 LAPACK  3.8.0 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)

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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.EQ.0 and ITER.GE.0, see description below), then A is unchanged, if double precision factorization has been used (INFO.EQ.0 and ITER.LT.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.EQ.0 and ITER.GE.0) or the double precision factorization (if INFO.EQ.0 and ITER.LT.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.```
Date
June 2016

Definition at line 197 of file dsgesv.f.

197 *
198 * -- LAPACK driver routine (version 3.8.0) --
199 * -- LAPACK is a software package provided by Univ. of Tennessee, --
200 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
201 * June 2016
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, dgetrf, dgetrs,
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 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:123
subroutine dgetrf(M, N, A, LDA, IPIV, INFO)
DGETRF
Definition: dgetrf.f:110
integer function idamax(N, DX, INCX)
IDAMAX
Definition: idamax.f:73
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:91
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:123
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:189
subroutine dlag2s(M, N, A, LDA, SA, LDSA, INFO)
DLAG2S converts a double precision matrix to a single precision matrix.
Definition: dlag2s.f:110
subroutine slag2d(M, N, SA, LDSA, A, LDA, INFO)
SLAG2D converts a single precision matrix to a double precision matrix.
Definition: slag2d.f:106
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:116
subroutine sgetrf(M, N, A, LDA, IPIV, INFO)
SGETRF
Definition: sgetrf.f:110
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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