Annotation of rpl/lapack/lapack/dlatdf.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DLATDF
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLATDF + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatdf.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatdf.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatdf.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DLATDF( 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: * DOUBLE PRECISION RHS( * ), Z( LDZ, * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *>
! 39: *> DLATDF uses the LU factorization of the n-by-n matrix Z computed by
! 40: *> DGETC2 and computes a contribution to the reciprocal Dif-estimate
! 41: *> by solving Z * x = b for x, and choosing the r.h.s. b such that
! 42: *> the norm of x is as large as possible. On entry RHS = b holds the
! 43: *> contribution from earlier solved sub-systems, and on return RHS = x.
! 44: *>
! 45: *> The factorization of Z returned by DGETC2 has the form Z = P*L*U*Q,
! 46: *> where P and Q are permutation matrices. L is lower triangular with
! 47: *> 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 DGECON, e is normalized and solve for
! 58: *> Zx = +-e - f with the sign giving the greater value
! 59: *> of 2-norm(x). About 5 times as expensive as Default.
! 60: *> IJOB .ne. 2: Local look ahead strategy where all entries of
! 61: *> the r.h.s. b is choosen as either +1 or -1 (Default).
! 62: *> \endverbatim
! 63: *>
! 64: *> \param[in] N
! 65: *> \verbatim
! 66: *> N is INTEGER
! 67: *> The number of columns of the matrix Z.
! 68: *> \endverbatim
! 69: *>
! 70: *> \param[in] Z
! 71: *> \verbatim
! 72: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
! 73: *> On entry, the LU part of the factorization of the n-by-n
! 74: *> matrix Z computed by DGETC2: Z = P * L * U * Q
! 75: *> \endverbatim
! 76: *>
! 77: *> \param[in] LDZ
! 78: *> \verbatim
! 79: *> LDZ is INTEGER
! 80: *> The leading dimension of the array Z. LDA >= max(1, N).
! 81: *> \endverbatim
! 82: *>
! 83: *> \param[in,out] RHS
! 84: *> \verbatim
! 85: *> RHS is DOUBLE PRECISION array, dimension (N)
! 86: *> On entry, RHS contains contributions from other subsystems.
! 87: *> On exit, RHS contains the solution of the subsystem with
! 88: *> entries acoording to the value of IJOB (see above).
! 89: *> \endverbatim
! 90: *>
! 91: *> \param[in,out] RDSUM
! 92: *> \verbatim
! 93: *> RDSUM is DOUBLE PRECISION
! 94: *> On entry, the sum of squares of computed contributions to
! 95: *> the Dif-estimate under computation by DTGSYL, where the
! 96: *> scaling factor RDSCAL (see below) has been factored out.
! 97: *> On exit, the corresponding sum of squares updated with the
! 98: *> contributions from the current sub-system.
! 99: *> If TRANS = 'T' RDSUM is not touched.
! 100: *> NOTE: RDSUM only makes sense when DTGSY2 is called by STGSYL.
! 101: *> \endverbatim
! 102: *>
! 103: *> \param[in,out] RDSCAL
! 104: *> \verbatim
! 105: *> RDSCAL is DOUBLE PRECISION
! 106: *> On entry, scaling factor used to prevent overflow in RDSUM.
! 107: *> On exit, RDSCAL is updated w.r.t. the current contributions
! 108: *> in RDSUM.
! 109: *> If TRANS = 'T', RDSCAL is not touched.
! 110: *> NOTE: RDSCAL only makes sense when DTGSY2 is called by
! 111: *> DTGSYL.
! 112: *> \endverbatim
! 113: *>
! 114: *> \param[in] IPIV
! 115: *> \verbatim
! 116: *> IPIV is INTEGER array, dimension (N).
! 117: *> The pivot indices; for 1 <= i <= N, row i of the
! 118: *> matrix has been interchanged with row IPIV(i).
! 119: *> \endverbatim
! 120: *>
! 121: *> \param[in] JPIV
! 122: *> \verbatim
! 123: *> JPIV is INTEGER array, dimension (N).
! 124: *> The pivot indices; for 1 <= j <= N, column j of the
! 125: *> matrix has been interchanged with column JPIV(j).
! 126: *> \endverbatim
! 127: *
! 128: * Authors:
! 129: * ========
! 130: *
! 131: *> \author Univ. of Tennessee
! 132: *> \author Univ. of California Berkeley
! 133: *> \author Univ. of Colorado Denver
! 134: *> \author NAG Ltd.
! 135: *
! 136: *> \date November 2011
! 137: *
! 138: *> \ingroup doubleOTHERauxiliary
! 139: *
! 140: *> \par Further Details:
! 141: * =====================
! 142: *>
! 143: *> This routine is a further developed implementation of algorithm
! 144: *> BSOLVE in [1] using complete pivoting in the LU factorization.
! 145: *
! 146: *> \par Contributors:
! 147: * ==================
! 148: *>
! 149: *> Bo Kagstrom and Peter Poromaa, Department of Computing Science,
! 150: *> Umea University, S-901 87 Umea, Sweden.
! 151: *
! 152: *> \par References:
! 153: * ================
! 154: *>
! 155: *> \verbatim
! 156: *>
! 157: *>
! 158: *> [1] Bo Kagstrom and Lars Westin,
! 159: *> Generalized Schur Methods with Condition Estimators for
! 160: *> Solving the Generalized Sylvester Equation, IEEE Transactions
! 161: *> on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
! 162: *>
! 163: *> [2] Peter Poromaa,
! 164: *> On Efficient and Robust Estimators for the Separation
! 165: *> between two Regular Matrix Pairs with Applications in
! 166: *> Condition Estimation. Report IMINF-95.05, Departement of
! 167: *> Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
! 168: *> \endverbatim
! 169: *>
! 170: * =====================================================================
1.1 bertrand 171: SUBROUTINE DLATDF( IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV,
172: $ JPIV )
173: *
1.9 ! bertrand 174: * -- LAPACK auxiliary routine (version 3.4.0) --
1.1 bertrand 175: * -- LAPACK is a software package provided by Univ. of Tennessee, --
176: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 177: * November 2011
1.1 bertrand 178: *
179: * .. Scalar Arguments ..
180: INTEGER IJOB, LDZ, N
181: DOUBLE PRECISION RDSCAL, RDSUM
182: * ..
183: * .. Array Arguments ..
184: INTEGER IPIV( * ), JPIV( * )
185: DOUBLE PRECISION RHS( * ), Z( LDZ, * )
186: * ..
187: *
188: * =====================================================================
189: *
190: * .. Parameters ..
191: INTEGER MAXDIM
192: PARAMETER ( MAXDIM = 8 )
193: DOUBLE PRECISION ZERO, ONE
194: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
195: * ..
196: * .. Local Scalars ..
197: INTEGER I, INFO, J, K
198: DOUBLE PRECISION BM, BP, PMONE, SMINU, SPLUS, TEMP
199: * ..
200: * .. Local Arrays ..
201: INTEGER IWORK( MAXDIM )
202: DOUBLE PRECISION WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
203: * ..
204: * .. External Subroutines ..
205: EXTERNAL DAXPY, DCOPY, DGECON, DGESC2, DLASSQ, DLASWP,
206: $ DSCAL
207: * ..
208: * .. External Functions ..
209: DOUBLE PRECISION DASUM, DDOT
210: EXTERNAL DASUM, DDOT
211: * ..
212: * .. Intrinsic Functions ..
213: INTRINSIC ABS, SQRT
214: * ..
215: * .. Executable Statements ..
216: *
217: IF( IJOB.NE.2 ) THEN
218: *
219: * Apply permutations IPIV to RHS
220: *
221: CALL DLASWP( 1, RHS, LDZ, 1, N-1, IPIV, 1 )
222: *
223: * Solve for L-part choosing RHS either to +1 or -1.
224: *
225: PMONE = -ONE
226: *
227: DO 10 J = 1, N - 1
228: BP = RHS( J ) + ONE
229: BM = RHS( J ) - ONE
230: SPLUS = ONE
231: *
232: * Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
233: * SMIN computed more efficiently than in BSOLVE [1].
234: *
235: SPLUS = SPLUS + DDOT( N-J, Z( J+1, J ), 1, Z( J+1, J ), 1 )
236: SMINU = DDOT( N-J, Z( J+1, J ), 1, RHS( J+1 ), 1 )
237: SPLUS = SPLUS*RHS( J )
238: IF( SPLUS.GT.SMINU ) THEN
239: RHS( J ) = BP
240: ELSE IF( SMINU.GT.SPLUS ) THEN
241: RHS( J ) = BM
242: ELSE
243: *
244: * In this case the updating sums are equal and we can
245: * choose RHS(J) +1 or -1. The first time this happens
246: * we choose -1, thereafter +1. This is a simple way to
247: * get good estimates of matrices like Byers well-known
248: * example (see [1]). (Not done in BSOLVE.)
249: *
250: RHS( J ) = RHS( J ) + PMONE
251: PMONE = ONE
252: END IF
253: *
254: * Compute the remaining r.h.s.
255: *
256: TEMP = -RHS( J )
257: CALL DAXPY( N-J, TEMP, Z( J+1, J ), 1, RHS( J+1 ), 1 )
258: *
259: 10 CONTINUE
260: *
261: * Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
262: * in BSOLVE and will hopefully give us a better estimate because
263: * any ill-conditioning of the original matrix is transfered to U
264: * and not to L. U(N, N) is an approximation to sigma_min(LU).
265: *
266: CALL DCOPY( N-1, RHS, 1, XP, 1 )
267: XP( N ) = RHS( N ) + ONE
268: RHS( N ) = RHS( N ) - ONE
269: SPLUS = ZERO
270: SMINU = ZERO
271: DO 30 I = N, 1, -1
272: TEMP = ONE / Z( I, I )
273: XP( I ) = XP( I )*TEMP
274: RHS( I ) = RHS( I )*TEMP
275: DO 20 K = I + 1, N
276: XP( I ) = XP( I ) - XP( K )*( Z( I, K )*TEMP )
277: RHS( I ) = RHS( I ) - RHS( K )*( Z( I, K )*TEMP )
278: 20 CONTINUE
279: SPLUS = SPLUS + ABS( XP( I ) )
280: SMINU = SMINU + ABS( RHS( I ) )
281: 30 CONTINUE
282: IF( SPLUS.GT.SMINU )
283: $ CALL DCOPY( N, XP, 1, RHS, 1 )
284: *
285: * Apply the permutations JPIV to the computed solution (RHS)
286: *
287: CALL DLASWP( 1, RHS, LDZ, 1, N-1, JPIV, -1 )
288: *
289: * Compute the sum of squares
290: *
291: CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
292: *
293: ELSE
294: *
295: * IJOB = 2, Compute approximate nullvector XM of Z
296: *
297: CALL DGECON( 'I', N, Z, LDZ, ONE, TEMP, WORK, IWORK, INFO )
298: CALL DCOPY( N, WORK( N+1 ), 1, XM, 1 )
299: *
300: * Compute RHS
301: *
302: CALL DLASWP( 1, XM, LDZ, 1, N-1, IPIV, -1 )
303: TEMP = ONE / SQRT( DDOT( N, XM, 1, XM, 1 ) )
304: CALL DSCAL( N, TEMP, XM, 1 )
305: CALL DCOPY( N, XM, 1, XP, 1 )
306: CALL DAXPY( N, ONE, RHS, 1, XP, 1 )
307: CALL DAXPY( N, -ONE, XM, 1, RHS, 1 )
308: CALL DGESC2( N, Z, LDZ, RHS, IPIV, JPIV, TEMP )
309: CALL DGESC2( N, Z, LDZ, XP, IPIV, JPIV, TEMP )
310: IF( DASUM( N, XP, 1 ).GT.DASUM( N, RHS, 1 ) )
311: $ CALL DCOPY( N, XP, 1, RHS, 1 )
312: *
313: * Compute the sum of squares
314: *
315: CALL DLASSQ( N, RHS, 1, RDSCAL, RDSUM )
316: *
317: END IF
318: *
319: RETURN
320: *
321: * End of DLATDF
322: *
323: END
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