 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dsgesv()

 subroutine dsgesv ( integer N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, 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 )

DSGESV computes the solution to system of linear equations A * X = B for GE matrices (mixed precision with iterative refinement)

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

DSGESV 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] 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 N-by-N coefficient matrix A. 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 factors L and U from the factorization A = P*L*U; the unit diagonal elements of L are not stored.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [out] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPIV(i). Corresponds either to the single precision factorization (if INFO = 0 and ITER >= 0) or the double precision factorization (if INFO = 0 and ITER < 0).``` [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 SGETRF -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, U(i,i) computed in DOUBLE PRECISION is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.```

Definition at line 193 of file dsgesv.f.

195 *
196 * -- LAPACK driver routine --
197 * -- LAPACK is a software package provided by Univ. of Tennessee, --
198 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
199 *
200 * .. Scalar Arguments ..
201  INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
202 * ..
203 * .. Array Arguments ..
204  INTEGER IPIV( * )
205  REAL SWORK( * )
206  DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( N, * ),
207  \$ X( LDX, * )
208 * ..
209 *
210 * =====================================================================
211 *
212 * .. Parameters ..
213  LOGICAL DOITREF
214  parameter( doitref = .true. )
215 *
216  INTEGER ITERMAX
217  parameter( itermax = 30 )
218 *
219  DOUBLE PRECISION BWDMAX
220  parameter( bwdmax = 1.0e+00 )
221 *
222  DOUBLE PRECISION NEGONE, ONE
223  parameter( negone = -1.0d+0, one = 1.0d+0 )
224 *
225 * .. Local Scalars ..
226  INTEGER I, IITER, PTSA, PTSX
227  DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM
228 *
229 * .. External Subroutines ..
230  EXTERNAL daxpy, dgemm, dlacpy, dlag2s, dgetrf, dgetrs,
232 * ..
233 * .. External Functions ..
234  INTEGER IDAMAX
235  DOUBLE PRECISION DLAMCH, DLANGE
236  EXTERNAL idamax, dlamch, dlange
237 * ..
238 * .. Intrinsic Functions ..
239  INTRINSIC abs, dble, max, sqrt
240 * ..
241 * .. Executable Statements ..
242 *
243  info = 0
244  iter = 0
245 *
246 * Test the input parameters.
247 *
248  IF( n.LT.0 ) THEN
249  info = -1
250  ELSE IF( nrhs.LT.0 ) THEN
251  info = -2
252  ELSE IF( lda.LT.max( 1, n ) ) THEN
253  info = -4
254  ELSE IF( ldb.LT.max( 1, n ) ) THEN
255  info = -7
256  ELSE IF( ldx.LT.max( 1, n ) ) THEN
257  info = -9
258  END IF
259  IF( info.NE.0 ) THEN
260  CALL xerbla( 'DSGESV', -info )
261  RETURN
262  END IF
263 *
264 * Quick return if (N.EQ.0).
265 *
266  IF( n.EQ.0 )
267  \$ RETURN
268 *
269 * Skip single precision iterative refinement if a priori slower
270 * than double precision factorization.
271 *
272  IF( .NOT.doitref ) THEN
273  iter = -1
274  GO TO 40
275  END IF
276 *
277 * Compute some constants.
278 *
279  anrm = dlange( 'I', n, n, a, lda, work )
280  eps = dlamch( 'Epsilon' )
281  cte = anrm*eps*sqrt( dble( n ) )*bwdmax
282 *
283 * Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
284 *
285  ptsa = 1
286  ptsx = ptsa + n*n
287 *
288 * Convert B from double precision to single precision and store the
289 * result in SX.
290 *
291  CALL dlag2s( n, nrhs, b, ldb, swork( ptsx ), n, info )
292 *
293  IF( info.NE.0 ) THEN
294  iter = -2
295  GO TO 40
296  END IF
297 *
298 * Convert A from double precision to single precision and store the
299 * result in SA.
300 *
301  CALL dlag2s( n, n, a, lda, swork( ptsa ), n, info )
302 *
303  IF( info.NE.0 ) THEN
304  iter = -2
305  GO TO 40
306  END IF
307 *
308 * Compute the LU factorization of SA.
309 *
310  CALL sgetrf( n, n, swork( ptsa ), n, ipiv, info )
311 *
312  IF( info.NE.0 ) THEN
313  iter = -3
314  GO TO 40
315  END IF
316 *
317 * Solve the system SA*SX = SB.
318 *
319  CALL sgetrs( 'No transpose', n, nrhs, swork( ptsa ), n, ipiv,
320  \$ swork( ptsx ), n, info )
321 *
322 * Convert SX back to double precision
323 *
324  CALL slag2d( n, nrhs, swork( ptsx ), n, x, ldx, info )
325 *
326 * Compute R = B - AX (R is WORK).
327 *
328  CALL dlacpy( 'All', n, nrhs, b, ldb, work, n )
329 *
330  CALL dgemm( 'No Transpose', 'No Transpose', n, nrhs, n, negone, a,
331  \$ lda, x, ldx, one, work, n )
332 *
333 * Check whether the NRHS normwise backward errors satisfy the
334 * stopping criterion. If yes, set ITER=0 and return.
335 *
336  DO i = 1, nrhs
337  xnrm = abs( x( idamax( n, x( 1, i ), 1 ), i ) )
338  rnrm = abs( work( idamax( n, work( 1, i ), 1 ), i ) )
339  IF( rnrm.GT.xnrm*cte )
340  \$ GO TO 10
341  END DO
342 *
343 * If we are here, the NRHS normwise backward errors satisfy the
344 * stopping criterion. We are good to exit.
345 *
346  iter = 0
347  RETURN
348 *
349  10 CONTINUE
350 *
351  DO 30 iiter = 1, itermax
352 *
353 * Convert R (in WORK) from double precision to single precision
354 * and store the result in SX.
355 *
356  CALL dlag2s( n, nrhs, work, n, swork( ptsx ), n, info )
357 *
358  IF( info.NE.0 ) THEN
359  iter = -2
360  GO TO 40
361  END IF
362 *
363 * Solve the system SA*SX = SR.
364 *
365  CALL sgetrs( 'No transpose', n, nrhs, swork( ptsa ), n, ipiv,
366  \$ swork( ptsx ), n, info )
367 *
368 * Convert SX back to double precision and update the current
369 * iterate.
370 *
371  CALL slag2d( n, nrhs, swork( ptsx ), n, work, n, info )
372 *
373  DO i = 1, nrhs
374  CALL daxpy( n, one, work( 1, i ), 1, x( 1, i ), 1 )
375  END DO
376 *
377 * Compute R = B - AX (R is WORK).
378 *
379  CALL dlacpy( 'All', n, nrhs, b, ldb, work, n )
380 *
381  CALL dgemm( 'No Transpose', 'No Transpose', n, nrhs, n, negone,
382  \$ a, lda, x, ldx, one, work, n )
383 *
384 * Check whether the NRHS normwise backward errors satisfy the
385 * stopping criterion. If yes, set ITER=IITER>0 and return.
386 *
387  DO i = 1, nrhs
388  xnrm = abs( x( idamax( n, x( 1, i ), 1 ), i ) )
389  rnrm = abs( work( idamax( n, work( 1, i ), 1 ), i ) )
390  IF( rnrm.GT.xnrm*cte )
391  \$ GO TO 20
392  END DO
393 *
394 * If we are here, the NRHS normwise backward errors satisfy the
395 * stopping criterion, we are good to exit.
396 *
397  iter = iiter
398 *
399  RETURN
400 *
401  20 CONTINUE
402 *
403  30 CONTINUE
404 *
405 * If we are at this place of the code, this is because we have
406 * performed ITER=ITERMAX iterations and never satisfied the
407 * stopping criterion, set up the ITER flag accordingly and follow up
408 * on double precision routine.
409 *
410  iter = -itermax - 1
411 *
412  40 CONTINUE
413 *
414 * Single-precision iterative refinement failed to converge to a
415 * satisfactory solution, so we resort to double precision.
416 *
417  CALL dgetrf( n, n, a, lda, ipiv, info )
418 *
419  IF( info.NE.0 )
420  \$ RETURN
421 *
422  CALL dlacpy( 'All', n, nrhs, b, ldb, x, ldx )
423  CALL dgetrs( 'No transpose', n, nrhs, a, lda, ipiv, x, ldx,
424  \$ info )
425 *
426  RETURN
427 *
428 * End of DSGESV
429 *
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
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dgetrf(M, N, A, LDA, IPIV, INFO)
DGETRF
Definition: dgetrf.f:108
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:121
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 sgetrf(M, N, A, LDA, IPIV, INFO)
SGETRF
Definition: sgetrf.f:108
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:121
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