 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ slatdf()

 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 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 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).```
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 169 of file slatdf.f.

171 *
172 * -- LAPACK auxiliary routine --
173 * -- LAPACK is a software package provided by Univ. of Tennessee, --
174 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175 *
176 * .. Scalar Arguments ..
177  INTEGER IJOB, LDZ, N
178  REAL RDSCAL, RDSUM
179 * ..
180 * .. Array Arguments ..
181  INTEGER IPIV( * ), JPIV( * )
182  REAL RHS( * ), Z( LDZ, * )
183 * ..
184 *
185 * =====================================================================
186 *
187 * .. Parameters ..
188  INTEGER MAXDIM
189  parameter( maxdim = 8 )
190  REAL ZERO, ONE
191  parameter( zero = 0.0e+0, one = 1.0e+0 )
192 * ..
193 * .. Local Scalars ..
194  INTEGER I, INFO, J, K
195  REAL BM, BP, PMONE, SMINU, SPLUS, TEMP
196 * ..
197 * .. Local Arrays ..
198  INTEGER IWORK( MAXDIM )
199  REAL WORK( 4*MAXDIM ), XM( MAXDIM ), XP( MAXDIM )
200 * ..
201 * .. External Subroutines ..
202  EXTERNAL saxpy, scopy, sgecon, sgesc2, slassq, slaswp,
203  \$ sscal
204 * ..
205 * .. External Functions ..
206  REAL SASUM, SDOT
207  EXTERNAL sasum, sdot
208 * ..
209 * .. Intrinsic Functions ..
210  INTRINSIC abs, sqrt
211 * ..
212 * .. Executable Statements ..
213 *
214  IF( ijob.NE.2 ) THEN
215 *
216 * Apply permutations IPIV to RHS
217 *
218  CALL slaswp( 1, rhs, ldz, 1, n-1, ipiv, 1 )
219 *
220 * Solve for L-part choosing RHS either to +1 or -1.
221 *
222  pmone = -one
223 *
224  DO 10 j = 1, n - 1
225  bp = rhs( j ) + one
226  bm = rhs( j ) - one
227  splus = one
228 *
229 * Look-ahead for L-part RHS(1:N-1) = + or -1, SPLUS and
230 * SMIN computed more efficiently than in BSOLVE .
231 *
232  splus = splus + sdot( n-j, z( j+1, j ), 1, z( j+1, j ), 1 )
233  sminu = sdot( n-j, z( j+1, j ), 1, rhs( j+1 ), 1 )
234  splus = splus*rhs( j )
235  IF( splus.GT.sminu ) THEN
236  rhs( j ) = bp
237  ELSE IF( sminu.GT.splus ) THEN
238  rhs( j ) = bm
239  ELSE
240 *
241 * In this case the updating sums are equal and we can
242 * choose RHS(J) +1 or -1. The first time this happens
243 * we choose -1, thereafter +1. This is a simple way to
244 * get good estimates of matrices like Byers well-known
245 * example (see ). (Not done in BSOLVE.)
246 *
247  rhs( j ) = rhs( j ) + pmone
248  pmone = one
249  END IF
250 *
251 * Compute the remaining r.h.s.
252 *
253  temp = -rhs( j )
254  CALL saxpy( n-j, temp, z( j+1, j ), 1, rhs( j+1 ), 1 )
255 *
256  10 CONTINUE
257 *
258 * Solve for U-part, look-ahead for RHS(N) = +-1. This is not done
259 * in BSOLVE and will hopefully give us a better estimate because
260 * any ill-conditioning of the original matrix is transferred to U
261 * and not to L. U(N, N) is an approximation to sigma_min(LU).
262 *
263  CALL scopy( n-1, rhs, 1, xp, 1 )
264  xp( n ) = rhs( n ) + one
265  rhs( n ) = rhs( n ) - one
266  splus = zero
267  sminu = zero
268  DO 30 i = n, 1, -1
269  temp = one / z( i, i )
270  xp( i ) = xp( i )*temp
271  rhs( i ) = rhs( i )*temp
272  DO 20 k = i + 1, n
273  xp( i ) = xp( i ) - xp( k )*( z( i, k )*temp )
274  rhs( i ) = rhs( i ) - rhs( k )*( z( i, k )*temp )
275  20 CONTINUE
276  splus = splus + abs( xp( i ) )
277  sminu = sminu + abs( rhs( i ) )
278  30 CONTINUE
279  IF( splus.GT.sminu )
280  \$ CALL scopy( n, xp, 1, rhs, 1 )
281 *
282 * Apply the permutations JPIV to the computed solution (RHS)
283 *
284  CALL slaswp( 1, rhs, ldz, 1, n-1, jpiv, -1 )
285 *
286 * Compute the sum of squares
287 *
288  CALL slassq( n, rhs, 1, rdscal, rdsum )
289 *
290  ELSE
291 *
292 * IJOB = 2, Compute approximate nullvector XM of Z
293 *
294  CALL sgecon( 'I', n, z, ldz, one, temp, work, iwork, info )
295  CALL scopy( n, work( n+1 ), 1, xm, 1 )
296 *
297 * Compute RHS
298 *
299  CALL slaswp( 1, xm, ldz, 1, n-1, ipiv, -1 )
300  temp = one / sqrt( sdot( n, xm, 1, xm, 1 ) )
301  CALL sscal( n, temp, xm, 1 )
302  CALL scopy( n, xm, 1, xp, 1 )
303  CALL saxpy( n, one, rhs, 1, xp, 1 )
304  CALL saxpy( n, -one, xm, 1, rhs, 1 )
305  CALL sgesc2( n, z, ldz, rhs, ipiv, jpiv, temp )
306  CALL sgesc2( n, z, ldz, xp, ipiv, jpiv, temp )
307  IF( sasum( n, xp, 1 ).GT.sasum( n, rhs, 1 ) )
308  \$ CALL scopy( n, xp, 1, rhs, 1 )
309 *
310 * Compute the sum of squares
311 *
312  CALL slassq( n, rhs, 1, rdscal, rdsum )
313 *
314  END IF
315 *
316  RETURN
317 *
318 * End of SLATDF
319 *
subroutine slassq(n, x, incx, scl, sumsq)
SLASSQ updates a sum of squares represented in scaled form.
Definition: slassq.f90:126
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:114
subroutine sgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SGECON
Definition: sgecon.f:124
subroutine slaswp(N, A, LDA, K1, K2, IPIV, INCX)
SLASWP performs a series of row interchanges on a general rectangular matrix.
Definition: slaswp.f:115
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
real function sasum(N, SX, INCX)
SASUM
Definition: sasum.f:72
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