LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ 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.

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

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).
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 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 [1].
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 [1]). (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*
sasum
real function sasum(n, sx, incx)
SASUM
Definition sasum.f:72
saxpy
subroutine saxpy(n, sa, sx, incx, sy, incy)
SAXPY
Definition saxpy.f:89
scopy
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
sdot
real function sdot(n, sx, incx, sy, incy)
SDOT
Definition sdot.f:82
sgecon
subroutine sgecon(norm, n, a, lda, anorm, rcond, work, iwork, info)
SGECON
Definition sgecon.f:132
sgesc2
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
slassq
subroutine slassq(n, x, incx, scale, sumsq)
SLASSQ updates a sum of squares represented in scaled form.
Definition slassq.f90:124
slaswp
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
sscal
subroutine sscal(n, sa, sx, incx)
SSCAL
Definition sscal.f:79
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