1: *> \brief <b> DPTSVX computes the solution to system of linear equations A * X = B for PT 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 DPTSVX + dependencies
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptsvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
22: * RCOND, FERR, BERR, WORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER FACT
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * DOUBLE PRECISION RCOND
28: * ..
29: * .. Array Arguments ..
30: * DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
31: * $ E( * ), EF( * ), FERR( * ), WORK( * ),
32: * $ X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DPTSVX uses the factorization A = L*D*L**T to compute the solution
42: *> to a real system of linear equations A*X = B, where A is an N-by-N
43: *> symmetric positive definite tridiagonal matrix and X and B are
44: *> N-by-NRHS matrices.
45: *>
46: *> Error bounds on the solution and a condition estimate are also
47: *> provided.
48: *> \endverbatim
49: *
50: *> \par Description:
51: * =================
52: *>
53: *> \verbatim
54: *>
55: *> The following steps are performed:
56: *>
57: *> 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L
58: *> is a unit lower bidiagonal matrix and D is diagonal. The
59: *> factorization can also be regarded as having the form
60: *> A = U**T*D*U.
61: *>
62: *> 2. If the leading i-by-i principal minor is not positive definite,
63: *> then the routine returns with INFO = i. Otherwise, the factored
64: *> form of A is used to estimate the condition number of the matrix
65: *> A. If the reciprocal of the condition number is less than machine
66: *> precision, INFO = N+1 is returned as a warning, but the routine
67: *> still goes on to solve for X and compute error bounds as
68: *> described below.
69: *>
70: *> 3. The system of equations is solved for X using the factored form
71: *> of A.
72: *>
73: *> 4. Iterative refinement is applied to improve the computed solution
74: *> matrix and calculate error bounds and backward error estimates
75: *> for it.
76: *> \endverbatim
77: *
78: * Arguments:
79: * ==========
80: *
81: *> \param[in] FACT
82: *> \verbatim
83: *> FACT is CHARACTER*1
84: *> Specifies whether or not the factored form of A has been
85: *> supplied on entry.
86: *> = 'F': On entry, DF and EF contain the factored form of A.
87: *> D, E, DF, and EF will not be modified.
88: *> = 'N': The matrix A will be copied to DF and EF and
89: *> factored.
90: *> \endverbatim
91: *>
92: *> \param[in] N
93: *> \verbatim
94: *> N is INTEGER
95: *> The order of the matrix A. N >= 0.
96: *> \endverbatim
97: *>
98: *> \param[in] NRHS
99: *> \verbatim
100: *> NRHS is INTEGER
101: *> The number of right hand sides, i.e., the number of columns
102: *> of the matrices B and X. NRHS >= 0.
103: *> \endverbatim
104: *>
105: *> \param[in] D
106: *> \verbatim
107: *> D is DOUBLE PRECISION array, dimension (N)
108: *> The n diagonal elements of the tridiagonal matrix A.
109: *> \endverbatim
110: *>
111: *> \param[in] E
112: *> \verbatim
113: *> E is DOUBLE PRECISION array, dimension (N-1)
114: *> The (n-1) subdiagonal elements of the tridiagonal matrix A.
115: *> \endverbatim
116: *>
117: *> \param[in,out] DF
118: *> \verbatim
119: *> DF is DOUBLE PRECISION array, dimension (N)
120: *> If FACT = 'F', then DF is an input argument and on entry
121: *> contains the n diagonal elements of the diagonal matrix D
122: *> from the L*D*L**T factorization of A.
123: *> If FACT = 'N', then DF is an output argument and on exit
124: *> contains the n diagonal elements of the diagonal matrix D
125: *> from the L*D*L**T factorization of A.
126: *> \endverbatim
127: *>
128: *> \param[in,out] EF
129: *> \verbatim
130: *> EF is DOUBLE PRECISION array, dimension (N-1)
131: *> If FACT = 'F', then EF is an input argument and on entry
132: *> contains the (n-1) subdiagonal elements of the unit
133: *> bidiagonal factor L from the L*D*L**T factorization of A.
134: *> If FACT = 'N', then EF is an output argument and on exit
135: *> contains the (n-1) subdiagonal elements of the unit
136: *> bidiagonal factor L from the L*D*L**T factorization of A.
137: *> \endverbatim
138: *>
139: *> \param[in] B
140: *> \verbatim
141: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
142: *> The N-by-NRHS right hand side matrix B.
143: *> \endverbatim
144: *>
145: *> \param[in] LDB
146: *> \verbatim
147: *> LDB is INTEGER
148: *> The leading dimension of the array B. LDB >= max(1,N).
149: *> \endverbatim
150: *>
151: *> \param[out] X
152: *> \verbatim
153: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
154: *> If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.
155: *> \endverbatim
156: *>
157: *> \param[in] LDX
158: *> \verbatim
159: *> LDX is INTEGER
160: *> The leading dimension of the array X. LDX >= max(1,N).
161: *> \endverbatim
162: *>
163: *> \param[out] RCOND
164: *> \verbatim
165: *> RCOND is DOUBLE PRECISION
166: *> The reciprocal condition number of the matrix A. If RCOND
167: *> is less than the machine precision (in particular, if
168: *> RCOND = 0), the matrix is singular to working precision.
169: *> This condition is indicated by a return code of INFO > 0.
170: *> \endverbatim
171: *>
172: *> \param[out] FERR
173: *> \verbatim
174: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
175: *> The forward error bound for each solution vector
176: *> X(j) (the j-th column of the solution matrix X).
177: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
178: *> is an estimated upper bound for the magnitude of the largest
179: *> element in (X(j) - XTRUE) divided by the magnitude of the
180: *> largest element in X(j).
181: *> \endverbatim
182: *>
183: *> \param[out] BERR
184: *> \verbatim
185: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
186: *> The componentwise relative backward error of each solution
187: *> vector X(j) (i.e., the smallest relative change in any
188: *> element of A or B that makes X(j) an exact solution).
189: *> \endverbatim
190: *>
191: *> \param[out] WORK
192: *> \verbatim
193: *> WORK is DOUBLE PRECISION array, dimension (2*N)
194: *> \endverbatim
195: *>
196: *> \param[out] INFO
197: *> \verbatim
198: *> INFO is INTEGER
199: *> = 0: successful exit
200: *> < 0: if INFO = -i, the i-th argument had an illegal value
201: *> > 0: if INFO = i, and i is
202: *> <= N: the leading minor of order i of A is
203: *> not positive definite, so the factorization
204: *> could not be completed, and the solution has not
205: *> been computed. RCOND = 0 is returned.
206: *> = N+1: U is nonsingular, but RCOND is less than machine
207: *> precision, meaning that the matrix is singular
208: *> to working precision. Nevertheless, the
209: *> solution and error bounds are computed because
210: *> there are a number of situations where the
211: *> computed solution can be more accurate than the
212: *> value of RCOND would suggest.
213: *> \endverbatim
214: *
215: * Authors:
216: * ========
217: *
218: *> \author Univ. of Tennessee
219: *> \author Univ. of California Berkeley
220: *> \author Univ. of Colorado Denver
221: *> \author NAG Ltd.
222: *
223: *> \ingroup doublePTsolve
224: *
225: * =====================================================================
226: SUBROUTINE DPTSVX( FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
227: $ RCOND, FERR, BERR, WORK, INFO )
228: *
229: * -- LAPACK driver routine --
230: * -- LAPACK is a software package provided by Univ. of Tennessee, --
231: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232: *
233: * .. Scalar Arguments ..
234: CHARACTER FACT
235: INTEGER INFO, LDB, LDX, N, NRHS
236: DOUBLE PRECISION RCOND
237: * ..
238: * .. Array Arguments ..
239: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
240: $ E( * ), EF( * ), FERR( * ), WORK( * ),
241: $ X( LDX, * )
242: * ..
243: *
244: * =====================================================================
245: *
246: * .. Parameters ..
247: DOUBLE PRECISION ZERO
248: PARAMETER ( ZERO = 0.0D+0 )
249: * ..
250: * .. Local Scalars ..
251: LOGICAL NOFACT
252: DOUBLE PRECISION ANORM
253: * ..
254: * .. External Functions ..
255: LOGICAL LSAME
256: DOUBLE PRECISION DLAMCH, DLANST
257: EXTERNAL LSAME, DLAMCH, DLANST
258: * ..
259: * .. External Subroutines ..
260: EXTERNAL DCOPY, DLACPY, DPTCON, DPTRFS, DPTTRF, DPTTRS,
261: $ XERBLA
262: * ..
263: * .. Intrinsic Functions ..
264: INTRINSIC MAX
265: * ..
266: * .. Executable Statements ..
267: *
268: * Test the input parameters.
269: *
270: INFO = 0
271: NOFACT = LSAME( FACT, 'N' )
272: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
273: INFO = -1
274: ELSE IF( N.LT.0 ) THEN
275: INFO = -2
276: ELSE IF( NRHS.LT.0 ) THEN
277: INFO = -3
278: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
279: INFO = -9
280: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
281: INFO = -11
282: END IF
283: IF( INFO.NE.0 ) THEN
284: CALL XERBLA( 'DPTSVX', -INFO )
285: RETURN
286: END IF
287: *
288: IF( NOFACT ) THEN
289: *
290: * Compute the L*D*L**T (or U**T*D*U) factorization of A.
291: *
292: CALL DCOPY( N, D, 1, DF, 1 )
293: IF( N.GT.1 )
294: $ CALL DCOPY( N-1, E, 1, EF, 1 )
295: CALL DPTTRF( N, DF, EF, INFO )
296: *
297: * Return if INFO is non-zero.
298: *
299: IF( INFO.GT.0 )THEN
300: RCOND = ZERO
301: RETURN
302: END IF
303: END IF
304: *
305: * Compute the norm of the matrix A.
306: *
307: ANORM = DLANST( '1', N, D, E )
308: *
309: * Compute the reciprocal of the condition number of A.
310: *
311: CALL DPTCON( N, DF, EF, ANORM, RCOND, WORK, INFO )
312: *
313: * Compute the solution vectors X.
314: *
315: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
316: CALL DPTTRS( N, NRHS, DF, EF, X, LDX, INFO )
317: *
318: * Use iterative refinement to improve the computed solutions and
319: * compute error bounds and backward error estimates for them.
320: *
321: CALL DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR,
322: $ WORK, INFO )
323: *
324: * Set INFO = N+1 if the matrix is singular to working precision.
325: *
326: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
327: $ INFO = N + 1
328: *
329: RETURN
330: *
331: * End of DPTSVX
332: *
333: END
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