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