Annotation of rpl/lapack/lapack/zlatdf.f, revision 1.8
1.8 ! bertrand 1: *> \brief \b ZLATDF
! 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
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlatdf.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlatdf.f">
! 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 choosen 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 November 2011
! 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: * =====================================================================
1.1 bertrand 169: SUBROUTINE ZLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
170: $ JPIV )
171: *
1.8 ! bertrand 172: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 173: * -- LAPACK is a software package provided by Univ. of Tennessee, --
174: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 175: * November 2011
1.1 bertrand 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 transfered 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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