 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dsposv()

 subroutine dsposv ( character UPLO, integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( n, * ) WORK, real, dimension( * ) SWORK, integer ITER, integer INFO )

DSPOSV computes the solution to system of linear equations A * X = B for PO matrices

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Purpose:
``` DSPOSV computes the solution to a real system of linear equations
A * X = B,
where A is an N-by-N symmetric positive definite matrix and X and B
are N-by-NRHS matrices.

DSPOSV first attempts to factorize the matrix in SINGLE PRECISION
and use this factorization within an iterative refinement procedure
to produce a solution with DOUBLE PRECISION normwise backward error
quality (see below). If the approach fails the method switches to a
DOUBLE PRECISION factorization and solve.

The iterative refinement is not going to be a winning strategy if
the ratio SINGLE PRECISION performance over DOUBLE PRECISION
performance is too small. A reasonable strategy should take the
number of right-hand sides and the size of the matrix into account.
This might be done with a call to ILAENV in the future. Up to now, we
always try iterative refinement.

The iterative refinement process is stopped if
ITER > ITERMAX
or for all the RHS we have:
RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
where
o ITER is the number of the current iteration in the iterative
refinement process
o RNRM is the infinity-norm of the residual
o XNRM is the infinity-norm of the solution
o ANRM is the infinity-operator-norm of the matrix A
o EPS is the machine epsilon returned by DLAMCH('Epsilon')
The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
respectively.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.``` [in,out] A ``` A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit, if iterative refinement has been successfully used (INFO = 0 and ITER >= 0, see description below), then A is unchanged, if double precision factorization has been used (INFO = 0 and ITER < 0, see description below), then the array A contains the factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] B ``` B is DOUBLE PRECISION array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] X ``` X is DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] WORK ``` WORK is DOUBLE PRECISION array, dimension (N,NRHS) This array is used to hold the residual vectors.``` [out] SWORK ``` SWORK is REAL array, dimension (N*(N+NRHS)) This array is used to use the single precision matrix and the right-hand sides or solutions in single precision.``` [out] ITER ``` ITER is INTEGER < 0: iterative refinement has failed, double precision factorization has been performed -1 : the routine fell back to full precision for implementation- or machine-specific reasons -2 : narrowing the precision induced an overflow, the routine fell back to full precision -3 : failure of SPOTRF -31: stop the iterative refinement after the 30th iterations > 0: iterative refinement has been successfully used. Returns the number of iterations``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i of (DOUBLE PRECISION) A is not positive definite, so the factorization could not be completed, and the solution has not been computed.```

Definition at line 197 of file dsposv.f.

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 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
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
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
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
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