1 SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
2 $ BERR, WORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDB, LDX, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
14 $ E( * ), EF( * ), FERR( * ), WORK( * ),
15 $ X( LDX, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DPTRFS improves the computed solution to a system of linear
22 * equations when the coefficient matrix is symmetric positive definite
23 * and tridiagonal, and provides error bounds and backward error
24 * estimates for the solution.
25 *
26 * Arguments
27 * =========
28 *
29 * N (input) INTEGER
30 * The order of the matrix A. N >= 0.
31 *
32 * NRHS (input) INTEGER
33 * The number of right hand sides, i.e., the number of columns
34 * of the matrix B. NRHS >= 0.
35 *
36 * D (input) DOUBLE PRECISION array, dimension (N)
37 * The n diagonal elements of the tridiagonal matrix A.
38 *
39 * E (input) DOUBLE PRECISION array, dimension (N-1)
40 * The (n-1) subdiagonal elements of the tridiagonal matrix A.
41 *
42 * DF (input) DOUBLE PRECISION array, dimension (N)
43 * The n diagonal elements of the diagonal matrix D from the
44 * factorization computed by DPTTRF.
45 *
46 * EF (input) DOUBLE PRECISION array, dimension (N-1)
47 * The (n-1) subdiagonal elements of the unit bidiagonal factor
48 * L from the factorization computed by DPTTRF.
49 *
50 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
51 * The right hand side matrix B.
52 *
53 * LDB (input) INTEGER
54 * The leading dimension of the array B. LDB >= max(1,N).
55 *
56 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
57 * On entry, the solution matrix X, as computed by DPTTRS.
58 * On exit, the improved solution matrix X.
59 *
60 * LDX (input) INTEGER
61 * The leading dimension of the array X. LDX >= max(1,N).
62 *
63 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
64 * The forward error bound for each solution vector
65 * X(j) (the j-th column of the solution matrix X).
66 * If XTRUE is the true solution corresponding to X(j), FERR(j)
67 * is an estimated upper bound for the magnitude of the largest
68 * element in (X(j) - XTRUE) divided by the magnitude of the
69 * largest element in X(j).
70 *
71 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
72 * The componentwise relative backward error of each solution
73 * vector X(j) (i.e., the smallest relative change in
74 * any element of A or B that makes X(j) an exact solution).
75 *
76 * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
77 *
78 * INFO (output) INTEGER
79 * = 0: successful exit
80 * < 0: if INFO = -i, the i-th argument had an illegal value
81 *
82 * Internal Parameters
83 * ===================
84 *
85 * ITMAX is the maximum number of steps of iterative refinement.
86 *
87 * =====================================================================
88 *
89 * .. Parameters ..
90 INTEGER ITMAX
91 PARAMETER ( ITMAX = 5 )
92 DOUBLE PRECISION ZERO
93 PARAMETER ( ZERO = 0.0D+0 )
94 DOUBLE PRECISION ONE
95 PARAMETER ( ONE = 1.0D+0 )
96 DOUBLE PRECISION TWO
97 PARAMETER ( TWO = 2.0D+0 )
98 DOUBLE PRECISION THREE
99 PARAMETER ( THREE = 3.0D+0 )
100 * ..
101 * .. Local Scalars ..
102 INTEGER COUNT, I, IX, J, NZ
103 DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
104 $ SAFMIN
105 * ..
106 * .. External Subroutines ..
107 EXTERNAL DAXPY, DPTTRS, XERBLA
108 * ..
109 * .. Intrinsic Functions ..
110 INTRINSIC ABS, MAX
111 * ..
112 * .. External Functions ..
113 INTEGER IDAMAX
114 DOUBLE PRECISION DLAMCH
115 EXTERNAL IDAMAX, DLAMCH
116 * ..
117 * .. Executable Statements ..
118 *
119 * Test the input parameters.
120 *
121 INFO = 0
122 IF( N.LT.0 ) THEN
123 INFO = -1
124 ELSE IF( NRHS.LT.0 ) THEN
125 INFO = -2
126 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
127 INFO = -8
128 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
129 INFO = -10
130 END IF
131 IF( INFO.NE.0 ) THEN
132 CALL XERBLA( 'DPTRFS', -INFO )
133 RETURN
134 END IF
135 *
136 * Quick return if possible
137 *
138 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
139 DO 10 J = 1, NRHS
140 FERR( J ) = ZERO
141 BERR( J ) = ZERO
142 10 CONTINUE
143 RETURN
144 END IF
145 *
146 * NZ = maximum number of nonzero elements in each row of A, plus 1
147 *
148 NZ = 4
149 EPS = DLAMCH( 'Epsilon' )
150 SAFMIN = DLAMCH( 'Safe minimum' )
151 SAFE1 = NZ*SAFMIN
152 SAFE2 = SAFE1 / EPS
153 *
154 * Do for each right hand side
155 *
156 DO 90 J = 1, NRHS
157 *
158 COUNT = 1
159 LSTRES = THREE
160 20 CONTINUE
161 *
162 * Loop until stopping criterion is satisfied.
163 *
164 * Compute residual R = B - A * X. Also compute
165 * abs(A)*abs(x) + abs(b) for use in the backward error bound.
166 *
167 IF( N.EQ.1 ) THEN
168 BI = B( 1, J )
169 DX = D( 1 )*X( 1, J )
170 WORK( N+1 ) = BI - DX
171 WORK( 1 ) = ABS( BI ) + ABS( DX )
172 ELSE
173 BI = B( 1, J )
174 DX = D( 1 )*X( 1, J )
175 EX = E( 1 )*X( 2, J )
176 WORK( N+1 ) = BI - DX - EX
177 WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
178 DO 30 I = 2, N - 1
179 BI = B( I, J )
180 CX = E( I-1 )*X( I-1, J )
181 DX = D( I )*X( I, J )
182 EX = E( I )*X( I+1, J )
183 WORK( N+I ) = BI - CX - DX - EX
184 WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
185 30 CONTINUE
186 BI = B( N, J )
187 CX = E( N-1 )*X( N-1, J )
188 DX = D( N )*X( N, J )
189 WORK( N+N ) = BI - CX - DX
190 WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
191 END IF
192 *
193 * Compute componentwise relative backward error from formula
194 *
195 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
196 *
197 * where abs(Z) is the componentwise absolute value of the matrix
198 * or vector Z. If the i-th component of the denominator is less
199 * than SAFE2, then SAFE1 is added to the i-th components of the
200 * numerator and denominator before dividing.
201 *
202 S = ZERO
203 DO 40 I = 1, N
204 IF( WORK( I ).GT.SAFE2 ) THEN
205 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
206 ELSE
207 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
208 $ ( WORK( I )+SAFE1 ) )
209 END IF
210 40 CONTINUE
211 BERR( J ) = S
212 *
213 * Test stopping criterion. Continue iterating if
214 * 1) The residual BERR(J) is larger than machine epsilon, and
215 * 2) BERR(J) decreased by at least a factor of 2 during the
216 * last iteration, and
217 * 3) At most ITMAX iterations tried.
218 *
219 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
220 $ COUNT.LE.ITMAX ) THEN
221 *
222 * Update solution and try again.
223 *
224 CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
225 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
226 LSTRES = BERR( J )
227 COUNT = COUNT + 1
228 GO TO 20
229 END IF
230 *
231 * Bound error from formula
232 *
233 * norm(X - XTRUE) / norm(X) .le. FERR =
234 * norm( abs(inv(A))*
235 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
236 *
237 * where
238 * norm(Z) is the magnitude of the largest component of Z
239 * inv(A) is the inverse of A
240 * abs(Z) is the componentwise absolute value of the matrix or
241 * vector Z
242 * NZ is the maximum number of nonzeros in any row of A, plus 1
243 * EPS is machine epsilon
244 *
245 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
246 * is incremented by SAFE1 if the i-th component of
247 * abs(A)*abs(X) + abs(B) is less than SAFE2.
248 *
249 DO 50 I = 1, N
250 IF( WORK( I ).GT.SAFE2 ) THEN
251 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
252 ELSE
253 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
254 END IF
255 50 CONTINUE
256 IX = IDAMAX( N, WORK, 1 )
257 FERR( J ) = WORK( IX )
258 *
259 * Estimate the norm of inv(A).
260 *
261 * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
262 *
263 * m(i,j) = abs(A(i,j)), i = j,
264 * m(i,j) = -abs(A(i,j)), i .ne. j,
265 *
266 * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
267 *
268 * Solve M(L) * x = e.
269 *
270 WORK( 1 ) = ONE
271 DO 60 I = 2, N
272 WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
273 60 CONTINUE
274 *
275 * Solve D * M(L)**T * x = b.
276 *
277 WORK( N ) = WORK( N ) / DF( N )
278 DO 70 I = N - 1, 1, -1
279 WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
280 70 CONTINUE
281 *
282 * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
283 *
284 IX = IDAMAX( N, WORK, 1 )
285 FERR( J ) = FERR( J )*ABS( WORK( IX ) )
286 *
287 * Normalize error.
288 *
289 LSTRES = ZERO
290 DO 80 I = 1, N
291 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
292 80 CONTINUE
293 IF( LSTRES.NE.ZERO )
294 $ FERR( J ) = FERR( J ) / LSTRES
295 *
296 90 CONTINUE
297 *
298 RETURN
299 *
300 * End of DPTRFS
301 *
302 END
2 $ BERR, WORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 INTEGER INFO, LDB, LDX, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
14 $ E( * ), EF( * ), FERR( * ), WORK( * ),
15 $ X( LDX, * )
16 * ..
17 *
18 * Purpose
19 * =======
20 *
21 * DPTRFS improves the computed solution to a system of linear
22 * equations when the coefficient matrix is symmetric positive definite
23 * and tridiagonal, and provides error bounds and backward error
24 * estimates for the solution.
25 *
26 * Arguments
27 * =========
28 *
29 * N (input) INTEGER
30 * The order of the matrix A. N >= 0.
31 *
32 * NRHS (input) INTEGER
33 * The number of right hand sides, i.e., the number of columns
34 * of the matrix B. NRHS >= 0.
35 *
36 * D (input) DOUBLE PRECISION array, dimension (N)
37 * The n diagonal elements of the tridiagonal matrix A.
38 *
39 * E (input) DOUBLE PRECISION array, dimension (N-1)
40 * The (n-1) subdiagonal elements of the tridiagonal matrix A.
41 *
42 * DF (input) DOUBLE PRECISION array, dimension (N)
43 * The n diagonal elements of the diagonal matrix D from the
44 * factorization computed by DPTTRF.
45 *
46 * EF (input) DOUBLE PRECISION array, dimension (N-1)
47 * The (n-1) subdiagonal elements of the unit bidiagonal factor
48 * L from the factorization computed by DPTTRF.
49 *
50 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
51 * The right hand side matrix B.
52 *
53 * LDB (input) INTEGER
54 * The leading dimension of the array B. LDB >= max(1,N).
55 *
56 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
57 * On entry, the solution matrix X, as computed by DPTTRS.
58 * On exit, the improved solution matrix X.
59 *
60 * LDX (input) INTEGER
61 * The leading dimension of the array X. LDX >= max(1,N).
62 *
63 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
64 * The forward error bound for each solution vector
65 * X(j) (the j-th column of the solution matrix X).
66 * If XTRUE is the true solution corresponding to X(j), FERR(j)
67 * is an estimated upper bound for the magnitude of the largest
68 * element in (X(j) - XTRUE) divided by the magnitude of the
69 * largest element in X(j).
70 *
71 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
72 * The componentwise relative backward error of each solution
73 * vector X(j) (i.e., the smallest relative change in
74 * any element of A or B that makes X(j) an exact solution).
75 *
76 * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
77 *
78 * INFO (output) INTEGER
79 * = 0: successful exit
80 * < 0: if INFO = -i, the i-th argument had an illegal value
81 *
82 * Internal Parameters
83 * ===================
84 *
85 * ITMAX is the maximum number of steps of iterative refinement.
86 *
87 * =====================================================================
88 *
89 * .. Parameters ..
90 INTEGER ITMAX
91 PARAMETER ( ITMAX = 5 )
92 DOUBLE PRECISION ZERO
93 PARAMETER ( ZERO = 0.0D+0 )
94 DOUBLE PRECISION ONE
95 PARAMETER ( ONE = 1.0D+0 )
96 DOUBLE PRECISION TWO
97 PARAMETER ( TWO = 2.0D+0 )
98 DOUBLE PRECISION THREE
99 PARAMETER ( THREE = 3.0D+0 )
100 * ..
101 * .. Local Scalars ..
102 INTEGER COUNT, I, IX, J, NZ
103 DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
104 $ SAFMIN
105 * ..
106 * .. External Subroutines ..
107 EXTERNAL DAXPY, DPTTRS, XERBLA
108 * ..
109 * .. Intrinsic Functions ..
110 INTRINSIC ABS, MAX
111 * ..
112 * .. External Functions ..
113 INTEGER IDAMAX
114 DOUBLE PRECISION DLAMCH
115 EXTERNAL IDAMAX, DLAMCH
116 * ..
117 * .. Executable Statements ..
118 *
119 * Test the input parameters.
120 *
121 INFO = 0
122 IF( N.LT.0 ) THEN
123 INFO = -1
124 ELSE IF( NRHS.LT.0 ) THEN
125 INFO = -2
126 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
127 INFO = -8
128 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
129 INFO = -10
130 END IF
131 IF( INFO.NE.0 ) THEN
132 CALL XERBLA( 'DPTRFS', -INFO )
133 RETURN
134 END IF
135 *
136 * Quick return if possible
137 *
138 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
139 DO 10 J = 1, NRHS
140 FERR( J ) = ZERO
141 BERR( J ) = ZERO
142 10 CONTINUE
143 RETURN
144 END IF
145 *
146 * NZ = maximum number of nonzero elements in each row of A, plus 1
147 *
148 NZ = 4
149 EPS = DLAMCH( 'Epsilon' )
150 SAFMIN = DLAMCH( 'Safe minimum' )
151 SAFE1 = NZ*SAFMIN
152 SAFE2 = SAFE1 / EPS
153 *
154 * Do for each right hand side
155 *
156 DO 90 J = 1, NRHS
157 *
158 COUNT = 1
159 LSTRES = THREE
160 20 CONTINUE
161 *
162 * Loop until stopping criterion is satisfied.
163 *
164 * Compute residual R = B - A * X. Also compute
165 * abs(A)*abs(x) + abs(b) for use in the backward error bound.
166 *
167 IF( N.EQ.1 ) THEN
168 BI = B( 1, J )
169 DX = D( 1 )*X( 1, J )
170 WORK( N+1 ) = BI - DX
171 WORK( 1 ) = ABS( BI ) + ABS( DX )
172 ELSE
173 BI = B( 1, J )
174 DX = D( 1 )*X( 1, J )
175 EX = E( 1 )*X( 2, J )
176 WORK( N+1 ) = BI - DX - EX
177 WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
178 DO 30 I = 2, N - 1
179 BI = B( I, J )
180 CX = E( I-1 )*X( I-1, J )
181 DX = D( I )*X( I, J )
182 EX = E( I )*X( I+1, J )
183 WORK( N+I ) = BI - CX - DX - EX
184 WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
185 30 CONTINUE
186 BI = B( N, J )
187 CX = E( N-1 )*X( N-1, J )
188 DX = D( N )*X( N, J )
189 WORK( N+N ) = BI - CX - DX
190 WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
191 END IF
192 *
193 * Compute componentwise relative backward error from formula
194 *
195 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
196 *
197 * where abs(Z) is the componentwise absolute value of the matrix
198 * or vector Z. If the i-th component of the denominator is less
199 * than SAFE2, then SAFE1 is added to the i-th components of the
200 * numerator and denominator before dividing.
201 *
202 S = ZERO
203 DO 40 I = 1, N
204 IF( WORK( I ).GT.SAFE2 ) THEN
205 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
206 ELSE
207 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
208 $ ( WORK( I )+SAFE1 ) )
209 END IF
210 40 CONTINUE
211 BERR( J ) = S
212 *
213 * Test stopping criterion. Continue iterating if
214 * 1) The residual BERR(J) is larger than machine epsilon, and
215 * 2) BERR(J) decreased by at least a factor of 2 during the
216 * last iteration, and
217 * 3) At most ITMAX iterations tried.
218 *
219 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
220 $ COUNT.LE.ITMAX ) THEN
221 *
222 * Update solution and try again.
223 *
224 CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
225 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
226 LSTRES = BERR( J )
227 COUNT = COUNT + 1
228 GO TO 20
229 END IF
230 *
231 * Bound error from formula
232 *
233 * norm(X - XTRUE) / norm(X) .le. FERR =
234 * norm( abs(inv(A))*
235 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
236 *
237 * where
238 * norm(Z) is the magnitude of the largest component of Z
239 * inv(A) is the inverse of A
240 * abs(Z) is the componentwise absolute value of the matrix or
241 * vector Z
242 * NZ is the maximum number of nonzeros in any row of A, plus 1
243 * EPS is machine epsilon
244 *
245 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
246 * is incremented by SAFE1 if the i-th component of
247 * abs(A)*abs(X) + abs(B) is less than SAFE2.
248 *
249 DO 50 I = 1, N
250 IF( WORK( I ).GT.SAFE2 ) THEN
251 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
252 ELSE
253 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
254 END IF
255 50 CONTINUE
256 IX = IDAMAX( N, WORK, 1 )
257 FERR( J ) = WORK( IX )
258 *
259 * Estimate the norm of inv(A).
260 *
261 * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
262 *
263 * m(i,j) = abs(A(i,j)), i = j,
264 * m(i,j) = -abs(A(i,j)), i .ne. j,
265 *
266 * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
267 *
268 * Solve M(L) * x = e.
269 *
270 WORK( 1 ) = ONE
271 DO 60 I = 2, N
272 WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
273 60 CONTINUE
274 *
275 * Solve D * M(L)**T * x = b.
276 *
277 WORK( N ) = WORK( N ) / DF( N )
278 DO 70 I = N - 1, 1, -1
279 WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
280 70 CONTINUE
281 *
282 * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
283 *
284 IX = IDAMAX( N, WORK, 1 )
285 FERR( J ) = FERR( J )*ABS( WORK( IX ) )
286 *
287 * Normalize error.
288 *
289 LSTRES = ZERO
290 DO 80 I = 1, N
291 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
292 80 CONTINUE
293 IF( LSTRES.NE.ZERO )
294 $ FERR( J ) = FERR( J ) / LSTRES
295 *
296 90 CONTINUE
297 *
298 RETURN
299 *
300 * End of DPTRFS
301 *
302 END