1: SUBROUTINE DGTSVX( FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF,
2: $ DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR,
3: $ WORK, IWORK, INFO )
4: *
5: * -- LAPACK routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: CHARACTER FACT, TRANS
12: INTEGER INFO, LDB, LDX, N, NRHS
13: DOUBLE PRECISION RCOND
14: * ..
15: * .. Array Arguments ..
16: INTEGER IPIV( * ), IWORK( * )
17: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
18: $ DL( * ), DLF( * ), DU( * ), DU2( * ), DUF( * ),
19: $ FERR( * ), WORK( * ), X( LDX, * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * DGTSVX uses the LU factorization to compute the solution to a real
26: * system of linear equations A * X = B or A**T * X = B,
27: * where A is a tridiagonal matrix of order N and X and B are N-by-NRHS
28: * matrices.
29: *
30: * Error bounds on the solution and a condition estimate are also
31: * provided.
32: *
33: * Description
34: * ===========
35: *
36: * The following steps are performed:
37: *
38: * 1. If FACT = 'N', the LU decomposition is used to factor the matrix A
39: * as A = L * U, where L is a product of permutation and unit lower
40: * bidiagonal matrices and U is upper triangular with nonzeros in
41: * only the main diagonal and first two superdiagonals.
42: *
43: * 2. If some U(i,i)=0, so that U is exactly singular, then the routine
44: * returns with INFO = i. Otherwise, the factored form of A is used
45: * to estimate the condition number of the matrix A. If the
46: * reciprocal of the condition number is less than machine precision,
47: * INFO = N+1 is returned as a warning, but the routine still goes on
48: * to solve for X and compute error bounds as described below.
49: *
50: * 3. The system of equations is solved for X using the factored form
51: * of A.
52: *
53: * 4. Iterative refinement is applied to improve the computed solution
54: * matrix and calculate error bounds and backward error estimates
55: * for it.
56: *
57: * Arguments
58: * =========
59: *
60: * FACT (input) CHARACTER*1
61: * Specifies whether or not the factored form of A has been
62: * supplied on entry.
63: * = 'F': DLF, DF, DUF, DU2, and IPIV contain the factored
64: * form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV
65: * will not be modified.
66: * = 'N': The matrix will be copied to DLF, DF, and DUF
67: * and factored.
68: *
69: * TRANS (input) CHARACTER*1
70: * Specifies the form of the system of equations:
71: * = 'N': A * X = B (No transpose)
72: * = 'T': A**T * X = B (Transpose)
73: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
74: *
75: * N (input) INTEGER
76: * The order of the matrix A. N >= 0.
77: *
78: * NRHS (input) INTEGER
79: * The number of right hand sides, i.e., the number of columns
80: * of the matrix B. NRHS >= 0.
81: *
82: * DL (input) DOUBLE PRECISION array, dimension (N-1)
83: * The (n-1) subdiagonal elements of A.
84: *
85: * D (input) DOUBLE PRECISION array, dimension (N)
86: * The n diagonal elements of A.
87: *
88: * DU (input) DOUBLE PRECISION array, dimension (N-1)
89: * The (n-1) superdiagonal elements of A.
90: *
91: * DLF (input or output) DOUBLE PRECISION array, dimension (N-1)
92: * If FACT = 'F', then DLF is an input argument and on entry
93: * contains the (n-1) multipliers that define the matrix L from
94: * the LU factorization of A as computed by DGTTRF.
95: *
96: * If FACT = 'N', then DLF is an output argument and on exit
97: * contains the (n-1) multipliers that define the matrix L from
98: * the LU factorization of A.
99: *
100: * DF (input or output) DOUBLE PRECISION array, dimension (N)
101: * If FACT = 'F', then DF is an input argument and on entry
102: * contains the n diagonal elements of the upper triangular
103: * matrix U from the LU factorization of A.
104: *
105: * If FACT = 'N', then DF is an output argument and on exit
106: * contains the n diagonal elements of the upper triangular
107: * matrix U from the LU factorization of A.
108: *
109: * DUF (input or output) DOUBLE PRECISION array, dimension (N-1)
110: * If FACT = 'F', then DUF is an input argument and on entry
111: * contains the (n-1) elements of the first superdiagonal of U.
112: *
113: * If FACT = 'N', then DUF is an output argument and on exit
114: * contains the (n-1) elements of the first superdiagonal of U.
115: *
116: * DU2 (input or output) DOUBLE PRECISION array, dimension (N-2)
117: * If FACT = 'F', then DU2 is an input argument and on entry
118: * contains the (n-2) elements of the second superdiagonal of
119: * U.
120: *
121: * If FACT = 'N', then DU2 is an output argument and on exit
122: * contains the (n-2) elements of the second superdiagonal of
123: * U.
124: *
125: * IPIV (input or output) INTEGER array, dimension (N)
126: * If FACT = 'F', then IPIV is an input argument and on entry
127: * contains the pivot indices from the LU factorization of A as
128: * computed by DGTTRF.
129: *
130: * If FACT = 'N', then IPIV is an output argument and on exit
131: * contains the pivot indices from the LU factorization of A;
132: * row i of the matrix was interchanged with row IPIV(i).
133: * IPIV(i) will always be either i or i+1; IPIV(i) = i indicates
134: * a row interchange was not required.
135: *
136: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
137: * The N-by-NRHS right hand side matrix B.
138: *
139: * LDB (input) INTEGER
140: * The leading dimension of the array B. LDB >= max(1,N).
141: *
142: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
143: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
144: *
145: * LDX (input) INTEGER
146: * The leading dimension of the array X. LDX >= max(1,N).
147: *
148: * RCOND (output) DOUBLE PRECISION
149: * The estimate of the reciprocal condition number of the matrix
150: * A. If RCOND is less than the machine precision (in
151: * particular, if RCOND = 0), the matrix is singular to working
152: * precision. This condition is indicated by a return code of
153: * INFO > 0.
154: *
155: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
156: * The estimated forward error bound for each solution vector
157: * X(j) (the j-th column of the solution matrix X).
158: * If XTRUE is the true solution corresponding to X(j), FERR(j)
159: * is an estimated upper bound for the magnitude of the largest
160: * element in (X(j) - XTRUE) divided by the magnitude of the
161: * largest element in X(j). The estimate is as reliable as
162: * the estimate for RCOND, and is almost always a slight
163: * overestimate of the true error.
164: *
165: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
166: * The componentwise relative backward error of each solution
167: * vector X(j) (i.e., the smallest relative change in
168: * any element of A or B that makes X(j) an exact solution).
169: *
170: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
171: *
172: * IWORK (workspace) INTEGER array, dimension (N)
173: *
174: * INFO (output) INTEGER
175: * = 0: successful exit
176: * < 0: if INFO = -i, the i-th argument had an illegal value
177: * > 0: if INFO = i, and i is
178: * <= N: U(i,i) is exactly zero. The factorization
179: * has not been completed unless i = N, but the
180: * factor U is exactly singular, so the solution
181: * and error bounds could not be computed.
182: * RCOND = 0 is returned.
183: * = N+1: U is nonsingular, but RCOND is less than machine
184: * precision, meaning that the matrix is singular
185: * to working precision. Nevertheless, the
186: * solution and error bounds are computed because
187: * there are a number of situations where the
188: * computed solution can be more accurate than the
189: * value of RCOND would suggest.
190: *
191: * =====================================================================
192: *
193: * .. Parameters ..
194: DOUBLE PRECISION ZERO
195: PARAMETER ( ZERO = 0.0D+0 )
196: * ..
197: * .. Local Scalars ..
198: LOGICAL NOFACT, NOTRAN
199: CHARACTER NORM
200: DOUBLE PRECISION ANORM
201: * ..
202: * .. External Functions ..
203: LOGICAL LSAME
204: DOUBLE PRECISION DLAMCH, DLANGT
205: EXTERNAL LSAME, DLAMCH, DLANGT
206: * ..
207: * .. External Subroutines ..
208: EXTERNAL DCOPY, DGTCON, DGTRFS, DGTTRF, DGTTRS, DLACPY,
209: $ XERBLA
210: * ..
211: * .. Intrinsic Functions ..
212: INTRINSIC MAX
213: * ..
214: * .. Executable Statements ..
215: *
216: INFO = 0
217: NOFACT = LSAME( FACT, 'N' )
218: NOTRAN = LSAME( TRANS, 'N' )
219: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
220: INFO = -1
221: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
222: $ LSAME( TRANS, 'C' ) ) THEN
223: INFO = -2
224: ELSE IF( N.LT.0 ) THEN
225: INFO = -3
226: ELSE IF( NRHS.LT.0 ) THEN
227: INFO = -4
228: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
229: INFO = -14
230: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
231: INFO = -16
232: END IF
233: IF( INFO.NE.0 ) THEN
234: CALL XERBLA( 'DGTSVX', -INFO )
235: RETURN
236: END IF
237: *
238: IF( NOFACT ) THEN
239: *
240: * Compute the LU factorization of A.
241: *
242: CALL DCOPY( N, D, 1, DF, 1 )
243: IF( N.GT.1 ) THEN
244: CALL DCOPY( N-1, DL, 1, DLF, 1 )
245: CALL DCOPY( N-1, DU, 1, DUF, 1 )
246: END IF
247: CALL DGTTRF( N, DLF, DF, DUF, DU2, IPIV, INFO )
248: *
249: * Return if INFO is non-zero.
250: *
251: IF( INFO.GT.0 )THEN
252: RCOND = ZERO
253: RETURN
254: END IF
255: END IF
256: *
257: * Compute the norm of the matrix A.
258: *
259: IF( NOTRAN ) THEN
260: NORM = '1'
261: ELSE
262: NORM = 'I'
263: END IF
264: ANORM = DLANGT( NORM, N, DL, D, DU )
265: *
266: * Compute the reciprocal of the condition number of A.
267: *
268: CALL DGTCON( NORM, N, DLF, DF, DUF, DU2, IPIV, ANORM, RCOND, WORK,
269: $ IWORK, INFO )
270: *
271: * Compute the solution vectors X.
272: *
273: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
274: CALL DGTTRS( TRANS, N, NRHS, DLF, DF, DUF, DU2, IPIV, X, LDX,
275: $ INFO )
276: *
277: * Use iterative refinement to improve the computed solutions and
278: * compute error bounds and backward error estimates for them.
279: *
280: CALL DGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV,
281: $ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
282: *
283: * Set INFO = N+1 if the matrix is singular to working precision.
284: *
285: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
286: $ INFO = N + 1
287: *
288: RETURN
289: *
290: * End of DGTSVX
291: *
292: END
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