LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cgtsvx.f
Go to the documentation of this file.
1 *> \brief <b> CGTSVX 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 CGTSVX( 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 * REAL RCOND
29 * ..
30 * .. Array Arguments ..
31 * INTEGER IPIV( * )
32 * REAL BERR( * ), FERR( * ), RWORK( * )
33 * COMPLEX 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 *> CGTSVX 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 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 array, dimension (N)
126 *> The n diagonal elements of A.
127 *> \endverbatim
128 *>
129 *> \param[in] DU
130 *> \verbatim
131 *> DU is COMPLEX 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 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 CGTTRF.
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 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 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 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 CGTTRF.
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 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 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 REAL
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 REAL 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 REAL 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 array, dimension (2*N)
253 *> \endverbatim
254 *>
255 *> \param[out] RWORK
256 *> \verbatim
257 *> RWORK is REAL 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 complexGTsolve
289 *
290 * =====================================================================
291  SUBROUTINE cgtsvx( 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  REAL RCOND
303 * ..
304 * .. Array Arguments ..
305  INTEGER IPIV( * )
306  REAL BERR( * ), FERR( * ), RWORK( * )
307  COMPLEX B( LDB, * ), D( * ), DF( * ), DL( * ),
308  $ dlf( * ), du( * ), du2( * ), duf( * ),
309  $ work( * ), x( ldx, * )
310 * ..
311 *
312 * =====================================================================
313 *
314 * .. Parameters ..
315  REAL ZERO
316  PARAMETER ( ZERO = 0.0e+0 )
317 * ..
318 * .. Local Scalars ..
319  LOGICAL NOFACT, NOTRAN
320  CHARACTER NORM
321  REAL ANORM
322 * ..
323 * .. External Functions ..
324  LOGICAL LSAME
325  REAL CLANGT, SLAMCH
326  EXTERNAL lsame, clangt, slamch
327 * ..
328 * .. External Subroutines ..
329  EXTERNAL ccopy, cgtcon, cgtrfs, cgttrf, cgttrs, clacpy,
330  $ xerbla
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( 'CGTSVX', -info )
356  RETURN
357  END IF
358 *
359  IF( nofact ) THEN
360 *
361 * Compute the LU factorization of A.
362 *
363  CALL ccopy( n, d, 1, df, 1 )
364  IF( n.GT.1 ) THEN
365  CALL ccopy( n-1, dl, 1, dlf, 1 )
366  CALL ccopy( n-1, du, 1, duf, 1 )
367  END IF
368  CALL cgttrf( 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 = clangt( norm, n, dl, d, du )
386 *
387 * Compute the reciprocal of the condition number of A.
388 *
389  CALL cgtcon( norm, n, dlf, df, duf, du2, ipiv, anorm, rcond, work,
390  $ info )
391 *
392 * Compute the solution vectors X.
393 *
394  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
395  CALL cgttrs( 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 cgtrfs( 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.slamch( 'Epsilon' ) )
407  $ info = n + 1
408 *
409  RETURN
410 *
411 * End of CGTSVX
412 *
413  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine cgttrf(N, DL, D, DU, DU2, IPIV, INFO)
CGTTRF
Definition: cgttrf.f:124
subroutine cgtcon(NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, INFO)
CGTCON
Definition: cgtcon.f:141
subroutine cgtrfs(TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CGTRFS
Definition: cgtrfs.f:210
subroutine cgttrs(TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)
CGTTRS
Definition: cgttrs.f:138
subroutine cgtsvx(FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CGTSVX computes the solution to system of linear equations A * X = B for GT matrices
Definition: cgtsvx.f:294
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103