LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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zgtsvx.f
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1*> \brief <b> ZGTSVX 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*
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15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
22* DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
23* WORK, RWORK, INFO )
24*
25* .. Scalar Arguments ..
26* CHARACTER FACT, TRANS
27* INTEGER INFO, LDB, LDX, N, NRHS
28* DOUBLE PRECISION RCOND
29* ..
30* .. Array Arguments ..
31* INTEGER IPIV( * )
32* DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
33* COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
34* \$ DLF( * ), DU( * ), DU2( * ), DUF( * ),
35* \$ WORK( * ), X( LDX, * )
36* ..
37*
38*
39*> \par Purpose:
40* =============
41*>
42*> \verbatim
43*>
44*> ZGTSVX uses the LU factorization to compute the solution to a complex
45*> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
46*> where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
47*> matrices.
48*>
49*> Error bounds on the solution and a condition estimate are also
50*> provided.
51*> \endverbatim
52*
53*> \par Description:
54* =================
55*>
56*> \verbatim
57*>
58*> The following steps are performed:
59*>
60*> 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
61*> as A = L * U, where L is a product of permutation and unit lower
62*> bidiagonal matrices and U is upper triangular with nonzeros in
63*> only the main diagonal and first two superdiagonals.
64*>
65*> 2. If some U(i,i)=0, so that U is exactly singular, then the routine
66*> returns with INFO = i. Otherwise, the factored form of A is used
67*> to estimate the condition number of the matrix A. If the
68*> reciprocal of the condition number is less than machine precision,
69*> INFO = N+1 is returned as a warning, but the routine still goes on
70*> to solve for X and compute error bounds as described below.
71*>
72*> 3. The system of equations is solved for X using the factored form
73*> of A.
74*>
75*> 4. Iterative refinement is applied to improve the computed solution
76*> matrix and calculate error bounds and backward error estimates
77*> for it.
78*> \endverbatim
79*
80* Arguments:
81* ==========
82*
83*> \param[in] FACT
84*> \verbatim
85*> FACT is CHARACTER*1
86*> Specifies whether or not the factored form of A has been
87*> supplied on entry.
88*> = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored form
89*> of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV will not
90*> be modified.
91*> = 'N': The matrix will be copied to DLF, DF, and DUF
92*> and factored.
93*> \endverbatim
94*>
95*> \param[in] TRANS
96*> \verbatim
97*> TRANS is CHARACTER*1
98*> Specifies the form of the system of equations:
99*> = 'N': A * X = B (No transpose)
100*> = 'T': A**T * X = B (Transpose)
101*> = 'C': A**H * X = B (Conjugate transpose)
102*> \endverbatim
103*>
104*> \param[in] N
105*> \verbatim
106*> N is INTEGER
107*> The order of the matrix A. N >= 0.
108*> \endverbatim
109*>
110*> \param[in] NRHS
111*> \verbatim
112*> NRHS is INTEGER
113*> The number of right hand sides, i.e., the number of columns
114*> of the matrix B. NRHS >= 0.
115*> \endverbatim
116*>
117*> \param[in] DL
118*> \verbatim
119*> DL is COMPLEX*16 array, dimension (N-1)
120*> The (n-1) subdiagonal elements of A.
121*> \endverbatim
122*>
123*> \param[in] D
124*> \verbatim
125*> D is COMPLEX*16 array, dimension (N)
126*> The n diagonal elements of A.
127*> \endverbatim
128*>
129*> \param[in] DU
130*> \verbatim
131*> DU is COMPLEX*16 array, dimension (N-1)
132*> The (n-1) superdiagonal elements of A.
133*> \endverbatim
134*>
135*> \param[in,out] DLF
136*> \verbatim
137*> DLF is COMPLEX*16 array, dimension (N-1)
138*> If FACT = 'F', then DLF is an input argument and on entry
139*> contains the (n-1) multipliers that define the matrix L from
140*> the LU factorization of A as computed by ZGTTRF.
141*>
142*> If FACT = 'N', then DLF is an output argument and on exit
143*> contains the (n-1) multipliers that define the matrix L from
144*> the LU factorization of A.
145*> \endverbatim
146*>
147*> \param[in,out] DF
148*> \verbatim
149*> DF is COMPLEX*16 array, dimension (N)
150*> If FACT = 'F', then DF is an input argument and on entry
151*> contains the n diagonal elements of the upper triangular
152*> matrix U from the LU factorization of A.
153*>
154*> If FACT = 'N', then DF is an output argument and on exit
155*> contains the n diagonal elements of the upper triangular
156*> matrix U from the LU factorization of A.
157*> \endverbatim
158*>
159*> \param[in,out] DUF
160*> \verbatim
161*> DUF is COMPLEX*16 array, dimension (N-1)
162*> If FACT = 'F', then DUF is an input argument and on entry
163*> contains the (n-1) elements of the first superdiagonal of U.
164*>
165*> If FACT = 'N', then DUF is an output argument and on exit
166*> contains the (n-1) elements of the first superdiagonal of U.
167*> \endverbatim
168*>
169*> \param[in,out] DU2
170*> \verbatim
171*> DU2 is COMPLEX*16 array, dimension (N-2)
172*> If FACT = 'F', then DU2 is an input argument and on entry
173*> contains the (n-2) elements of the second superdiagonal of
174*> U.
175*>
176*> If FACT = 'N', then DU2 is an output argument and on exit
177*> contains the (n-2) elements of the second superdiagonal of
178*> U.
179*> \endverbatim
180*>
181*> \param[in,out] IPIV
182*> \verbatim
183*> IPIV is INTEGER array, dimension (N)
184*> If FACT = 'F', then IPIV is an input argument and on entry
185*> contains the pivot indices from the LU factorization of A as
186*> computed by ZGTTRF.
187*>
188*> If FACT = 'N', then IPIV is an output argument and on exit
189*> contains the pivot indices from the LU factorization of A;
190*> row i of the matrix was interchanged with row IPIV(i).
191*> IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
192*> a row interchange was not required.
193*> \endverbatim
194*>
195*> \param[in] B
196*> \verbatim
197*> B is COMPLEX*16 array, dimension (LDB,NRHS)
198*> The N-by-NRHS right hand side matrix B.
199*> \endverbatim
200*>
201*> \param[in] LDB
202*> \verbatim
203*> LDB is INTEGER
204*> The leading dimension of the array B. LDB >= max(1,N).
205*> \endverbatim
206*>
207*> \param[out] X
208*> \verbatim
209*> X is COMPLEX*16 array, dimension (LDX,NRHS)
210*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
211*> \endverbatim
212*>
213*> \param[in] LDX
214*> \verbatim
215*> LDX is INTEGER
216*> The leading dimension of the array X. LDX >= max(1,N).
217*> \endverbatim
218*>
219*> \param[out] RCOND
220*> \verbatim
221*> RCOND is DOUBLE PRECISION
222*> The estimate of the reciprocal condition number of the matrix
223*> A. If RCOND is less than the machine precision (in
224*> particular, if RCOND = 0), the matrix is singular to working
225*> precision. This condition is indicated by a return code of
226*> INFO > 0.
227*> \endverbatim
228*>
229*> \param[out] FERR
230*> \verbatim
231*> FERR is DOUBLE PRECISION array, dimension (NRHS)
232*> The estimated forward error bound for each solution vector
233*> X(j) (the j-th column of the solution matrix X).
234*> If XTRUE is the true solution corresponding to X(j), FERR(j)
235*> is an estimated upper bound for the magnitude of the largest
236*> element in (X(j) - XTRUE) divided by the magnitude of the
237*> largest element in X(j). The estimate is as reliable as
238*> the estimate for RCOND, and is almost always a slight
239*> overestimate of the true error.
240*> \endverbatim
241*>
242*> \param[out] BERR
243*> \verbatim
244*> BERR is DOUBLE PRECISION array, dimension (NRHS)
245*> The componentwise relative backward error of each solution
246*> vector X(j) (i.e., the smallest relative change in
247*> any element of A or B that makes X(j) an exact solution).
248*> \endverbatim
249*>
250*> \param[out] WORK
251*> \verbatim
252*> WORK is COMPLEX*16 array, dimension (2*N)
253*> \endverbatim
254*>
255*> \param[out] RWORK
256*> \verbatim
257*> RWORK is DOUBLE PRECISION array, dimension (N)
258*> \endverbatim
259*>
260*> \param[out] INFO
261*> \verbatim
262*> INFO is INTEGER
263*> = 0: successful exit
264*> < 0: if INFO = -i, the i-th argument had an illegal value
265*> > 0: if INFO = i, and i is
266*> <= N: U(i,i) is exactly zero. The factorization
267*> has not been completed unless i = N, but the
268*> factor U is exactly singular, so the solution
269*> and error bounds could not be computed.
270*> RCOND = 0 is returned.
271*> = N+1: U is nonsingular, but RCOND is less than machine
272*> precision, meaning that the matrix is singular
273*> to working precision. Nevertheless, the
274*> solution and error bounds are computed because
275*> there are a number of situations where the
276*> computed solution can be more accurate than the
277*> value of RCOND would suggest.
278*> \endverbatim
279*
280* Authors:
281* ========
282*
283*> \author Univ. of Tennessee
284*> \author Univ. of California Berkeley
285*> \author Univ. of Colorado Denver
286*> \author NAG Ltd.
287*
288*> \ingroup gtsvx
289*
290* =====================================================================
291 SUBROUTINE zgtsvx( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
292 \$ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
293 \$ WORK, RWORK, INFO )
294*
295* -- LAPACK driver routine --
296* -- LAPACK is a software package provided by Univ. of Tennessee, --
297* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
298*
299* .. Scalar Arguments ..
300 CHARACTER FACT, TRANS
301 INTEGER INFO, LDB, LDX, N, NRHS
302 DOUBLE PRECISION RCOND
303* ..
304* .. Array Arguments ..
305 INTEGER IPIV( * )
306 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
307 COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
308 \$ dlf( * ), du( * ), du2( * ), duf( * ),
309 \$ work( * ), x( ldx, * )
310* ..
311*
312* =====================================================================
313*
314* .. Parameters ..
315 DOUBLE PRECISION ZERO
316 PARAMETER ( ZERO = 0.0d+0 )
317* ..
318* .. Local Scalars ..
319 LOGICAL NOFACT, NOTRAN
320 CHARACTER NORM
321 DOUBLE PRECISION ANORM
322* ..
323* .. External Functions ..
324 LOGICAL LSAME
325 DOUBLE PRECISION DLAMCH, ZLANGT
326 EXTERNAL lsame, dlamch, zlangt
327* ..
328* .. External Subroutines ..
329 EXTERNAL xerbla, zcopy, zgtcon, zgtrfs, zgttrf, zgttrs,
330 \$ zlacpy
331* ..
332* .. Intrinsic Functions ..
333 INTRINSIC max
334* ..
335* .. Executable Statements ..
336*
337 info = 0
338 nofact = lsame( fact, 'N' )
339 notran = lsame( trans, 'N' )
340 IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
341 info = -1
342 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
343 \$ lsame( trans, 'C' ) ) THEN
344 info = -2
345 ELSE IF( n.LT.0 ) THEN
346 info = -3
347 ELSE IF( nrhs.LT.0 ) THEN
348 info = -4
349 ELSE IF( ldb.LT.max( 1, n ) ) THEN
350 info = -14
351 ELSE IF( ldx.LT.max( 1, n ) ) THEN
352 info = -16
353 END IF
354 IF( info.NE.0 ) THEN
355 CALL xerbla( 'ZGTSVX', -info )
356 RETURN
357 END IF
358*
359 IF( nofact ) THEN
360*
361* Compute the LU factorization of A.
362*
363 CALL zcopy( n, d, 1, df, 1 )
364 IF( n.GT.1 ) THEN
365 CALL zcopy( n-1, dl, 1, dlf, 1 )
366 CALL zcopy( n-1, du, 1, duf, 1 )
367 END IF
368 CALL zgttrf( n, dlf, df, duf, du2, ipiv, info )
369*
370* Return if INFO is non-zero.
371*
372 IF( info.GT.0 )THEN
373 rcond = zero
374 RETURN
375 END IF
376 END IF
377*
378* Compute the norm of the matrix A.
379*
380 IF( notran ) THEN
381 norm = '1'
382 ELSE
383 norm = 'I'
384 END IF
385 anorm = zlangt( norm, n, dl, d, du )
386*
387* Compute the reciprocal of the condition number of A.
388*
389 CALL zgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
390 \$ info )
391*
392* Compute the solution vectors X.
393*
394 CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
395 CALL zgttrs( trans, n, nrhs, dlf, df, duf, du2, ipiv, x, ldx,
396 \$ info )
397*
398* Use iterative refinement to improve the computed solutions and
399* compute error bounds and backward error estimates for them.
400*
401 CALL zgtrfs( trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv,
402 \$ b, ldb, x, ldx, ferr, berr, work, rwork, info )
403*
404* Set INFO = N+1 if the matrix is singular to working precision.
405*
406 IF( rcond.LT.dlamch( 'Epsilon' ) )
407 \$ info = n + 1
408*
409 RETURN
410*
411* End of ZGTSVX
412*
413 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine zcopy(n, zx, incx, zy, incy)
ZCOPY
Definition zcopy.f:81
subroutine zgtcon(norm, n, dl, d, du, du2, ipiv, anorm, rcond, work, info)
ZGTCON
Definition zgtcon.f:141
subroutine zgtrfs(trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZGTRFS
Definition zgtrfs.f:210
subroutine zgtsvx(fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZGTSVX computes the solution to system of linear equations A * X = B for GT matrices
Definition zgtsvx.f:294
subroutine zgttrf(n, dl, d, du, du2, ipiv, info)
ZGTTRF
Definition zgttrf.f:124
subroutine zgttrs(trans, n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
ZGTTRS
Definition zgttrs.f:138
subroutine zlacpy(uplo, m, n, a, lda, b, ldb)
ZLACPY copies all or part of one two-dimensional array to another.
Definition zlacpy.f:103