 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ clatdf()

 subroutine clatdf ( integer IJOB, integer N, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV )

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

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Purpose:
``` CLATDF 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 CGETC2. On entry RHS = f holds the
contribution from earlier solved sub-systems, and on return RHS = x.

The factorization of Z returned by CGETC2 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 CGECON, 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 array, dimension (LDZ, N) On entry, the LU part of the factorization of the n-by-n matrix Z computed by CGETC2: 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 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 REAL On entry, the sum of squares of computed contributions to the Dif-estimate under computation by CTGSYL, 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 CTGSY2 is called by CTGSYL.``` [in,out] RDSCAL ``` RDSCAL is REAL 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 CTGSY2 is called by CTGSYL.``` [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 clatdf.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  REAL RDSCAL, RDSUM
177 * ..
178 * .. Array Arguments ..
179  INTEGER IPIV( * ), JPIV( * )
180  COMPLEX RHS( * ), Z( LDZ, * )
181 * ..
182 *
183 * =====================================================================
184 *
185 * .. Parameters ..
186  INTEGER MAXDIM
187  parameter( maxdim = 2 )
188  REAL ZERO, ONE
189  parameter( zero = 0.0e+0, one = 1.0e+0 )
190  COMPLEX CONE
191  parameter( cone = ( 1.0e+0, 0.0e+0 ) )
192 * ..
193 * .. Local Scalars ..
194  INTEGER I, INFO, J, K
195  REAL RTEMP, SCALE, SMINU, SPLUS
196  COMPLEX BM, BP, PMONE, TEMP
197 * ..
198 * .. Local Arrays ..
199  REAL RWORK( MAXDIM )
200  COMPLEX WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
201 * ..
202 * .. External Subroutines ..
203  EXTERNAL caxpy, ccopy, cgecon, cgesc2, classq, claswp,
204  \$ cscal
205 * ..
206 * .. External Functions ..
207  REAL SCASUM
208  COMPLEX CDOTC
209  EXTERNAL scasum, cdotc
210 * ..
211 * .. Intrinsic Functions ..
212  INTRINSIC abs, real, sqrt
213 * ..
214 * .. Executable Statements ..
215 *
216  IF( ijob.NE.2 ) THEN
217 *
218 * Apply permutations IPIV to RHS
219 *
220  CALL claswp( 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 + real( cdotc( n-j, z( j+1, j ), 1, z( j+1,
234  \$ j ), 1 ) )
235  sminu = real( cdotc( n-j, z( j+1, j ), 1, rhs( j+1 ), 1 ) )
236  splus = splus*real( 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 caxpy( 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 ccopy( 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 ccopy( n, work, 1, rhs, 1 )
282 *
283 * Apply the permutations JPIV to the computed solution (RHS)
284 *
285  CALL claswp( 1, rhs, ldz, 1, n-1, jpiv, -1 )
286 *
287 * Compute the sum of squares
288 *
289  CALL classq( 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 cgecon( 'I', n, z, ldz, one, rtemp, work, rwork, info )
298  CALL ccopy( n, work( n+1 ), 1, xm, 1 )
299 *
300 * Compute RHS
301 *
302  CALL claswp( 1, xm, ldz, 1, n-1, ipiv, -1 )
303  temp = cone / sqrt( cdotc( n, xm, 1, xm, 1 ) )
304  CALL cscal( n, temp, xm, 1 )
305  CALL ccopy( n, xm, 1, xp, 1 )
306  CALL caxpy( n, cone, rhs, 1, xp, 1 )
307  CALL caxpy( n, -cone, xm, 1, rhs, 1 )
308  CALL cgesc2( n, z, ldz, rhs, ipiv, jpiv, scale )
309  CALL cgesc2( n, z, ldz, xp, ipiv, jpiv, scale )
310  IF( scasum( n, xp, 1 ).GT.scasum( n, rhs, 1 ) )
311  \$ CALL ccopy( n, xp, 1, rhs, 1 )
312 *
313 * Compute the sum of squares
314 *
315  CALL classq( n, rhs, 1, rdscal, rdsum )
316  RETURN
317 *
318 * End of CLATDF
319 *
subroutine classq(n, x, incx, scl, sumsq)
CLASSQ updates a sum of squares represented in scaled form.
Definition: classq.f90:137
complex function cdotc(N, CX, INCX, CY, INCY)
CDOTC
Definition: cdotc.f:83
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine cscal(N, CA, CX, INCX)
CSCAL
Definition: cscal.f:78
subroutine cgesc2(N, A, LDA, RHS, IPIV, JPIV, SCALE)
CGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed...
Definition: cgesc2.f:115
subroutine cgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
CGECON
Definition: cgecon.f:124
subroutine claswp(N, A, LDA, K1, K2, IPIV, INCX)
CLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: claswp.f:115
real function scasum(N, CX, INCX)
SCASUM
Definition: scasum.f:72
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