LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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sgtsvx.f
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1*> \brief <b> SGTSVX computes the solution to system of linear equations A * X = B for GT matrices </b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download SGTSVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgtsvx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgtsvx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgtsvx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
22* DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
23* WORK, IWORK, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER FACT, TRANS
27* INTEGER INFO, LDB, LDX, N, NRHS
28* REAL RCOND
29* ..
30* .. Array Arguments ..
31* INTEGER IPIV( * ), IWORK( * )
32* REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
33* \$ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
34* \$ FERR( * ), WORK( * ), X( LDX, * )
35* ..
36*
37*
38*> \par Purpose:
39* =============
40*>
41*> \verbatim
42*>
43*> SGTSVX uses the LU factorization to compute the solution to a real
44*> system of linear equations A * X = B or A**T * X = B,
45*> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
46*> 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 = 'N', the LU decomposition is used to factor the matrix A
60*> as A = L * U, where L is a product of permutation and unit lower
61*> bidiagonal matrices and U is upper triangular with nonzeros in
62*> only the main diagonal and first two superdiagonals.
63*>
64*> 2. If some U(i,i)=0, so that U is exactly singular, then the routine
65*> returns with INFO = i. Otherwise, the factored form of A is used
66*> to estimate the condition number of the matrix A. If the
67*> reciprocal of the condition number is less than machine precision,
68*> INFO = N+1 is returned as a warning, but the routine still goes on
69*> to solve for X and compute error bounds as described below.
70*>
71*> 3. The system of equations is solved for X using the factored form
72*> of A.
73*>
74*> 4. Iterative refinement is applied to improve the computed solution
75*> matrix and calculate error bounds and backward error estimates
76*> for it.
77*> \endverbatim
78*
79* Arguments:
80* ==========
81*
82*> \param[in] FACT
83*> \verbatim
84*> FACT is CHARACTER*1
85*> Specifies whether or not the factored form of A has been
86*> supplied on entry.
87*> = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored
88*> form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
89*> will not be modified.
90*> = 'N': The matrix will be copied to DLF, DF, and DUF
91*> and factored.
92*> \endverbatim
93*>
94*> \param[in] TRANS
95*> \verbatim
96*> TRANS is CHARACTER*1
97*> Specifies the form of the system of equations:
98*> = 'N': A * X = B (No transpose)
99*> = 'T': A**T * X = B (Transpose)
100*> = 'C': A**H * X = B (Conjugate transpose = Transpose)
101*> \endverbatim
102*>
103*> \param[in] N
104*> \verbatim
105*> N is INTEGER
106*> The order of the matrix A. N >= 0.
107*> \endverbatim
108*>
109*> \param[in] NRHS
110*> \verbatim
111*> NRHS is INTEGER
112*> The number of right hand sides, i.e., the number of columns
113*> of the matrix B. NRHS >= 0.
114*> \endverbatim
115*>
116*> \param[in] DL
117*> \verbatim
118*> DL is REAL array, dimension (N-1)
119*> The (n-1) subdiagonal elements of A.
120*> \endverbatim
121*>
122*> \param[in] D
123*> \verbatim
124*> D is REAL array, dimension (N)
125*> The n diagonal elements of A.
126*> \endverbatim
127*>
128*> \param[in] DU
129*> \verbatim
130*> DU is REAL array, dimension (N-1)
131*> The (n-1) superdiagonal elements of A.
132*> \endverbatim
133*>
134*> \param[in,out] DLF
135*> \verbatim
136*> DLF is REAL array, dimension (N-1)
137*> If FACT = 'F', then DLF is an input argument and on entry
138*> contains the (n-1) multipliers that define the matrix L from
139*> the LU factorization of A as computed by SGTTRF.
140*>
141*> If FACT = 'N', then DLF is an output argument and on exit
142*> contains the (n-1) multipliers that define the matrix L from
143*> the LU factorization of A.
144*> \endverbatim
145*>
146*> \param[in,out] DF
147*> \verbatim
148*> DF is REAL array, dimension (N)
149*> If FACT = 'F', then DF is an input argument and on entry
150*> contains the n diagonal elements of the upper triangular
151*> matrix U from the LU factorization of A.
152*>
153*> If FACT = 'N', then DF is an output argument and on exit
154*> contains the n diagonal elements of the upper triangular
155*> matrix U from the LU factorization of A.
156*> \endverbatim
157*>
158*> \param[in,out] DUF
159*> \verbatim
160*> DUF is REAL array, dimension (N-1)
161*> If FACT = 'F', then DUF is an input argument and on entry
162*> contains the (n-1) elements of the first superdiagonal of U.
163*>
164*> If FACT = 'N', then DUF is an output argument and on exit
165*> contains the (n-1) elements of the first superdiagonal of U.
166*> \endverbatim
167*>
168*> \param[in,out] DU2
169*> \verbatim
170*> DU2 is REAL array, dimension (N-2)
171*> If FACT = 'F', then DU2 is an input argument and on entry
172*> contains the (n-2) elements of the second superdiagonal of
173*> U.
174*>
175*> If FACT = 'N', then DU2 is an output argument and on exit
176*> contains the (n-2) elements of the second superdiagonal of
177*> U.
178*> \endverbatim
179*>
180*> \param[in,out] IPIV
181*> \verbatim
182*> IPIV is INTEGER array, dimension (N)
183*> If FACT = 'F', then IPIV is an input argument and on entry
184*> contains the pivot indices from the LU factorization of A as
185*> computed by SGTTRF.
186*>
187*> If FACT = 'N', then IPIV is an output argument and on exit
188*> contains the pivot indices from the LU factorization of A;
189*> row i of the matrix was interchanged with row IPIV(i).
190*> IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
191*> a row interchange was not required.
192*> \endverbatim
193*>
194*> \param[in] B
195*> \verbatim
196*> B is REAL array, dimension (LDB,NRHS)
197*> The N-by-NRHS right hand side matrix B.
198*> \endverbatim
199*>
200*> \param[in] LDB
201*> \verbatim
202*> LDB is INTEGER
203*> The leading dimension of the array B. LDB >= max(1,N).
204*> \endverbatim
205*>
206*> \param[out] X
207*> \verbatim
208*> X is REAL array, dimension (LDX,NRHS)
209*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
210*> \endverbatim
211*>
212*> \param[in] LDX
213*> \verbatim
214*> LDX is INTEGER
215*> The leading dimension of the array X. LDX >= max(1,N).
216*> \endverbatim
217*>
218*> \param[out] RCOND
219*> \verbatim
220*> RCOND is REAL
221*> The estimate of the reciprocal condition number of the matrix
222*> A. If RCOND is less than the machine precision (in
223*> particular, if RCOND = 0), the matrix is singular to working
224*> precision. This condition is indicated by a return code of
225*> INFO > 0.
226*> \endverbatim
227*>
228*> \param[out] FERR
229*> \verbatim
230*> FERR is REAL array, dimension (NRHS)
231*> The estimated forward error bound for each solution vector
232*> X(j) (the j-th column of the solution matrix X).
233*> If XTRUE is the true solution corresponding to X(j), FERR(j)
234*> is an estimated upper bound for the magnitude of the largest
235*> element in (X(j) - XTRUE) divided by the magnitude of the
236*> largest element in X(j). The estimate is as reliable as
237*> the estimate for RCOND, and is almost always a slight
238*> overestimate of the true error.
239*> \endverbatim
240*>
241*> \param[out] BERR
242*> \verbatim
243*> BERR is REAL array, dimension (NRHS)
244*> The componentwise relative backward error of each solution
245*> vector X(j) (i.e., the smallest relative change in
246*> any element of A or B that makes X(j) an exact solution).
247*> \endverbatim
248*>
249*> \param[out] WORK
250*> \verbatim
251*> WORK is REAL array, dimension (3*N)
252*> \endverbatim
253*>
254*> \param[out] IWORK
255*> \verbatim
256*> IWORK is INTEGER array, dimension (N)
257*> \endverbatim
258*>
259*> \param[out] INFO
260*> \verbatim
261*> INFO is INTEGER
262*> = 0: successful exit
263*> < 0: if INFO = -i, the i-th argument had an illegal value
264*> > 0: if INFO = i, and i is
265*> <= N: U(i,i) is exactly zero. The factorization
266*> has not been completed unless i = N, but the
267*> factor U is exactly singular, so the solution
268*> and error bounds could not be computed.
269*> RCOND = 0 is returned.
270*> = N+1: U is nonsingular, but RCOND is less than machine
271*> precision, meaning that the matrix is singular
272*> to working precision. Nevertheless, the
273*> solution and error bounds are computed because
274*> there are a number of situations where the
275*> computed solution can be more accurate than the
276*> value of RCOND would suggest.
277*> \endverbatim
278*
279* Authors:
280* ========
281*
282*> \author Univ. of Tennessee
283*> \author Univ. of California Berkeley
284*> \author Univ. of Colorado Denver
285*> \author NAG Ltd.
286*
287*> \ingroup gtsvx
288*
289* =====================================================================
290 SUBROUTINE sgtsvx( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
291 \$ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
292 \$ WORK, IWORK, INFO )
293*
294* -- LAPACK driver routine --
295* -- LAPACK is a software package provided by Univ. of Tennessee, --
296* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
297*
298* .. Scalar Arguments ..
299 CHARACTER FACT, TRANS
300 INTEGER INFO, LDB, LDX, N, NRHS
301 REAL RCOND
302* ..
303* .. Array Arguments ..
304 INTEGER IPIV( * ), IWORK( * )
305 REAL B( LDB, * ), BERR( * ), D( * ), DF( * ),
306 \$ dl( * ), dlf( * ), du( * ), du2( * ), duf( * ),
307 \$ ferr( * ), work( * ), x( ldx, * )
308* ..
309*
310* =====================================================================
311*
312* .. Parameters ..
313 REAL ZERO
314 PARAMETER ( ZERO = 0.0e+0 )
315* ..
316* .. Local Scalars ..
317 LOGICAL NOFACT, NOTRAN
318 CHARACTER NORM
319 REAL ANORM
320* ..
321* .. External Functions ..
322 LOGICAL LSAME
323 REAL SLAMCH, SLANGT
324 EXTERNAL lsame, slamch, slangt
325* ..
326* .. External Subroutines ..
327 EXTERNAL scopy, sgtcon, sgtrfs, sgttrf, sgttrs, slacpy,
328 \$ xerbla
329* ..
330* .. Intrinsic Functions ..
331 INTRINSIC max
332* ..
333* .. Executable Statements ..
334*
335 info = 0
336 nofact = lsame( fact, 'N' )
337 notran = lsame( trans, 'N' )
338 IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
339 info = -1
340 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
341 \$ lsame( trans, 'C' ) ) THEN
342 info = -2
343 ELSE IF( n.LT.0 ) THEN
344 info = -3
345 ELSE IF( nrhs.LT.0 ) THEN
346 info = -4
347 ELSE IF( ldb.LT.max( 1, n ) ) THEN
348 info = -14
349 ELSE IF( ldx.LT.max( 1, n ) ) THEN
350 info = -16
351 END IF
352 IF( info.NE.0 ) THEN
353 CALL xerbla( 'SGTSVX', -info )
354 RETURN
355 END IF
356*
357 IF( nofact ) THEN
358*
359* Compute the LU factorization of A.
360*
361 CALL scopy( n, d, 1, df, 1 )
362 IF( n.GT.1 ) THEN
363 CALL scopy( n-1, dl, 1, dlf, 1 )
364 CALL scopy( n-1, du, 1, duf, 1 )
365 END IF
366 CALL sgttrf( n, dlf, df, duf, du2, ipiv, info )
367*
368* Return if INFO is non-zero.
369*
370 IF( info.GT.0 )THEN
371 rcond = zero
372 RETURN
373 END IF
374 END IF
375*
376* Compute the norm of the matrix A.
377*
378 IF( notran ) THEN
379 norm = '1'
380 ELSE
381 norm = 'I'
382 END IF
383 anorm = slangt( norm, n, dl, d, du )
384*
385* Compute the reciprocal of the condition number of A.
386*
387 CALL sgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
388 \$ iwork, info )
389*
390* Compute the solution vectors X.
391*
392 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
393 CALL sgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
394 \$ info )
395*
396* Use iterative refinement to improve the computed solutions and
397* compute error bounds and backward error estimates for them.
398*
399 CALL sgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv,
400 \$ b, ldb, x, ldx, ferr, berr, work, iwork, info )
401*
402* Set INFO = N+1 if the matrix is singular to working precision.
403*
404 IF( rcond.LT.slamch( 'Epsilon' ) )
405 \$ info = n + 1
406*
407 RETURN
408*
409* End of SGTSVX
410*
411 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine scopy(n, sx, incx, sy, incy)
SCOPY
Definition scopy.f:82
subroutine sgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, iwork, info)
SGTCON
Definition sgtcon.f:146
subroutine sgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, iwork, info)
SGTRFS
Definition sgtrfs.f:209
subroutine sgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
SGTSVX computes the solution to system of linear equations A * X = B for GT matrices
Definition sgtsvx.f:293
subroutine sgttrf(n, dl, d, du, du2, ipiv, info)
SGTTRF
Definition sgttrf.f:124
subroutine sgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
SGTTRS
Definition sgttrs.f:138
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