LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
sgesvx.f
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1 *> \brief <b> SGESVX computes the solution to system of linear equations A * X = B for GE matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
22 * EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
23 * WORK, IWORK, INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER EQUED, FACT, TRANS
27 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
28 * REAL RCOND
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IPIV( * ), IWORK( * )
32 * REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
33 * $ BERR( * ), C( * ), FERR( * ), R( * ),
34 * $ WORK( * ), X( LDX, * )
35 * ..
36 *
37 *
38 *> \par Purpose:
39 * =============
40 *>
41 *> \verbatim
42 *>
43 *> SGESVX uses the LU factorization to compute the solution to a real
44 *> system of linear equations
45 *> A * X = B,
46 *> where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
47 *>
48 *> Error bounds on the solution and a condition estimate are also
49 *> provided.
50 *> \endverbatim
51 *
52 *> \par Description:
53 * =================
54 *>
55 *> \verbatim
56 *>
57 *> The following steps are performed:
58 *>
59 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
60 *> the system:
61 *> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
62 *> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
63 *> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
64 *> Whether or not the system will be equilibrated depends on the
65 *> scaling of the matrix A, but if equilibration is used, A is
66 *> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
67 *> or diag(C)*B (if TRANS = 'T' or 'C').
68 *>
69 *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
70 *> matrix A (after equilibration if FACT = 'E') as
71 *> A = P * L * U,
72 *> where P is a permutation matrix, L is a unit lower triangular
73 *> matrix, and U is upper triangular.
74 *>
75 *> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
76 *> returns with INFO = i. Otherwise, the factored form of A is used
77 *> to estimate the condition number of the matrix A. If the
78 *> reciprocal of the condition number is less than machine precision,
79 *> INFO = N+1 is returned as a warning, but the routine still goes on
80 *> to solve for X and compute error bounds as described below.
81 *>
82 *> 4. The system of equations is solved for X using the factored form
83 *> of A.
84 *>
85 *> 5. Iterative refinement is applied to improve the computed solution
86 *> matrix and calculate error bounds and backward error estimates
87 *> for it.
88 *>
89 *> 6. If equilibration was used, the matrix X is premultiplied by
90 *> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
91 *> that it solves the original system before equilibration.
92 *> \endverbatim
93 *
94 * Arguments:
95 * ==========
96 *
97 *> \param[in] FACT
98 *> \verbatim
99 *> FACT is CHARACTER*1
100 *> Specifies whether or not the factored form of the matrix A is
101 *> supplied on entry, and if not, whether the matrix A should be
102 *> equilibrated before it is factored.
103 *> = 'F': On entry, AF and IPIV contain the factored form of A.
104 *> If EQUED is not 'N', the matrix A has been
105 *> equilibrated with scaling factors given by R and C.
106 *> A, AF, and IPIV are not modified.
107 *> = 'N': The matrix A will be copied to AF and factored.
108 *> = 'E': The matrix A will be equilibrated if necessary, then
109 *> copied to AF and factored.
110 *> \endverbatim
111 *>
112 *> \param[in] TRANS
113 *> \verbatim
114 *> TRANS is CHARACTER*1
115 *> Specifies the form of the system of equations:
116 *> = 'N': A * X = B (No transpose)
117 *> = 'T': A**T * X = B (Transpose)
118 *> = 'C': A**H * X = B (Transpose)
119 *> \endverbatim
120 *>
121 *> \param[in] N
122 *> \verbatim
123 *> N is INTEGER
124 *> The number of linear equations, i.e., the order of the
125 *> matrix A. N >= 0.
126 *> \endverbatim
127 *>
128 *> \param[in] NRHS
129 *> \verbatim
130 *> NRHS is INTEGER
131 *> The number of right hand sides, i.e., the number of columns
132 *> of the matrices B and X. NRHS >= 0.
133 *> \endverbatim
134 *>
135 *> \param[in,out] A
136 *> \verbatim
137 *> A is REAL array, dimension (LDA,N)
138 *> On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is
139 *> not 'N', then A must have been equilibrated by the scaling
140 *> factors in R and/or C. A is not modified if FACT = 'F' or
141 *> 'N', or if FACT = 'E' and EQUED = 'N' on exit.
142 *>
143 *> On exit, if EQUED .ne. 'N', A is scaled as follows:
144 *> EQUED = 'R': A := diag(R) * A
145 *> EQUED = 'C': A := A * diag(C)
146 *> EQUED = 'B': A := diag(R) * A * diag(C).
147 *> \endverbatim
148 *>
149 *> \param[in] LDA
150 *> \verbatim
151 *> LDA is INTEGER
152 *> The leading dimension of the array A. LDA >= max(1,N).
153 *> \endverbatim
154 *>
155 *> \param[in,out] AF
156 *> \verbatim
157 *> AF is REAL array, dimension (LDAF,N)
158 *> If FACT = 'F', then AF is an input argument and on entry
159 *> contains the factors L and U from the factorization
160 *> A = P*L*U as computed by SGETRF. If EQUED .ne. 'N', then
161 *> AF is the factored form of the equilibrated matrix A.
162 *>
163 *> If FACT = 'N', then AF is an output argument and on exit
164 *> returns the factors L and U from the factorization A = P*L*U
165 *> of the original matrix A.
166 *>
167 *> If FACT = 'E', then AF is an output argument and on exit
168 *> returns the factors L and U from the factorization A = P*L*U
169 *> of the equilibrated matrix A (see the description of A for
170 *> the form of the equilibrated matrix).
171 *> \endverbatim
172 *>
173 *> \param[in] LDAF
174 *> \verbatim
175 *> LDAF is INTEGER
176 *> The leading dimension of the array AF. LDAF >= max(1,N).
177 *> \endverbatim
178 *>
179 *> \param[in,out] IPIV
180 *> \verbatim
181 *> IPIV is INTEGER array, dimension (N)
182 *> If FACT = 'F', then IPIV is an input argument and on entry
183 *> contains the pivot indices from the factorization A = P*L*U
184 *> as computed by SGETRF; row i of the matrix was interchanged
185 *> with row IPIV(i).
186 *>
187 *> If FACT = 'N', then IPIV is an output argument and on exit
188 *> contains the pivot indices from the factorization A = P*L*U
189 *> of the original matrix A.
190 *>
191 *> If FACT = 'E', then IPIV is an output argument and on exit
192 *> contains the pivot indices from the factorization A = P*L*U
193 *> of the equilibrated matrix A.
194 *> \endverbatim
195 *>
196 *> \param[in,out] EQUED
197 *> \verbatim
198 *> EQUED is CHARACTER*1
199 *> Specifies the form of equilibration that was done.
200 *> = 'N': No equilibration (always true if FACT = 'N').
201 *> = 'R': Row equilibration, i.e., A has been premultiplied by
202 *> diag(R).
203 *> = 'C': Column equilibration, i.e., A has been postmultiplied
204 *> by diag(C).
205 *> = 'B': Both row and column equilibration, i.e., A has been
206 *> replaced by diag(R) * A * diag(C).
207 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
208 *> output argument.
209 *> \endverbatim
210 *>
211 *> \param[in,out] R
212 *> \verbatim
213 *> R is REAL array, dimension (N)
214 *> The row scale factors for A. If EQUED = 'R' or 'B', A is
215 *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R
216 *> is not accessed. R is an input argument if FACT = 'F';
217 *> otherwise, R is an output argument. If FACT = 'F' and
218 *> EQUED = 'R' or 'B', each element of R must be positive.
219 *> \endverbatim
220 *>
221 *> \param[in,out] C
222 *> \verbatim
223 *> C is REAL array, dimension (N)
224 *> The column scale factors for A. If EQUED = 'C' or 'B', A is
225 *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C
226 *> is not accessed. C is an input argument if FACT = 'F';
227 *> otherwise, C is an output argument. If FACT = 'F' and
228 *> EQUED = 'C' or 'B', each element of C must be positive.
229 *> \endverbatim
230 *>
231 *> \param[in,out] B
232 *> \verbatim
233 *> B is REAL array, dimension (LDB,NRHS)
234 *> On entry, the N-by-NRHS right hand side matrix B.
235 *> On exit,
236 *> if EQUED = 'N', B is not modified;
237 *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by
238 *> diag(R)*B;
239 *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is
240 *> overwritten by diag(C)*B.
241 *> \endverbatim
242 *>
243 *> \param[in] LDB
244 *> \verbatim
245 *> LDB is INTEGER
246 *> The leading dimension of the array B. LDB >= max(1,N).
247 *> \endverbatim
248 *>
249 *> \param[out] X
250 *> \verbatim
251 *> X is REAL array, dimension (LDX,NRHS)
252 *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X
253 *> to the original system of equations. Note that A and B are
254 *> modified on exit if EQUED .ne. 'N', and the solution to the
255 *> equilibrated system is inv(diag(C))*X if TRANS = 'N' and
256 *> EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C'
257 *> and EQUED = 'R' or 'B'.
258 *> \endverbatim
259 *>
260 *> \param[in] LDX
261 *> \verbatim
262 *> LDX is INTEGER
263 *> The leading dimension of the array X. LDX >= max(1,N).
264 *> \endverbatim
265 *>
266 *> \param[out] RCOND
267 *> \verbatim
268 *> RCOND is REAL
269 *> The estimate of the reciprocal condition number of the matrix
270 *> A after equilibration (if done). If RCOND is less than the
271 *> machine precision (in particular, if RCOND = 0), the matrix
272 *> is singular to working precision. This condition is
273 *> indicated by a return code of INFO > 0.
274 *> \endverbatim
275 *>
276 *> \param[out] FERR
277 *> \verbatim
278 *> FERR is REAL array, dimension (NRHS)
279 *> The estimated forward error bound for each solution vector
280 *> X(j) (the j-th column of the solution matrix X).
281 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
282 *> is an estimated upper bound for the magnitude of the largest
283 *> element in (X(j) - XTRUE) divided by the magnitude of the
284 *> largest element in X(j). The estimate is as reliable as
285 *> the estimate for RCOND, and is almost always a slight
286 *> overestimate of the true error.
287 *> \endverbatim
288 *>
289 *> \param[out] BERR
290 *> \verbatim
291 *> BERR is REAL array, dimension (NRHS)
292 *> The componentwise relative backward error of each solution
293 *> vector X(j) (i.e., the smallest relative change in
294 *> any element of A or B that makes X(j) an exact solution).
295 *> \endverbatim
296 *>
297 *> \param[out] WORK
298 *> \verbatim
299 *> WORK is REAL array, dimension (4*N)
300 *> On exit, WORK(1) contains the reciprocal pivot growth
301 *> factor norm(A)/norm(U). The "max absolute element" norm is
302 *> used. If WORK(1) is much less than 1, then the stability
303 *> of the LU factorization of the (equilibrated) matrix A
304 *> could be poor. This also means that the solution X, condition
305 *> estimator RCOND, and forward error bound FERR could be
306 *> unreliable. If factorization fails with 0<INFO<=N, then
307 *> WORK(1) contains the reciprocal pivot growth factor for the
308 *> leading INFO columns of A.
309 *> \endverbatim
310 *>
311 *> \param[out] IWORK
312 *> \verbatim
313 *> IWORK is INTEGER array, dimension (N)
314 *> \endverbatim
315 *>
316 *> \param[out] INFO
317 *> \verbatim
318 *> INFO is INTEGER
319 *> = 0: successful exit
320 *> < 0: if INFO = -i, the i-th argument had an illegal value
321 *> > 0: if INFO = i, and i is
322 *> <= N: U(i,i) is exactly zero. The factorization has
323 *> been completed, but the factor U is exactly
324 *> singular, so the solution and error bounds
325 *> could not be computed. RCOND = 0 is returned.
326 *> = N+1: U is nonsingular, but RCOND is less than machine
327 *> precision, meaning that the matrix is singular
328 *> to working precision. Nevertheless, the
329 *> solution and error bounds are computed because
330 *> there are a number of situations where the
331 *> computed solution can be more accurate than the
332 *> value of RCOND would suggest.
333 *> \endverbatim
334 *
335 * Authors:
336 * ========
337 *
338 *> \author Univ. of Tennessee
339 *> \author Univ. of California Berkeley
340 *> \author Univ. of Colorado Denver
341 *> \author NAG Ltd.
342 *
343 *> \ingroup realGEsolve
344 *
345 * =====================================================================
346  SUBROUTINE sgesvx( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
347  $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
348  $ WORK, IWORK, INFO )
349 *
350 * -- LAPACK driver routine --
351 * -- LAPACK is a software package provided by Univ. of Tennessee, --
352 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
353 *
354 * .. Scalar Arguments ..
355  CHARACTER EQUED, FACT, TRANS
356  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
357  REAL RCOND
358 * ..
359 * .. Array Arguments ..
360  INTEGER IPIV( * ), IWORK( * )
361  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
362  $ berr( * ), c( * ), ferr( * ), r( * ),
363  $ work( * ), x( ldx, * )
364 * ..
365 *
366 * =====================================================================
367 *
368 * .. Parameters ..
369  REAL ZERO, ONE
370  PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
371 * ..
372 * .. Local Scalars ..
373  LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
374  CHARACTER NORM
375  INTEGER I, INFEQU, J
376  REAL AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
377  $ rowcnd, rpvgrw, smlnum
378 * ..
379 * .. External Functions ..
380  LOGICAL LSAME
381  REAL SLAMCH, SLANGE, SLANTR
382  EXTERNAL lsame, slamch, slange, slantr
383 * ..
384 * .. External Subroutines ..
385  EXTERNAL sgecon, sgeequ, sgerfs, sgetrf, sgetrs, slacpy,
386  $ slaqge, xerbla
387 * ..
388 * .. Intrinsic Functions ..
389  INTRINSIC max, min
390 * ..
391 * .. Executable Statements ..
392 *
393  info = 0
394  nofact = lsame( fact, 'N' )
395  equil = lsame( fact, 'E' )
396  notran = lsame( trans, 'N' )
397  IF( nofact .OR. equil ) THEN
398  equed = 'N'
399  rowequ = .false.
400  colequ = .false.
401  ELSE
402  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
403  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
404  smlnum = slamch( 'Safe minimum' )
405  bignum = one / smlnum
406  END IF
407 *
408 * Test the input parameters.
409 *
410  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
411  $ THEN
412  info = -1
413  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
414  $ lsame( trans, 'C' ) ) THEN
415  info = -2
416  ELSE IF( n.LT.0 ) THEN
417  info = -3
418  ELSE IF( nrhs.LT.0 ) THEN
419  info = -4
420  ELSE IF( lda.LT.max( 1, n ) ) THEN
421  info = -6
422  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
423  info = -8
424  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
425  $ ( rowequ .OR. colequ .OR. lsame( equed, 'N' ) ) ) THEN
426  info = -10
427  ELSE
428  IF( rowequ ) THEN
429  rcmin = bignum
430  rcmax = zero
431  DO 10 j = 1, n
432  rcmin = min( rcmin, r( j ) )
433  rcmax = max( rcmax, r( j ) )
434  10 CONTINUE
435  IF( rcmin.LE.zero ) THEN
436  info = -11
437  ELSE IF( n.GT.0 ) THEN
438  rowcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
439  ELSE
440  rowcnd = one
441  END IF
442  END IF
443  IF( colequ .AND. info.EQ.0 ) THEN
444  rcmin = bignum
445  rcmax = zero
446  DO 20 j = 1, n
447  rcmin = min( rcmin, c( j ) )
448  rcmax = max( rcmax, c( j ) )
449  20 CONTINUE
450  IF( rcmin.LE.zero ) THEN
451  info = -12
452  ELSE IF( n.GT.0 ) THEN
453  colcnd = max( rcmin, smlnum ) / min( rcmax, bignum )
454  ELSE
455  colcnd = one
456  END IF
457  END IF
458  IF( info.EQ.0 ) THEN
459  IF( ldb.LT.max( 1, n ) ) THEN
460  info = -14
461  ELSE IF( ldx.LT.max( 1, n ) ) THEN
462  info = -16
463  END IF
464  END IF
465  END IF
466 *
467  IF( info.NE.0 ) THEN
468  CALL xerbla( 'SGESVX', -info )
469  RETURN
470  END IF
471 *
472  IF( equil ) THEN
473 *
474 * Compute row and column scalings to equilibrate the matrix A.
475 *
476  CALL sgeequ( n, n, a, lda, r, c, rowcnd, colcnd, amax, infequ )
477  IF( infequ.EQ.0 ) THEN
478 *
479 * Equilibrate the matrix.
480 *
481  CALL slaqge( n, n, a, lda, r, c, rowcnd, colcnd, amax,
482  $ equed )
483  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
484  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
485  END IF
486  END IF
487 *
488 * Scale the right hand side.
489 *
490  IF( notran ) THEN
491  IF( rowequ ) THEN
492  DO 40 j = 1, nrhs
493  DO 30 i = 1, n
494  b( i, j ) = r( i )*b( i, j )
495  30 CONTINUE
496  40 CONTINUE
497  END IF
498  ELSE IF( colequ ) THEN
499  DO 60 j = 1, nrhs
500  DO 50 i = 1, n
501  b( i, j ) = c( i )*b( i, j )
502  50 CONTINUE
503  60 CONTINUE
504  END IF
505 *
506  IF( nofact .OR. equil ) THEN
507 *
508 * Compute the LU factorization of A.
509 *
510  CALL slacpy( 'Full', n, n, a, lda, af, ldaf )
511  CALL sgetrf( n, n, af, ldaf, ipiv, info )
512 *
513 * Return if INFO is non-zero.
514 *
515  IF( info.GT.0 ) THEN
516 *
517 * Compute the reciprocal pivot growth factor of the
518 * leading rank-deficient INFO columns of A.
519 *
520  rpvgrw = slantr( 'M', 'U', 'N', info, info, af, ldaf,
521  $ work )
522  IF( rpvgrw.EQ.zero ) THEN
523  rpvgrw = one
524  ELSE
525  rpvgrw = slange( 'M', n, info, a, lda, work ) / rpvgrw
526  END IF
527  work( 1 ) = rpvgrw
528  rcond = zero
529  RETURN
530  END IF
531  END IF
532 *
533 * Compute the norm of the matrix A and the
534 * reciprocal pivot growth factor RPVGRW.
535 *
536  IF( notran ) THEN
537  norm = '1'
538  ELSE
539  norm = 'I'
540  END IF
541  anorm = slange( norm, n, n, a, lda, work )
542  rpvgrw = slantr( 'M', 'U', 'N', n, n, af, ldaf, work )
543  IF( rpvgrw.EQ.zero ) THEN
544  rpvgrw = one
545  ELSE
546  rpvgrw = slange( 'M', n, n, a, lda, work ) / rpvgrw
547  END IF
548 *
549 * Compute the reciprocal of the condition number of A.
550 *
551  CALL sgecon( norm, n, af, ldaf, anorm, rcond, work, iwork, info )
552 *
553 * Compute the solution matrix X.
554 *
555  CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
556  CALL sgetrs( trans, n, nrhs, af, ldaf, ipiv, x, ldx, info )
557 *
558 * Use iterative refinement to improve the computed solution and
559 * compute error bounds and backward error estimates for it.
560 *
561  CALL sgerfs( trans, n, nrhs, a, lda, af, ldaf, ipiv, b, ldb, x,
562  $ ldx, ferr, berr, work, iwork, info )
563 *
564 * Transform the solution matrix X to a solution of the original
565 * system.
566 *
567  IF( notran ) THEN
568  IF( colequ ) THEN
569  DO 80 j = 1, nrhs
570  DO 70 i = 1, n
571  x( i, j ) = c( i )*x( i, j )
572  70 CONTINUE
573  80 CONTINUE
574  DO 90 j = 1, nrhs
575  ferr( j ) = ferr( j ) / colcnd
576  90 CONTINUE
577  END IF
578  ELSE IF( rowequ ) THEN
579  DO 110 j = 1, nrhs
580  DO 100 i = 1, n
581  x( i, j ) = r( i )*x( i, j )
582  100 CONTINUE
583  110 CONTINUE
584  DO 120 j = 1, nrhs
585  ferr( j ) = ferr( j ) / rowcnd
586  120 CONTINUE
587  END IF
588 *
589 * Set INFO = N+1 if the matrix is singular to working precision.
590 *
591  IF( rcond.LT.slamch( 'Epsilon' ) )
592  $ info = n + 1
593 *
594  work( 1 ) = rpvgrw
595  RETURN
596 *
597 * End of SGESVX
598 *
599  END
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine slaqge(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, EQUED)
SLAQGE scales a general rectangular matrix, using row and column scaling factors computed by sgeequ.
Definition: slaqge.f:142
subroutine sgeequ(M, N, A, LDA, R, C, ROWCND, COLCND, AMAX, INFO)
SGEEQU
Definition: sgeequ.f:139
subroutine sgerfs(TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGERFS
Definition: sgerfs.f:185
subroutine sgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SGECON
Definition: sgecon.f:124
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
subroutine sgesvx(FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SGESVX computes the solution to system of linear equations A * X = B for GE matrices
Definition: sgesvx.f:349