contains the scale factors,
42: *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
43: *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
44: *> choice of S puts the condition number of B within a factor N of the
45: *> smallest possible condition number over all possible diagonal
46: *> scalings.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] UPLO
53: *> \verbatim
54: *> UPLO is CHARACTER*1
55: *> = 'U': Upper triangles of A and B are stored;
56: *> = 'L': Lower triangles of A and B are stored.
57: *> \endverbatim
58: *>
59: *> \param[in] N
60: *> \verbatim
61: *> N is INTEGER
62: *> The order of the matrix A. N >= 0.
63: *> \endverbatim
64: *>
65: *> \param[in] A
66: *> \verbatim
67: *> A is COMPLEX*16 array, dimension (LDA,N)
68: *> The N-by-N Hermitian matrix whose scaling
69: *> factors are to be computed. Only the diagonal elements of A
70: *> are referenced.
71: *> \endverbatim
72: *>
73: *> \param[in] LDA
74: *> \verbatim
75: *> LDA is INTEGER
76: *> The leading dimension of the array A. LDA >= max(1,N).
77: *> \endverbatim
78: *>
79: *> \param[out] S
80: *> \verbatim
81: *> S is DOUBLE PRECISION array, dimension (N)
82: *> If INFO = 0, S contains the scale factors for A.
83: *> \endverbatim
84: *>
85: *> \param[out] SCOND
86: *> \verbatim
87: *> SCOND is DOUBLE PRECISION
88: *> If INFO = 0, S contains the ratio of the smallest S(i) to
89: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
90: *> large nor too small, it is not worth scaling by S.
91: *> \endverbatim
92: *>
93: *> \param[out] AMAX
94: *> \verbatim
95: *> AMAX is DOUBLE PRECISION
96: *> Absolute value of largest matrix element. If AMAX is very
97: *> close to overflow or very close to underflow, the matrix
98: *> should be scaled.
99: *> \endverbatim
100: *>
101: *> \param[out] WORK
102: *> \verbatim
103: *> WORK is COMPLEX*16 array, dimension (3*N)
104: *> \endverbatim
105: *>
106: *> \param[out] INFO
107: *> \verbatim
108: *> INFO is INTEGER
109: *> = 0: successful exit
110: *> < 0: if INFO = -i, the i-th argument had an illegal value
111: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
112: *> \endverbatim
113: *
114: * Authors:
115: * ========
116: *
117: *> \author Univ. of Tennessee
118: *> \author Univ. of California Berkeley
119: *> \author Univ. of Colorado Denver
120: *> \author NAG Ltd.
121: *
122: *> \date April 2012
123: *
124: *> \ingroup complex16HEcomputational
125: *
126: * =====================================================================
127: SUBROUTINE ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
128: *
129: * -- LAPACK computational routine (version 3.4.1) --
130: * -- LAPACK is a software package provided by Univ. of Tennessee, --
131: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
132: * April 2012
133: *
134: * .. Scalar Arguments ..
135: INTEGER INFO, LDA, N
136: DOUBLE PRECISION AMAX, SCOND
137: CHARACTER UPLO
138: * ..
139: * .. Array Arguments ..
140: COMPLEX*16 A( LDA, * ), WORK( * )
141: DOUBLE PRECISION S( * )
142: * ..
143: *
144: * =====================================================================
145: *
146: * .. Parameters ..
147: DOUBLE PRECISION ONE, ZERO
148: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
149: INTEGER MAX_ITER
150: PARAMETER ( MAX_ITER = 100 )
151: * ..
152: * .. Local Scalars ..
153: INTEGER I, J, ITER
154: DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D,
155: $ BASE, SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
156: LOGICAL UP
157: COMPLEX*16 ZDUM
158: * ..
159: * .. External Functions ..
160: DOUBLE PRECISION DLAMCH
161: LOGICAL LSAME
162: EXTERNAL DLAMCH, LSAME
163: * ..
164: * .. External Subroutines ..
165: EXTERNAL ZLASSQ
166: * ..
167: * .. Intrinsic Functions ..
168: INTRINSIC ABS, DBLE, DIMAG, INT, LOG, MAX, MIN, SQRT
169: * ..
170: * .. Statement Functions ..
171: DOUBLE PRECISION CABS1
172: * ..
173: * .. Statement Function Definitions ..
174: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
175: *
176: * Test input parameters.
177: *
178: INFO = 0
179: IF (.NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
180: INFO = -1
181: ELSE IF ( N .LT. 0 ) THEN
182: INFO = -2
183: ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
184: INFO = -4
185: END IF
186: IF ( INFO .NE. 0 ) THEN
187: CALL XERBLA( 'ZHEEQUB', -INFO )
188: RETURN
189: END IF
190:
191: UP = LSAME( UPLO, 'U' )
192: AMAX = ZERO
193: *
194: * Quick return if possible.
195: *
196: IF ( N .EQ. 0 ) THEN
197: SCOND = ONE
198: RETURN
199: END IF
200:
201: DO I = 1, N
202: S( I ) = ZERO
203: END DO
204:
205: AMAX = ZERO
206: IF ( UP ) THEN
207: DO J = 1, N
208: DO I = 1, J-1
209: S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
210: S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
211: AMAX = MAX( AMAX, CABS1( A( I, J ) ) )
212: END DO
213: S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
214: AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
215: END DO
216: ELSE
217: DO J = 1, N
218: S( J ) = MAX( S( J ), CABS1( A( J, J ) ) )
219: AMAX = MAX( AMAX, CABS1( A( J, J ) ) )
220: DO I = J+1, N
221: S( I ) = MAX( S( I ), CABS1( A( I, J ) ) )
222: S( J ) = MAX( S( J ), CABS1( A( I, J ) ) )
223: AMAX = MAX( AMAX, CABS1( A(I, J ) ) )
224: END DO
225: END DO
226: END IF
227: DO J = 1, N
228: S( J ) = 1.0D+0 / S( J )
229: END DO
230:
231: TOL = ONE / SQRT( 2.0D0 * N )
232:
233: DO ITER = 1, MAX_ITER
234: SCALE = 0.0D+0
235: SUMSQ = 0.0D+0
236: * beta = |A|s
237: DO I = 1, N
238: WORK( I ) = ZERO
239: END DO
240: IF ( UP ) THEN
241: DO J = 1, N
242: DO I = 1, J-1
243: T = CABS1( A( I, J ) )
244: WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
245: WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
246: END DO
247: WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
248: END DO
249: ELSE
250: DO J = 1, N
251: WORK( J ) = WORK( J ) + CABS1( A( J, J ) ) * S( J )
252: DO I = J+1, N
253: T = CABS1( A( I, J ) )
254: WORK( I ) = WORK( I ) + CABS1( A( I, J ) ) * S( J )
255: WORK( J ) = WORK( J ) + CABS1( A( I, J ) ) * S( I )
256: END DO
257: END DO
258: END IF
259:
260: * avg = s^T beta / n
261: AVG = 0.0D+0
262: DO I = 1, N
263: AVG = AVG + S( I )*WORK( I )
264: END DO
265: AVG = AVG / N
266:
267: STD = 0.0D+0
268: DO I = 2*N+1, 3*N
269: WORK( I ) = S( I-2*N ) * WORK( I-2*N ) - AVG
270: END DO
271: CALL ZLASSQ( N, WORK( 2*N+1 ), 1, SCALE, SUMSQ )
272: STD = SCALE * SQRT( SUMSQ / N )
273:
274: IF ( STD .LT. TOL * AVG ) GOTO 999
275:
276: DO I = 1, N
277: T = CABS1( A( I, I ) )
278: SI = S( I )
279: C2 = ( N-1 ) * T
280: C1 = ( N-2 ) * ( WORK( I ) - T*SI )
281: C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
282:
283: D = C1*C1 - 4*C0*C2
284: IF ( D .LE. 0 ) THEN
285: INFO = -1
286: RETURN
287: END IF
288: SI = -2*C0 / ( C1 + SQRT( D ) )
289:
290: D = SI - S(I)
291: U = ZERO
292: IF ( UP ) THEN
293: DO J = 1, I
294: T = CABS1( A( J, I ) )
295: U = U + S( J )*T
296: WORK( J ) = WORK( J ) + D*T
297: END DO
298: DO J = I+1,N
299: T = CABS1( A( I, J ) )
300: U = U + S( J )*T
301: WORK( J ) = WORK( J ) + D*T
302: END DO
303: ELSE
304: DO J = 1, I
305: T = CABS1( A( I, J ) )
306: U = U + S( J )*T
307: WORK( J ) = WORK( J ) + D*T
308: END DO
309: DO J = I+1,N
310: T = CABS1( A( J, I ) )
311: U = U + S( J )*T
312: WORK( J ) = WORK( J ) + D*T
313: END DO
314: END IF
315: AVG = AVG + ( U + WORK( I ) ) * D / N
316: S( I ) = SI
317: END DO
318:
319: END DO
320:
321: 999 CONTINUE
322:
323: SMLNUM = DLAMCH( 'SAFEMIN' )
324: BIGNUM = ONE / SMLNUM
325: SMIN = BIGNUM
326: SMAX = ZERO
327: T = ONE / SQRT( AVG )
328: BASE = DLAMCH( 'B' )
329: U = ONE / LOG( BASE )
330: DO I = 1, N
331: S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
332: SMIN = MIN( SMIN, S( I ) )
333: SMAX = MAX( SMAX, S( I ) )
334: END DO
335: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
336:
337: END
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