 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zlatdf()

 subroutine zlatdf ( integer IJOB, integer N, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension( * ) RHS, double precision RDSUM, double precision RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV )

ZLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.

Purpose:
``` ZLATDF computes the contribution to the reciprocal Dif-estimate
by solving for x in Z * x = b, where b is chosen such that the norm
of x is as large as possible. It is assumed that LU decomposition
of Z has been computed by ZGETC2. On entry RHS = f holds the
contribution from earlier solved sub-systems, and on return RHS = x.

The factorization of Z returned by ZGETC2 has the form
Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
triangular with unit diagonal elements and U is upper triangular.```
Parameters
 [in] IJOB ``` IJOB is INTEGER IJOB = 2: First compute an approximative null-vector e of Z using ZGECON, e is normalized and solve for Zx = +-e - f with the sign giving the greater value of 2-norm(x). About 5 times as expensive as Default. IJOB .ne. 2: Local look ahead strategy where all entries of the r.h.s. b is chosen as either +1 or -1. Default.``` [in] N ``` N is INTEGER The number of columns of the matrix Z.``` [in] Z ``` Z is COMPLEX*16 array, dimension (LDZ, N) On entry, the LU part of the factorization of the n-by-n matrix Z computed by ZGETC2: Z = P * L * U * Q``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDA >= max(1, N).``` [in,out] RHS ``` RHS is COMPLEX*16 array, dimension (N). On entry, RHS contains contributions from other subsystems. On exit, RHS contains the solution of the subsystem with entries according to the value of IJOB (see above).``` [in,out] RDSUM ``` RDSUM is DOUBLE PRECISION On entry, the sum of squares of computed contributions to the Dif-estimate under computation by ZTGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.``` [in,out] RDSCAL ``` RDSCAL is DOUBLE PRECISION On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when ZTGSY2 is called by ZTGSYL.``` [in] IPIV ``` IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i).``` [in] JPIV ``` JPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j).```
Further Details:
This routine is a further developed implementation of algorithm BSOLVE in  using complete pivoting in the LU factorization.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.
References:
 Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.
 Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report UMINF-95.05, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.

Definition at line 167 of file zlatdf.f.

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.
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 ). (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*
subroutine zlassq(n, x, incx, scl, sumsq)
ZLASSQ updates a sum of squares represented in scaled form.
Definition: zlassq.f90:137
complex *16 function zdotc(N, ZX, INCX, ZY, INCY)
ZDOTC
Definition: zdotc.f:83
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zscal(N, ZA, ZX, INCX)
ZSCAL
Definition: zscal.f:78
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zgesc2(N, A, LDA, RHS, IPIV, JPIV, SCALE)
ZGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed...
Definition: zgesc2.f:115
subroutine zgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
ZGECON
Definition: zgecon.f:124
subroutine zlaswp(N, A, LDA, K1, K2, IPIV, INCX)
ZLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: zlaswp.f:115
double precision function dzasum(N, ZX, INCX)
DZASUM
Definition: dzasum.f:72
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