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: *> \date June 2016
138: *
139: *> \ingroup complex16OTHERauxiliary
140: *
141: *> \par Further Details:
142: * =====================
143: *>
144: *> This routine is a further developed implementation of algorithm
145: *> BSOLVE in [1] using complete pivoting in the LU factorization.
146: *
147: *> \par Contributors:
148: * ==================
149: *>
150: *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
151: *> Umea University, S-901 87 Umea, Sweden.
152: *
153: *> \par References:
154: * ================
155: *>
156: *> [1] Bo Kagstrom and Lars Westin,
157: *> Generalized Schur Methods with Condition Estimators for
158: *> Solving the Generalized Sylvester Equation, IEEE Transactions
159: *> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
160: *>\n
161: *> [2] Peter Poromaa,
162: *> On Efficient and Robust Estimators for the Separation
163: *> between two Regular Matrix Pairs with Applications in
164: *> Condition Estimation. Report UMINF-95.05, Department of
165: *> Computing Science, Umea University, S-901 87 Umea, Sweden,
166: *> 1995.
167: *
168: * =====================================================================
169: SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
170: $ JPIV )
171: *
172: * -- LAPACK auxiliary routine (version 3.7.0) --
173: * -- LAPACK is a software package provided by Univ. of Tennessee, --
174: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175: * June 2016
176: *
177: * .. Scalar Arguments ..
178: INTEGER IJOB, LDZ, N
179: DOUBLE PRECISION RDSCAL, RDSUM
180: * ..
181: * .. Array Arguments ..
182: INTEGER IPIV( * ), JPIV( * )
183: COMPLEX*16 RHS( * ), Z( LDZ, * )
184: * ..
185: *
186: * =====================================================================
187: *
188: * .. Parameters ..
189: INTEGER MAXDIM
190: PARAMETER ( MAXDIM = 2 )
191: DOUBLE PRECISION ZERO, ONE
192: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
193: COMPLEX*16 CONE
194: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
195: * ..
196: * .. Local Scalars ..
197: INTEGER I, INFO, J, K
198: DOUBLE PRECISION RTEMP, SCALE, SMINU, SPLUS
199: COMPLEX*16 BM, BP, PMONE, TEMP
200: * ..
201: * .. Local Arrays ..
202: DOUBLE PRECISION RWORK( MAXDIM )
203: COMPLEX*16 WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
204: * ..
205: * .. External Subroutines ..
206: EXTERNAL ZAXPY, ZCOPY, ZGECON, ZGESC2, ZLASSQ, ZLASWP,
207: $ ZSCAL
208: * ..
209: * .. External Functions ..
210: DOUBLE PRECISION DZASUM
211: COMPLEX*16 ZDOTC
212: EXTERNAL DZASUM, ZDOTC
213: * ..
214: * .. Intrinsic Functions ..
215: INTRINSIC ABS, DBLE, SQRT
216: * ..
217: * .. Executable Statements ..
218: *
219: IF( IJOB.NE.2 ) THEN
220: *
221: * Apply permutations IPIV to RHS
222: *
223: CALL ZLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
224: *
225: * Solve for L-part choosing RHS either to +1 or -1.
226: *
227: PMONE = -CONE
228: DO 10 J = 1, N - 1
229: BP = RHS( J ) + CONE
230: BM = RHS( J ) - CONE
231: SPLUS = ONE
232: *
233: * Lockahead for L- part RHS(1:N-1) = +-1
234: * SPLUS and SMIN computed more efficiently than in BSOLVE[1].
235: *
236: SPLUS = SPLUS + DBLE( ZDOTC( N-J, Z( J+1, J ), 1, Z( J+1,
237: $ J ), 1 ) )
238: SMINU = DBLE( ZDOTC( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 ) )
239: SPLUS = SPLUS*DBLE( RHS( J ) )
240: IF( SPLUS.GT.SMINU ) THEN
241: RHS( J ) = BP
242: ELSE IF( SMINU.GT.SPLUS ) THEN
243: RHS( J ) = BM
244: ELSE
245: *
246: * In this case the updating sums are equal and we can
247: * choose RHS(J) +1 or -1. The first time this happens we
248: * choose -1, thereafter +1. This is a simple way to get
249: * good estimates of matrices like Byers well-known example
250: * (see [1]). (Not done in BSOLVE.)
251: *
252: RHS( J ) = RHS( J ) + PMONE
253: PMONE = CONE
254: END IF
255: *
256: * Compute the remaining r.h.s.
257: *
258: TEMP = -RHS( J )
259: CALL ZAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
260: 10 CONTINUE
261: *
262: * Solve for U- part, lockahead for RHS(N) = +-1. This is not done
263: * In BSOLVE and will hopefully give us a better estimate because
264: * any ill-conditioning of the original matrix is transferred to U
265: * and not to L. U(N, N) is an approximation to sigma_min(LU).
266: *
267: CALL ZCOPY( N-1, RHS, 1, WORK, 1 )
268: WORK( N ) = RHS( N ) + CONE
269: RHS( N ) = RHS( N ) - CONE
270: SPLUS = ZERO
271: SMINU = ZERO
272: DO 30 I = N, 1, -1
273: TEMP = CONE / Z( I, I )
274: WORK( I ) = WORK( I )*TEMP
275: RHS( I ) = RHS( I )*TEMP
276: DO 20 K = I + 1, N
277: WORK( I ) = WORK( I ) - WORK( K )*( Z( I, K )*TEMP )
278: RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
279: 20 CONTINUE
280: SPLUS = SPLUS + ABS( WORK( I ) )
281: SMINU = SMINU + ABS( RHS( I ) )
282: 30 CONTINUE
283: IF( SPLUS.GT.SMINU )
284: $ CALL ZCOPY( N, WORK, 1, RHS, 1 )
285: *
286: * Apply the permutations JPIV to the computed solution (RHS)
287: *
288: CALL ZLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
289: *
290: * Compute the sum of squares
291: *
292: CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
293: RETURN
294: END IF
295: *
296: * ENTRY IJOB = 2
297: *
298: * Compute approximate nullvector XM of Z
299: *
300: CALL ZGECON( 'I', N, Z, LDZ, ONE, RTEMP, WORK, RWORK, INFO )
301: CALL ZCOPY( N, WORK( N+1 ), 1, XM, 1 )
302: *
303: * Compute RHS
304: *
305: CALL ZLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
306: TEMP = CONE / SQRT( ZDOTC( N, XM, 1, XM, 1 ) )
307: CALL ZSCAL( N, TEMP, XM, 1 )
308: CALL ZCOPY( N, XM, 1, XP, 1 )
309: CALL ZAXPY( N, CONE, RHS, 1, XP, 1 )
310: CALL ZAXPY( N, -CONE, XM, 1, RHS, 1 )
311: CALL ZGESC2( N, Z, LDZ, RHS, IPIV, JPIV, SCALE )
312: CALL ZGESC2( N, Z, LDZ, XP, IPIV, JPIV, SCALE )
313: IF( DZASUM( N, XP, 1 ).GT.DZASUM( N, RHS, 1 ) )
314: $ CALL ZCOPY( N, XP, 1, RHS, 1 )
315: *
316: * Compute the sum of squares
317: *
318: CALL ZLASSQ( N, RHS, 1, RDSCAL, RDSUM )
319: RETURN
320: *
321: * End of ZLATDF
322: *
323: END
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