LAPACK 3.12.0
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.

Download ZLATDF + dependencies [TGZ] [ZIP] [TXT]

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).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
This routine is a further developed implementation of algorithm BSOLVE in [1] 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:
[1] 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.
[2] 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* Look-ahead 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*
dzasum
double precision function dzasum(n, zx, incx)
DZASUM
Definition dzasum.f:72
zaxpy
subroutine zaxpy(n, za, zx, incx, zy, incy)
ZAXPY
Definition zaxpy.f:88
zcopy
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
zdotc
complex *16 function zdotc(n, zx, incx, zy, incy)
ZDOTC
Definition zdotc.f:83
zgecon
subroutine zgecon(norm, n, a, lda, anorm, rcond, work, rwork, info)
ZGECON
Definition zgecon.f:132
zgesc2
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
zlassq
subroutine zlassq(n, x, incx, scale, sumsq)
ZLASSQ updates a sum of squares represented in scaled form.
Definition zlassq.f90:124
zlaswp
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
zscal
subroutine zscal(n, za, zx, incx)
ZSCAL
Definition zscal.f:78
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