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dsgesv.f
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1 *> \brief <b> DSGESV computes the solution to system of linear equations A * X = B for GE matrices</b> (mixed precision with iterative refinement)
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsgesv.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSGESV( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
22 * SWORK, ITER, INFO )
23 *
24 * .. Scalar Arguments ..
25 * INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
26 * ..
27 * .. Array Arguments ..
28 * INTEGER IPIV( * )
29 * REAL SWORK( * )
30 * DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),
31 * $ X( LDX, * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> DSGESV computes the solution to a real system of linear equations
41 *> A * X = B,
42 *> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
43 *>
44 *> DSGESV first attempts to factorize the matrix in SINGLE PRECISION
45 *> and use this factorization within an iterative refinement procedure
46 *> to produce a solution with DOUBLE PRECISION normwise backward error
47 *> quality (see below). If the approach fails the method switches to a
48 *> DOUBLE PRECISION factorization and solve.
49 *>
50 *> The iterative refinement is not going to be a winning strategy if
51 *> the ratio SINGLE PRECISION performance over DOUBLE PRECISION
52 *> performance is too small. A reasonable strategy should take the
53 *> number of right-hand sides and the size of the matrix into account.
54 *> This might be done with a call to ILAENV in the future. Up to now, we
55 *> always try iterative refinement.
56 *>
57 *> The iterative refinement process is stopped if
58 *> ITER > ITERMAX
59 *> or for all the RHS we have:
60 *> RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
61 *> where
62 *> o ITER is the number of the current iteration in the iterative
63 *> refinement process
64 *> o RNRM is the infinity-norm of the residual
65 *> o XNRM is the infinity-norm of the solution
66 *> o ANRM is the infinity-operator-norm of the matrix A
67 *> o EPS is the machine epsilon returned by DLAMCH('Epsilon')
68 *> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
69 *> respectively.
70 *> \endverbatim
71 *
72 * Arguments:
73 * ==========
74 *
75 *> \param[in] N
76 *> \verbatim
77 *> N is INTEGER
78 *> The number of linear equations, i.e., the order of the
79 *> matrix A. N >= 0.
80 *> \endverbatim
81 *>
82 *> \param[in] NRHS
83 *> \verbatim
84 *> NRHS is INTEGER
85 *> The number of right hand sides, i.e., the number of columns
86 *> of the matrix B. NRHS >= 0.
87 *> \endverbatim
88 *>
89 *> \param[in,out] A
90 *> \verbatim
91 *> A is DOUBLE PRECISION array,
92 *> dimension (LDA,N)
93 *> On entry, the N-by-N coefficient matrix A.
94 *> On exit, if iterative refinement has been successfully used
95 *> (INFO.EQ.0 and ITER.GE.0, see description below), then A is
96 *> unchanged, if double precision factorization has been used
97 *> (INFO.EQ.0 and ITER.LT.0, see description below), then the
98 *> array A contains the factors L and U from the factorization
99 *> A = P*L*U; the unit diagonal elements of L are not stored.
100 *> \endverbatim
101 *>
102 *> \param[in] LDA
103 *> \verbatim
104 *> LDA is INTEGER
105 *> The leading dimension of the array A. LDA >= max(1,N).
106 *> \endverbatim
107 *>
108 *> \param[out] IPIV
109 *> \verbatim
110 *> IPIV is INTEGER array, dimension (N)
111 *> The pivot indices that define the permutation matrix P;
112 *> row i of the matrix was interchanged with row IPIV(i).
113 *> Corresponds either to the single precision factorization
114 *> (if INFO.EQ.0 and ITER.GE.0) or the double precision
115 *> factorization (if INFO.EQ.0 and ITER.LT.0).
116 *> \endverbatim
117 *>
118 *> \param[in] B
119 *> \verbatim
120 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
121 *> The N-by-NRHS right hand side matrix B.
122 *> \endverbatim
123 *>
124 *> \param[in] LDB
125 *> \verbatim
126 *> LDB is INTEGER
127 *> The leading dimension of the array B. LDB >= max(1,N).
128 *> \endverbatim
129 *>
130 *> \param[out] X
131 *> \verbatim
132 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
133 *> If INFO = 0, the N-by-NRHS solution matrix X.
134 *> \endverbatim
135 *>
136 *> \param[in] LDX
137 *> \verbatim
138 *> LDX is INTEGER
139 *> The leading dimension of the array X. LDX >= max(1,N).
140 *> \endverbatim
141 *>
142 *> \param[out] WORK
143 *> \verbatim
144 *> WORK is DOUBLE PRECISION array, dimension (N,NRHS)
145 *> This array is used to hold the residual vectors.
146 *> \endverbatim
147 *>
148 *> \param[out] SWORK
149 *> \verbatim
150 *> SWORK is REAL array, dimension (N*(N+NRHS))
151 *> This array is used to use the single precision matrix and the
152 *> right-hand sides or solutions in single precision.
153 *> \endverbatim
154 *>
155 *> \param[out] ITER
156 *> \verbatim
157 *> ITER is INTEGER
158 *> < 0: iterative refinement has failed, double precision
159 *> factorization has been performed
160 *> -1 : the routine fell back to full precision for
161 *> implementation- or machine-specific reasons
162 *> -2 : narrowing the precision induced an overflow,
163 *> the routine fell back to full precision
164 *> -3 : failure of SGETRF
165 *> -31: stop the iterative refinement after the 30th
166 *> iterations
167 *> > 0: iterative refinement has been sucessfully used.
168 *> Returns the number of iterations
169 *> \endverbatim
170 *>
171 *> \param[out] INFO
172 *> \verbatim
173 *> INFO is INTEGER
174 *> = 0: successful exit
175 *> < 0: if INFO = -i, the i-th argument had an illegal value
176 *> > 0: if INFO = i, U(i,i) computed in DOUBLE PRECISION is
177 *> exactly zero. The factorization has been completed,
178 *> but the factor U is exactly singular, so the solution
179 *> could not be computed.
180 *> \endverbatim
181 *
182 * Authors:
183 * ========
184 *
185 *> \author Univ. of Tennessee
186 *> \author Univ. of California Berkeley
187 *> \author Univ. of Colorado Denver
188 *> \author NAG Ltd.
189 *
190 *> \date November 2011
191 *
192 *> \ingroup doubleGEsolve
193 *
194 * =====================================================================
195  SUBROUTINE dsgesv( N, NRHS, A, LDA, IPIV, B, LDB, X, LDX, WORK,
196  $ swork, iter, info )
197 *
198 * -- LAPACK driver routine (version 3.4.0) --
199 * -- LAPACK is a software package provided by Univ. of Tennessee, --
200 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
201 * November 2011
202 *
203 * .. Scalar Arguments ..
204  INTEGER info, iter, lda, ldb, ldx, n, nrhs
205 * ..
206 * .. Array Arguments ..
207  INTEGER ipiv( * )
208  REAL swork( * )
209  DOUBLE PRECISION a( lda, * ), b( ldb, * ), work( n, * ),
210  $ x( ldx, * )
211 * ..
212 *
213 * =====================================================================
214 *
215 * .. Parameters ..
216  LOGICAL doitref
217  parameter( doitref = .true. )
218 *
219  INTEGER itermax
220  parameter( itermax = 30 )
221 *
222  DOUBLE PRECISION bwdmax
223  parameter( bwdmax = 1.0e+00 )
224 *
225  DOUBLE PRECISION negone, one
226  parameter( negone = -1.0d+0, one = 1.0d+0 )
227 *
228 * .. Local Scalars ..
229  INTEGER i, iiter, ptsa, ptsx
230  DOUBLE PRECISION anrm, cte, eps, rnrm, xnrm
231 *
232 * .. External Subroutines ..
233  EXTERNAL daxpy, dgemm, dlacpy, dlag2s, slag2d, sgetrf,
234  $ sgetrs, xerbla
235 * ..
236 * .. External Functions ..
237  INTEGER idamax
238  DOUBLE PRECISION dlamch, dlange
239  EXTERNAL idamax, dlamch, dlange
240 * ..
241 * .. Intrinsic Functions ..
242  INTRINSIC abs, dble, max, sqrt
243 * ..
244 * .. Executable Statements ..
245 *
246  info = 0
247  iter = 0
248 *
249 * Test the input parameters.
250 *
251  IF( n.LT.0 ) THEN
252  info = -1
253  ELSE IF( nrhs.LT.0 ) THEN
254  info = -2
255  ELSE IF( lda.LT.max( 1, n ) ) THEN
256  info = -4
257  ELSE IF( ldb.LT.max( 1, n ) ) THEN
258  info = -7
259  ELSE IF( ldx.LT.max( 1, n ) ) THEN
260  info = -9
261  END IF
262  IF( info.NE.0 ) THEN
263  CALL xerbla( 'DSGESV', -info )
264  RETURN
265  END IF
266 *
267 * Quick return if (N.EQ.0).
268 *
269  IF( n.EQ.0 )
270  $ RETURN
271 *
272 * Skip single precision iterative refinement if a priori slower
273 * than double precision factorization.
274 *
275  IF( .NOT.doitref ) THEN
276  iter = -1
277  go to 40
278  END IF
279 *
280 * Compute some constants.
281 *
282  anrm = dlange( 'I', n, n, a, lda, work )
283  eps = dlamch( 'Epsilon' )
284  cte = anrm*eps*sqrt( dble( n ) )*bwdmax
285 *
286 * Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
287 *
288  ptsa = 1
289  ptsx = ptsa + n*n
290 *
291 * Convert B from double precision to single precision and store the
292 * result in SX.
293 *
294  CALL dlag2s( n, nrhs, b, ldb, swork( ptsx ), n, info )
295 *
296  IF( info.NE.0 ) THEN
297  iter = -2
298  go to 40
299  END IF
300 *
301 * Convert A from double precision to single precision and store the
302 * result in SA.
303 *
304  CALL dlag2s( n, n, a, lda, swork( ptsa ), n, info )
305 *
306  IF( info.NE.0 ) THEN
307  iter = -2
308  go to 40
309  END IF
310 *
311 * Compute the LU factorization of SA.
312 *
313  CALL sgetrf( n, n, swork( ptsa ), n, ipiv, info )
314 *
315  IF( info.NE.0 ) THEN
316  iter = -3
317  go to 40
318  END IF
319 *
320 * Solve the system SA*SX = SB.
321 *
322  CALL sgetrs( 'No transpose', n, nrhs, swork( ptsa ), n, ipiv,
323  $ swork( ptsx ), n, info )
324 *
325 * Convert SX back to double precision
326 *
327  CALL slag2d( n, nrhs, swork( ptsx ), n, x, ldx, info )
328 *
329 * Compute R = B - AX (R is WORK).
330 *
331  CALL dlacpy( 'All', n, nrhs, b, ldb, work, n )
332 *
333  CALL dgemm( 'No Transpose', 'No Transpose', n, nrhs, n, negone, a,
334  $ lda, x, ldx, one, work, n )
335 *
336 * Check whether the NRHS normwise backward errors satisfy the
337 * stopping criterion. If yes, set ITER=0 and return.
338 *
339  DO i = 1, nrhs
340  xnrm = abs( x( idamax( n, x( 1, i ), 1 ), i ) )
341  rnrm = abs( work( idamax( n, work( 1, i ), 1 ), i ) )
342  IF( rnrm.GT.xnrm*cte )
343  $ go to 10
344  END DO
345 *
346 * If we are here, the NRHS normwise backward errors satisfy the
347 * stopping criterion. We are good to exit.
348 *
349  iter = 0
350  RETURN
351 *
352  10 CONTINUE
353 *
354  DO 30 iiter = 1, itermax
355 *
356 * Convert R (in WORK) from double precision to single precision
357 * and store the result in SX.
358 *
359  CALL dlag2s( n, nrhs, work, n, swork( ptsx ), n, info )
360 *
361  IF( info.NE.0 ) THEN
362  iter = -2
363  go to 40
364  END IF
365 *
366 * Solve the system SA*SX = SR.
367 *
368  CALL sgetrs( 'No transpose', n, nrhs, swork( ptsa ), n, ipiv,
369  $ swork( ptsx ), n, info )
370 *
371 * Convert SX back to double precision and update the current
372 * iterate.
373 *
374  CALL slag2d( n, nrhs, swork( ptsx ), n, work, n, info )
375 *
376  DO i = 1, nrhs
377  CALL daxpy( n, one, work( 1, i ), 1, x( 1, i ), 1 )
378  END DO
379 *
380 * Compute R = B - AX (R is WORK).
381 *
382  CALL dlacpy( 'All', n, nrhs, b, ldb, work, n )
383 *
384  CALL dgemm( 'No Transpose', 'No Transpose', n, nrhs, n, negone,
385  $ a, lda, x, ldx, one, work, n )
386 *
387 * Check whether the NRHS normwise backward errors satisfy the
388 * stopping criterion. If yes, set ITER=IITER>0 and return.
389 *
390  DO i = 1, nrhs
391  xnrm = abs( x( idamax( n, x( 1, i ), 1 ), i ) )
392  rnrm = abs( work( idamax( n, work( 1, i ), 1 ), i ) )
393  IF( rnrm.GT.xnrm*cte )
394  $ go to 20
395  END DO
396 *
397 * If we are here, the NRHS normwise backward errors satisfy the
398 * stopping criterion, we are good to exit.
399 *
400  iter = iiter
401 *
402  RETURN
403 *
404  20 CONTINUE
405 *
406  30 CONTINUE
407 *
408 * If we are at this place of the code, this is because we have
409 * performed ITER=ITERMAX iterations and never satisified the
410 * stopping criterion, set up the ITER flag accordingly and follow up
411 * on double precision routine.
412 *
413  iter = -itermax - 1
414 *
415  40 CONTINUE
416 *
417 * Single-precision iterative refinement failed to converge to a
418 * satisfactory solution, so we resort to double precision.
419 *
420  CALL dgetrf( n, n, a, lda, ipiv, info )
421 *
422  IF( info.NE.0 )
423  $ RETURN
424 *
425  CALL dlacpy( 'All', n, nrhs, b, ldb, x, ldx )
426  CALL dgetrs( 'No transpose', n, nrhs, a, lda, ipiv, x, ldx,
427  $ info )
428 *
429  RETURN
430 *
431 * End of DSGESV.
432 *
433  END