LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
dsposv.f
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1 *> \brief <b> DSPOSV computes the solution to system of linear equations A * X = B for PO matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DSPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
22 * SWORK, ITER, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
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 *> DSPOSV computes the solution to a real system of linear equations
41 *> A * X = B,
42 *> where A is an N-by-N symmetric positive definite matrix and X and B
43 *> are N-by-NRHS matrices.
44 *>
45 *> DSPOSV first attempts to factorize the matrix in SINGLE PRECISION
46 *> and use this factorization within an iterative refinement procedure
47 *> to produce a solution with DOUBLE PRECISION normwise backward error
48 *> quality (see below). If the approach fails the method switches to a
49 *> DOUBLE PRECISION factorization and solve.
50 *>
51 *> The iterative refinement is not going to be a winning strategy if
52 *> the ratio SINGLE PRECISION performance over DOUBLE PRECISION
53 *> performance is too small. A reasonable strategy should take the
54 *> number of right-hand sides and the size of the matrix into account.
55 *> This might be done with a call to ILAENV in the future. Up to now, we
56 *> always try iterative refinement.
57 *>
58 *> The iterative refinement process is stopped if
59 *> ITER > ITERMAX
60 *> or for all the RHS we have:
61 *> RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
62 *> where
63 *> o ITER is the number of the current iteration in the iterative
64 *> refinement process
65 *> o RNRM is the infinity-norm of the residual
66 *> o XNRM is the infinity-norm of the solution
67 *> o ANRM is the infinity-operator-norm of the matrix A
68 *> o EPS is the machine epsilon returned by DLAMCH('Epsilon')
69 *> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
70 *> respectively.
71 *> \endverbatim
72 *
73 * Arguments:
74 * ==========
75 *
76 *> \param[in] UPLO
77 *> \verbatim
78 *> UPLO is CHARACTER*1
79 *> = 'U': Upper triangle of A is stored;
80 *> = 'L': Lower triangle of A is stored.
81 *> \endverbatim
82 *>
83 *> \param[in] N
84 *> \verbatim
85 *> N is INTEGER
86 *> The number of linear equations, i.e., the order of the
87 *> matrix A. N >= 0.
88 *> \endverbatim
89 *>
90 *> \param[in] NRHS
91 *> \verbatim
92 *> NRHS is INTEGER
93 *> The number of right hand sides, i.e., the number of columns
94 *> of the matrix B. NRHS >= 0.
95 *> \endverbatim
96 *>
97 *> \param[in,out] A
98 *> \verbatim
99 *> A is DOUBLE PRECISION array,
100 *> dimension (LDA,N)
101 *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
102 *> N-by-N upper triangular part of A contains the upper
103 *> triangular part of the matrix A, and the strictly lower
104 *> triangular part of A is not referenced. If UPLO = 'L', the
105 *> leading N-by-N lower triangular part of A contains the lower
106 *> triangular part of the matrix A, and the strictly upper
107 *> triangular part of A is not referenced.
108 *> On exit, if iterative refinement has been successfully used
109 *> (INFO = 0 and ITER >= 0, see description below), then A is
110 *> unchanged, if double precision factorization has been used
111 *> (INFO = 0 and ITER < 0, see description below), then the
112 *> array A contains the factor U or L from the Cholesky
113 *> factorization A = U**T*U or A = L*L**T.
114 *> \endverbatim
115 *>
116 *> \param[in] LDA
117 *> \verbatim
118 *> LDA is INTEGER
119 *> The leading dimension of the array A. LDA >= max(1,N).
120 *> \endverbatim
121 *>
122 *> \param[in] B
123 *> \verbatim
124 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
125 *> The N-by-NRHS right hand side matrix B.
126 *> \endverbatim
127 *>
128 *> \param[in] LDB
129 *> \verbatim
130 *> LDB is INTEGER
131 *> The leading dimension of the array B. LDB >= max(1,N).
132 *> \endverbatim
133 *>
134 *> \param[out] X
135 *> \verbatim
136 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
137 *> If INFO = 0, the N-by-NRHS solution matrix X.
138 *> \endverbatim
139 *>
140 *> \param[in] LDX
141 *> \verbatim
142 *> LDX is INTEGER
143 *> The leading dimension of the array X. LDX >= max(1,N).
144 *> \endverbatim
145 *>
146 *> \param[out] WORK
147 *> \verbatim
148 *> WORK is DOUBLE PRECISION array, dimension (N,NRHS)
149 *> This array is used to hold the residual vectors.
150 *> \endverbatim
151 *>
152 *> \param[out] SWORK
153 *> \verbatim
154 *> SWORK is REAL array, dimension (N*(N+NRHS))
155 *> This array is used to use the single precision matrix and the
156 *> right-hand sides or solutions in single precision.
157 *> \endverbatim
158 *>
159 *> \param[out] ITER
160 *> \verbatim
161 *> ITER is INTEGER
162 *> < 0: iterative refinement has failed, double precision
163 *> factorization has been performed
164 *> -1 : the routine fell back to full precision for
165 *> implementation- or machine-specific reasons
166 *> -2 : narrowing the precision induced an overflow,
167 *> the routine fell back to full precision
168 *> -3 : failure of SPOTRF
169 *> -31: stop the iterative refinement after the 30th
170 *> iterations
171 *> > 0: iterative refinement has been successfully used.
172 *> Returns the number of iterations
173 *> \endverbatim
174 *>
175 *> \param[out] INFO
176 *> \verbatim
177 *> INFO is INTEGER
178 *> = 0: successful exit
179 *> < 0: if INFO = -i, the i-th argument had an illegal value
180 *> > 0: if INFO = i, the leading minor of order i of (DOUBLE
181 *> PRECISION) A is not positive definite, so the
182 *> factorization could not be completed, and the solution
183 *> has not been computed.
184 *> \endverbatim
185 *
186 * Authors:
187 * ========
188 *
189 *> \author Univ. of Tennessee
190 *> \author Univ. of California Berkeley
191 *> \author Univ. of Colorado Denver
192 *> \author NAG Ltd.
193 *
194 *> \ingroup doublePOsolve
195 *
196 * =====================================================================
197  SUBROUTINE dsposv( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
198  $ SWORK, ITER, INFO )
199 *
200 * -- LAPACK driver routine --
201 * -- LAPACK is a software package provided by Univ. of Tennessee, --
202 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
203 *
204 * .. Scalar Arguments ..
205  CHARACTER UPLO
206  INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
207 * ..
208 * .. Array Arguments ..
209  REAL SWORK( * )
210  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),
211  $ x( ldx, * )
212 * ..
213 *
214 * =====================================================================
215 *
216 * .. Parameters ..
217  LOGICAL DOITREF
218  parameter( doitref = .true. )
219 *
220  INTEGER ITERMAX
221  parameter( itermax = 30 )
222 *
223  DOUBLE PRECISION BWDMAX
224  parameter( bwdmax = 1.0e+00 )
225 *
226  DOUBLE PRECISION NEGONE, ONE
227  parameter( negone = -1.0d+0, one = 1.0d+0 )
228 *
229 * .. Local Scalars ..
230  INTEGER I, IITER, PTSA, PTSX
231  DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM
232 *
233 * .. External Subroutines ..
234  EXTERNAL daxpy, dsymm, dlacpy, dlat2s, dlag2s, slag2d,
236 * ..
237 * .. External Functions ..
238  INTEGER IDAMAX
239  DOUBLE PRECISION DLAMCH, DLANSY
240  LOGICAL LSAME
241  EXTERNAL idamax, dlamch, dlansy, lsame
242 * ..
243 * .. Intrinsic Functions ..
244  INTRINSIC abs, dble, max, sqrt
245 * ..
246 * .. Executable Statements ..
247 *
248  info = 0
249  iter = 0
250 *
251 * Test the input parameters.
252 *
253  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
254  info = -1
255  ELSE IF( n.LT.0 ) THEN
256  info = -2
257  ELSE IF( nrhs.LT.0 ) THEN
258  info = -3
259  ELSE IF( lda.LT.max( 1, n ) ) THEN
260  info = -5
261  ELSE IF( ldb.LT.max( 1, n ) ) THEN
262  info = -7
263  ELSE IF( ldx.LT.max( 1, n ) ) THEN
264  info = -9
265  END IF
266  IF( info.NE.0 ) THEN
267  CALL xerbla( 'DSPOSV', -info )
268  RETURN
269  END IF
270 *
271 * Quick return if (N.EQ.0).
272 *
273  IF( n.EQ.0 )
274  $ RETURN
275 *
276 * Skip single precision iterative refinement if a priori slower
277 * than double precision factorization.
278 *
279  IF( .NOT.doitref ) THEN
280  iter = -1
281  GO TO 40
282  END IF
283 *
284 * Compute some constants.
285 *
286  anrm = dlansy( 'I', uplo, n, a, lda, work )
287  eps = dlamch( 'Epsilon' )
288  cte = anrm*eps*sqrt( dble( n ) )*bwdmax
289 *
290 * Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
291 *
292  ptsa = 1
293  ptsx = ptsa + n*n
294 *
295 * Convert B from double precision to single precision and store the
296 * result in SX.
297 *
298  CALL dlag2s( n, nrhs, b, ldb, swork( ptsx ), n, info )
299 *
300  IF( info.NE.0 ) THEN
301  iter = -2
302  GO TO 40
303  END IF
304 *
305 * Convert A from double precision to single precision and store the
306 * result in SA.
307 *
308  CALL dlat2s( uplo, n, a, lda, swork( ptsa ), n, info )
309 *
310  IF( info.NE.0 ) THEN
311  iter = -2
312  GO TO 40
313  END IF
314 *
315 * Compute the Cholesky factorization of SA.
316 *
317  CALL spotrf( uplo, n, swork( ptsa ), n, info )
318 *
319  IF( info.NE.0 ) THEN
320  iter = -3
321  GO TO 40
322  END IF
323 *
324 * Solve the system SA*SX = SB.
325 *
326  CALL spotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
327  $ info )
328 *
329 * Convert SX back to double precision
330 *
331  CALL slag2d( n, nrhs, swork( ptsx ), n, x, ldx, info )
332 *
333 * Compute R = B - AX (R is WORK).
334 *
335  CALL dlacpy( 'All', n, nrhs, b, ldb, work, n )
336 *
337  CALL dsymm( 'Left', uplo, n, nrhs, negone, a, lda, x, ldx, one,
338  $ work, n )
339 *
340 * Check whether the NRHS normwise backward errors satisfy the
341 * stopping criterion. If yes, set ITER=0 and return.
342 *
343  DO i = 1, nrhs
344  xnrm = abs( x( idamax( n, x( 1, i ), 1 ), i ) )
345  rnrm = abs( work( idamax( n, work( 1, i ), 1 ), i ) )
346  IF( rnrm.GT.xnrm*cte )
347  $ GO TO 10
348  END DO
349 *
350 * If we are here, the NRHS normwise backward errors satisfy the
351 * stopping criterion. We are good to exit.
352 *
353  iter = 0
354  RETURN
355 *
356  10 CONTINUE
357 *
358  DO 30 iiter = 1, itermax
359 *
360 * Convert R (in WORK) from double precision to single precision
361 * and store the result in SX.
362 *
363  CALL dlag2s( n, nrhs, work, n, swork( ptsx ), n, info )
364 *
365  IF( info.NE.0 ) THEN
366  iter = -2
367  GO TO 40
368  END IF
369 *
370 * Solve the system SA*SX = SR.
371 *
372  CALL spotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
373  $ info )
374 *
375 * Convert SX back to double precision and update the current
376 * iterate.
377 *
378  CALL slag2d( n, nrhs, swork( ptsx ), n, work, n, info )
379 *
380  DO i = 1, nrhs
381  CALL daxpy( n, one, work( 1, i ), 1, x( 1, i ), 1 )
382  END DO
383 *
384 * Compute R = B - AX (R is WORK).
385 *
386  CALL dlacpy( 'All', n, nrhs, b, ldb, work, n )
387 *
388  CALL dsymm( 'L', uplo, n, nrhs, negone, a, lda, x, ldx, one,
389  $ work, n )
390 *
391 * Check whether the NRHS normwise backward errors satisfy the
392 * stopping criterion. If yes, set ITER=IITER>0 and return.
393 *
394  DO i = 1, nrhs
395  xnrm = abs( x( idamax( n, x( 1, i ), 1 ), i ) )
396  rnrm = abs( work( idamax( n, work( 1, i ), 1 ), i ) )
397  IF( rnrm.GT.xnrm*cte )
398  $ GO TO 20
399  END DO
400 *
401 * If we are here, the NRHS normwise backward errors satisfy the
402 * stopping criterion, we are good to exit.
403 *
404  iter = iiter
405 *
406  RETURN
407 *
408  20 CONTINUE
409 *
410  30 CONTINUE
411 *
412 * If we are at this place of the code, this is because we have
413 * performed ITER=ITERMAX iterations and never satisfied the
414 * stopping criterion, set up the ITER flag accordingly and follow
415 * up on double precision routine.
416 *
417  iter = -itermax - 1
418 *
419  40 CONTINUE
420 *
421 * Single-precision iterative refinement failed to converge to a
422 * satisfactory solution, so we resort to double precision.
423 *
424  CALL dpotrf( uplo, n, a, lda, info )
425 *
426  IF( info.NE.0 )
427  $ RETURN
428 *
429  CALL dlacpy( 'All', n, nrhs, b, ldb, x, ldx )
430  CALL dpotrs( uplo, n, nrhs, a, lda, x, ldx, info )
431 *
432  RETURN
433 *
434 * End of DSPOSV
435 *
436  END
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine slag2d(M, N, SA, LDSA, A, LDA, INFO)
SLAG2D converts a single precision matrix to a double precision matrix.
Definition: slag2d.f:104
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dsymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DSYMM
Definition: dsymm.f:189
subroutine dlat2s(UPLO, N, A, LDA, SA, LDSA, INFO)
DLAT2S converts a double-precision triangular matrix to a single-precision triangular matrix.
Definition: dlat2s.f:111
subroutine dlag2s(M, N, A, LDA, SA, LDSA, INFO)
DLAG2S converts a double precision matrix to a single precision matrix.
Definition: dlag2s.f:108
subroutine dpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
DPOTRS
Definition: dpotrs.f:110
subroutine dpotrf(UPLO, N, A, LDA, INFO)
DPOTRF
Definition: dpotrf.f:107
subroutine dsposv(UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, SWORK, ITER, INFO)
DSPOSV computes the solution to system of linear equations A * X = B for PO matrices
Definition: dsposv.f:199
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:107
subroutine spotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
SPOTRS
Definition: spotrs.f:110