 LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine slatdf ( integer IJOB, integer N, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV )

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

Purpose:
``` SLATDF uses the LU factorization of the n-by-n matrix Z computed by
SGETC2 and computes a contribution to the reciprocal Dif-estimate
by solving Z * x = b for x, and choosing the r.h.s. b such that
the norm of x is as large as possible. On entry RHS = b holds the
contribution from earlier solved sub-systems, and on return RHS = x.

The factorization of Z returned by SGETC2 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 SGECON, 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 REAL array, dimension (LDZ, N) On entry, the LU part of the factorization of the n-by-n matrix Z computed by SGETC2: 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 REAL array, dimension N. On entry, RHS contains contributions from other subsystems. On exit, RHS contains the solution of the subsystem with entries acoording 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 STGSYL, 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 STGSY2 is called by STGSYL.``` [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 STGSY2 is called by STGSYL.``` [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).```
Date
June 2016
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 IMINF-95.05, Departement of
Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.```

Definition at line 173 of file slatdf.f.

173 *
174 * -- LAPACK auxiliary routine (version 3.6.1) --
175 * -- LAPACK is a software package provided by Univ. of Tennessee, --
176 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177 * June 2016
178 *
179 * .. Scalar Arguments ..
180  INTEGER ijob, ldz, n
181  REAL rdscal, rdsum
182 * ..
183 * .. Array Arguments ..
184  INTEGER ipiv( * ), jpiv( * )
185  REAL rhs( * ), z( ldz, * )
186 * ..
187 *
188 * =====================================================================
189 *
190 * .. Parameters ..
191  INTEGER maxdim
192  parameter ( maxdim = 8 )
193  REAL zero, one
194  parameter ( zero = 0.0e+0, one = 1.0e+0 )
195 * ..
196 * .. Local Scalars ..
197  INTEGER i, info, j, k
198  REAL bm, bp, pmone, sminu, splus, temp
199 * ..
200 * .. Local Arrays ..
201  INTEGER iwork( maxdim )
202  REAL work( 4*maxdim ), xm( maxdim ), xp( maxdim )
203 * ..
204 * .. External Subroutines ..
205  EXTERNAL saxpy, scopy, sgecon, sgesc2, slassq, slaswp,
206  \$ sscal
207 * ..
208 * .. External Functions ..
209  REAL sasum, sdot
210  EXTERNAL sasum, sdot
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 slaswp( 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 .
234 *
235  splus = splus + sdot( n-j, z( j+1, j ), 1, z( j+1, j ), 1 )
236  sminu = sdot( 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 ). (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 saxpy( 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 scopy( 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 scopy( n, xp, 1, rhs, 1 )
284 *
285 * Apply the permutations JPIV to the computed solution (RHS)
286 *
287  CALL slaswp( 1, rhs, ldz, 1, n-1, jpiv, -1 )
288 *
289 * Compute the sum of squares
290 *
291  CALL slassq( n, rhs, 1, rdscal, rdsum )
292 *
293  ELSE
294 *
295 * IJOB = 2, Compute approximate nullvector XM of Z
296 *
297  CALL sgecon( 'I', n, z, ldz, one, temp, work, iwork, info )
298  CALL scopy( n, work( n+1 ), 1, xm, 1 )
299 *
300 * Compute RHS
301 *
302  CALL slaswp( 1, xm, ldz, 1, n-1, ipiv, -1 )
303  temp = one / sqrt( sdot( n, xm, 1, xm, 1 ) )
304  CALL sscal( n, temp, xm, 1 )
305  CALL scopy( n, xm, 1, xp, 1 )
306  CALL saxpy( n, one, rhs, 1, xp, 1 )
307  CALL saxpy( n, -one, xm, 1, rhs, 1 )
308  CALL sgesc2( n, z, ldz, rhs, ipiv, jpiv, temp )
309  CALL sgesc2( n, z, ldz, xp, ipiv, jpiv, temp )
310  IF( sasum( n, xp, 1 ).GT.sasum( n, rhs, 1 ) )
311  \$ CALL scopy( n, xp, 1, rhs, 1 )
312 *
313 * Compute the sum of squares
314 *
315  CALL slassq( n, rhs, 1, rdscal, rdsum )
316 *
317  END IF
318 *
319  RETURN
320 *
321 * End of SLATDF
322 *
subroutine slassq(N, X, INCX, SCALE, SUMSQ)
SLASSQ updates a sum of squares represented in scaled form.
Definition: slassq.f:105
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:53
subroutine sgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SGECON
Definition: sgecon.f:126
subroutine slaswp(N, A, LDA, K1, K2, IPIV, INCX)
SLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: slaswp.f:116
subroutine sgesc2(N, A, LDA, RHS, IPIV, JPIV, SCALE)
SGESC2 solves a system of linear equations using the LU factorization with complete pivoting computed...
Definition: sgesc2.f:116
real function sasum(N, SX, INCX)
SASUM
Definition: sasum.f:54
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:54
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:55
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:53

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