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
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