1: *> \brief \b ZLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLATDF + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlatdf.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
22: * JPIV )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER IJOB, LDZ, N
26: * DOUBLE PRECISION RDSCAL, RDSUM
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * ), JPIV( * )
30: * COMPLEX*16 RHS( * ), Z( LDZ, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZLATDF computes the contribution to the reciprocal Dif-estimate
40: *> by solving for x in Z * x = b, where b is chosen such that the norm
41: *> of x is as large as possible. It is assumed that LU decomposition
42: *> of Z has been computed by ZGETC2. On entry RHS = f holds the
43: *> contribution from earlier solved sub-systems, and on return RHS = x.
44: *>
45: *> The factorization of Z returned by ZGETC2 has the form
46: *> Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
47: *> triangular with unit diagonal elements and U is upper triangular.
48: *> \endverbatim
49: *
50: * Arguments:
51: * ==========
52: *
53: *> \param[in] IJOB
54: *> \verbatim
55: *> IJOB is INTEGER
56: *> IJOB = 2: First compute an approximative null-vector e
57: *> of Z using ZGECON, e is normalized and solve for
58: *> Zx = +-e - f with the sign giving the greater value of
59: *> 2-norm(x). About 5 times as expensive as Default.
60: *> IJOB .ne. 2: Local look ahead strategy where
61: *> all entries of the r.h.s. b is chosen as either +1 or
62: *> -1. Default.
63: *> \endverbatim
64: *>
65: *> \param[in] N
66: *> \verbatim
67: *> N is INTEGER
68: *> The number of columns of the matrix Z.
69: *> \endverbatim
70: *>
71: *> \param[in] Z
72: *> \verbatim
73: *> Z is COMPLEX*16 array, dimension (LDZ, N)
74: *> On entry, the LU part of the factorization of the n-by-n
75: *> matrix Z computed by ZGETC2: Z = P * L * U * Q
76: *> \endverbatim
77: *>
78: *> \param[in] LDZ
79: *> \verbatim
80: *> LDZ is INTEGER
81: *> The leading dimension of the array Z. LDA >= max(1, N).
82: *> \endverbatim
83: *>
84: *> \param[in,out] RHS
85: *> \verbatim
86: *> RHS is COMPLEX*16 array, dimension (N).
87: *> On entry, RHS contains contributions from other subsystems.
88: *> On exit, RHS contains the solution of the subsystem with
89: *> entries according to the value of IJOB (see above).
90: *> \endverbatim
91: *>
92: *> \param[in,out] RDSUM
93: *> \verbatim
94: *> RDSUM is DOUBLE PRECISION
95: *> On entry, the sum of squares of computed contributions to
96: *> the Dif-estimate under computation by ZTGSYL, where the
97: *> scaling factor RDSCAL (see below) has been factored out.
98: *> On exit, the corresponding sum of squares updated with the
99: *> contributions from the current sub-system.
100: *> If TRANS = 'T' RDSUM is not touched.
101: *> NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.
102: *> \endverbatim
103: *>
104: *> \param[in,out] RDSCAL
105: *> \verbatim
106: *> RDSCAL is DOUBLE PRECISION
107: *> On entry, scaling factor used to prevent overflow in RDSUM.
108: *> On exit, RDSCAL is updated w.r.t. the current contributions
109: *> in RDSUM.
110: *> If TRANS = 'T', RDSCAL is not touched.
111: *> NOTE: RDSCAL only makes sense when ZTGSY2 is called by
112: *> ZTGSYL.
113: *> \endverbatim
114: *>
115: *> \param[in] IPIV
116: *> \verbatim
117: *> IPIV is INTEGER array, dimension (N).
118: *> The pivot indices; for 1 <= i <= N, row i of the
119: *> matrix has been interchanged with row IPIV(i).
120: *> \endverbatim
121: *>
122: *> \param[in] JPIV
123: *> \verbatim
124: *> JPIV is INTEGER array, dimension (N).
125: *> The pivot indices; for 1 <= j <= N, column j of the
126: *> matrix has been interchanged with column JPIV(j).
127: *> \endverbatim
128: *
129: * Authors:
130: * ========
131: *
132: *> \author Univ. of Tennessee
133: *> \author Univ. of California Berkeley
134: *> \author Univ. of Colorado Denver
135: *> \author NAG Ltd.
136: *
137: *> \ingroup complex16OTHERauxiliary
138: *
139: *> \par Further Details:
140: * =====================
141: *>
142: *> This routine is a further developed implementation of algorithm
143: *> BSOLVE in [1] using complete pivoting in the LU factorization.
144: *
145: *> \par Contributors:
146: * ==================
147: *>
148: *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
149: *> Umea University, S-901 87 Umea, Sweden.
150: *
151: *> \par References:
152: * ================
153: *>
154: *> [1] Bo Kagstrom and Lars Westin,
155: *> Generalized Schur Methods with Condition Estimators for
156: *> Solving the Generalized Sylvester Equation, IEEE Transactions
157: *> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
158: *>\n
159: *> [2] Peter Poromaa,
160: *> On Efficient and Robust Estimators for the Separation
161: *> between two Regular Matrix Pairs with Applications in
162: *> Condition Estimation. Report UMINF-95.05, Department of
163: *> Computing Science, Umea University, S-901 87 Umea, Sweden,
164: *> 1995.
165: *
166: * =====================================================================
167: SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
168: $ JPIV )
169: *
170: * -- LAPACK auxiliary routine --
171: * -- LAPACK is a software package provided by Univ. of Tennessee, --
172: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
173: *
174: * .. Scalar Arguments ..
175: INTEGER IJOB, LDZ, N
176: DOUBLE PRECISION RDSCAL, RDSUM
177: * ..
178: * .. Array Arguments ..
179: INTEGER IPIV( * ), JPIV( * )
180: COMPLEX*16 RHS( * ), Z( LDZ, * )
181: * ..
182: *
183: * =====================================================================
184: *
185: * .. Parameters ..
186: INTEGER MAXDIM
187: PARAMETER ( MAXDIM = 2 )
188: DOUBLE PRECISION ZERO, ONE
189: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
190: COMPLEX*16 CONE
191: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
192: * ..
193: * .. Local Scalars ..
194: INTEGER I, INFO, J, K
195: DOUBLE PRECISION RTEMP, SCALE, SMINU, SPLUS
196: COMPLEX*16 BM, BP, PMONE, TEMP
197: * ..
198: * .. Local Arrays ..
199: DOUBLE PRECISION RWORK( MAXDIM )
200: COMPLEX*16 WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
201: * ..
202: * .. External Subroutines ..
203: EXTERNAL ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
204: $ ZSCAL
205: * ..
206: * .. External Functions ..
207: DOUBLE PRECISION DZASUM
208: COMPLEX*16 ZDOTC
209: EXTERNAL DZASUM, ZDOTC
210: * ..
211: * .. Intrinsic Functions ..
212: INTRINSIC ABS, DBLE, SQRT
213: * ..
214: * .. Executable Statements ..
215: *
216: IF( IJOB.NE.2 ) THEN
217: *
218: * Apply permutations IPIV to RHS
219: *
220: CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
221: *
222: * Solve for L-part choosing RHS either to +1 or -1.
223: *
224: PMONE = -CONE
225: DO 10 J = 1, N - 1
226: BP = RHS( J ) + CONE
227: BM = RHS( J ) - CONE
228: SPLUS = ONE
229: *
230: * Lockahead for L- part RHS(1:N-1) = +-1
231: * SPLUS and SMIN computed more efficiently than in BSOLVE[1].
232: *
233: SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
234: $ J ), 1 ) )
235: SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
236: SPLUS = SPLUS*DBLE( RHS( J ) )
237: IF( SPLUS.GT.SMINU ) THEN
238: RHS( J ) = BP
239: ELSE IF( SMINU.GT.SPLUS ) THEN
240: RHS( J ) = BM
241: ELSE
242: *
243: * In this case the updating sums are equal and we can
244: * choose RHS(J) +1 or -1. The first time this happens we
245: * choose -1, thereafter +1. This is a simple way to get
246: * good estimates of matrices like Byers well-known example
247: * (see [1]). (Not done in BSOLVE.)
248: *
249: RHS( J ) = RHS( J ) + PMONE
250: PMONE = CONE
251: END IF
252: *
253: * Compute the remaining r.h.s.
254: *
255: TEMP = -RHS( J )
256: CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
257: 10 CONTINUE
258: *
259: * Solve for U- part, lockahead for RHS(N) = +-1. This is not done
260: * In BSOLVE and will hopefully give us a better estimate because
261: * any ill-conditioning of the original matrix is transferred to U
262: * and not to L. U(N, N) is an approximation to sigma_min(LU).
263: *
264: CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
265: WORK( N ) = RHS( N ) + CONE
266: RHS( N ) = RHS( N ) - CONE
267: SPLUS = ZERO
268: SMINU = ZERO
269: DO 30 I = N, 1, -1
270: TEMP = CONE / Z( I, I )
271: WORK( I ) = WORK( I )*TEMP
272: RHS( I ) = RHS( I )*TEMP
273: DO 20 K = I + 1, N
274: WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
275: RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
276: 20 CONTINUE
277: SPLUS = SPLUS + ABS( WORK( I ) )
278: SMINU = SMINU + ABS( RHS( I ) )
279: 30 CONTINUE
280: IF( SPLUS.GT.SMINU )
281: $ CALL ZCOPY( N, WORK, 1, RHS, 1 )
282: *
283: * Apply the permutations JPIV to the computed solution (RHS)
284: *
285: CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
286: *
287: * Compute the sum of squares
288: *
289: CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
290: RETURN
291: END IF
292: *
293: * ENTRY IJOB = 2
294: *
295: * Compute approximate nullvector XM of Z
296: *
297: CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
298: CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
299: *
300: * Compute RHS
301: *
302: CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
303: TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
304: CALL ZSCAL( N, TEMP, XM, 1 )
305: CALL ZCOPY( N, XM, 1, XP, 1 )
306: CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
307: CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
308: CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
309: CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
310: IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
311: $ CALL ZCOPY( N, XP, 1, RHS, 1 )
312: *
313: * Compute the sum of squares
314: *
315: CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
316: RETURN
317: *
318: * End of ZLATDF
319: *
320: END
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