1: SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
2: $ LDX, FERR, BERR, WORK, IWORK, INFO )
3: *
4: * -- LAPACK routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
10: *
11: * .. Scalar Arguments ..
12: CHARACTER UPLO
13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: INTEGER IWORK( * )
17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DPORFS improves the computed solution to a system of linear
25: * equations when the coefficient matrix is symmetric positive definite,
26: * and provides error bounds and backward error estimates for the
27: * solution.
28: *
29: * Arguments
30: * =========
31: *
32: * UPLO (input) CHARACTER*1
33: * = 'U': Upper triangle of A is stored;
34: * = 'L': Lower triangle of A is stored.
35: *
36: * N (input) INTEGER
37: * The order of the matrix A. N >= 0.
38: *
39: * NRHS (input) INTEGER
40: * The number of right hand sides, i.e., the number of columns
41: * of the matrices B and X. NRHS >= 0.
42: *
43: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
44: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
45: * upper triangular part of A contains the upper triangular part
46: * of the matrix A, and the strictly lower triangular part of A
47: * is not referenced. If UPLO = 'L', the leading N-by-N lower
48: * triangular part of A contains the lower triangular part of
49: * the matrix A, and the strictly upper triangular part of A is
50: * not referenced.
51: *
52: * LDA (input) INTEGER
53: * The leading dimension of the array A. LDA >= max(1,N).
54: *
55: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
56: * The triangular factor U or L from the Cholesky factorization
57: * A = U**T*U or A = L*L**T, as computed by DPOTRF.
58: *
59: * LDAF (input) INTEGER
60: * The leading dimension of the array AF. LDAF >= max(1,N).
61: *
62: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
63: * The right hand side matrix B.
64: *
65: * LDB (input) INTEGER
66: * The leading dimension of the array B. LDB >= max(1,N).
67: *
68: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
69: * On entry, the solution matrix X, as computed by DPOTRS.
70: * On exit, the improved solution matrix X.
71: *
72: * LDX (input) INTEGER
73: * The leading dimension of the array X. LDX >= max(1,N).
74: *
75: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
76: * The estimated forward error bound for each solution vector
77: * X(j) (the j-th column of the solution matrix X).
78: * If XTRUE is the true solution corresponding to X(j), FERR(j)
79: * is an estimated upper bound for the magnitude of the largest
80: * element in (X(j) - XTRUE) divided by the magnitude of the
81: * largest element in X(j). The estimate is as reliable as
82: * the estimate for RCOND, and is almost always a slight
83: * overestimate of the true error.
84: *
85: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
86: * The componentwise relative backward error of each solution
87: * vector X(j) (i.e., the smallest relative change in
88: * any element of A or B that makes X(j) an exact solution).
89: *
90: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
91: *
92: * IWORK (workspace) INTEGER array, dimension (N)
93: *
94: * INFO (output) INTEGER
95: * = 0: successful exit
96: * < 0: if INFO = -i, the i-th argument had an illegal value
97: *
98: * Internal Parameters
99: * ===================
100: *
101: * ITMAX is the maximum number of steps of iterative refinement.
102: *
103: * =====================================================================
104: *
105: * .. Parameters ..
106: INTEGER ITMAX
107: PARAMETER ( ITMAX = 5 )
108: DOUBLE PRECISION ZERO
109: PARAMETER ( ZERO = 0.0D+0 )
110: DOUBLE PRECISION ONE
111: PARAMETER ( ONE = 1.0D+0 )
112: DOUBLE PRECISION TWO
113: PARAMETER ( TWO = 2.0D+0 )
114: DOUBLE PRECISION THREE
115: PARAMETER ( THREE = 3.0D+0 )
116: * ..
117: * .. Local Scalars ..
118: LOGICAL UPPER
119: INTEGER COUNT, I, J, K, KASE, NZ
120: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
121: * ..
122: * .. Local Arrays ..
123: INTEGER ISAVE( 3 )
124: * ..
125: * .. External Subroutines ..
126: EXTERNAL DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV, XERBLA
127: * ..
128: * .. Intrinsic Functions ..
129: INTRINSIC ABS, MAX
130: * ..
131: * .. External Functions ..
132: LOGICAL LSAME
133: DOUBLE PRECISION DLAMCH
134: EXTERNAL LSAME, DLAMCH
135: * ..
136: * .. Executable Statements ..
137: *
138: * Test the input parameters.
139: *
140: INFO = 0
141: UPPER = LSAME( UPLO, 'U' )
142: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
143: INFO = -1
144: ELSE IF( N.LT.0 ) THEN
145: INFO = -2
146: ELSE IF( NRHS.LT.0 ) THEN
147: INFO = -3
148: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149: INFO = -5
150: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
151: INFO = -7
152: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
153: INFO = -9
154: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
155: INFO = -11
156: END IF
157: IF( INFO.NE.0 ) THEN
158: CALL XERBLA( 'DPORFS', -INFO )
159: RETURN
160: END IF
161: *
162: * Quick return if possible
163: *
164: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
165: DO 10 J = 1, NRHS
166: FERR( J ) = ZERO
167: BERR( J ) = ZERO
168: 10 CONTINUE
169: RETURN
170: END IF
171: *
172: * NZ = maximum number of nonzero elements in each row of A, plus 1
173: *
174: NZ = N + 1
175: EPS = DLAMCH( 'Epsilon' )
176: SAFMIN = DLAMCH( 'Safe minimum' )
177: SAFE1 = NZ*SAFMIN
178: SAFE2 = SAFE1 / EPS
179: *
180: * Do for each right hand side
181: *
182: DO 140 J = 1, NRHS
183: *
184: COUNT = 1
185: LSTRES = THREE
186: 20 CONTINUE
187: *
188: * Loop until stopping criterion is satisfied.
189: *
190: * Compute residual R = B - A * X
191: *
192: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
193: CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
194: $ WORK( N+1 ), 1 )
195: *
196: * Compute componentwise relative backward error from formula
197: *
198: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
199: *
200: * where abs(Z) is the componentwise absolute value of the matrix
201: * or vector Z. If the i-th component of the denominator is less
202: * than SAFE2, then SAFE1 is added to the i-th components of the
203: * numerator and denominator before dividing.
204: *
205: DO 30 I = 1, N
206: WORK( I ) = ABS( B( I, J ) )
207: 30 CONTINUE
208: *
209: * Compute abs(A)*abs(X) + abs(B).
210: *
211: IF( UPPER ) THEN
212: DO 50 K = 1, N
213: S = ZERO
214: XK = ABS( X( K, J ) )
215: DO 40 I = 1, K - 1
216: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
217: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
218: 40 CONTINUE
219: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
220: 50 CONTINUE
221: ELSE
222: DO 70 K = 1, N
223: S = ZERO
224: XK = ABS( X( K, J ) )
225: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
226: DO 60 I = K + 1, N
227: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
228: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
229: 60 CONTINUE
230: WORK( K ) = WORK( K ) + S
231: 70 CONTINUE
232: END IF
233: S = ZERO
234: DO 80 I = 1, N
235: IF( WORK( I ).GT.SAFE2 ) THEN
236: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
237: ELSE
238: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
239: $ ( WORK( I )+SAFE1 ) )
240: END IF
241: 80 CONTINUE
242: BERR( J ) = S
243: *
244: * Test stopping criterion. Continue iterating if
245: * 1) The residual BERR(J) is larger than machine epsilon, and
246: * 2) BERR(J) decreased by at least a factor of 2 during the
247: * last iteration, and
248: * 3) At most ITMAX iterations tried.
249: *
250: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
251: $ COUNT.LE.ITMAX ) THEN
252: *
253: * Update solution and try again.
254: *
255: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
256: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
257: LSTRES = BERR( J )
258: COUNT = COUNT + 1
259: GO TO 20
260: END IF
261: *
262: * Bound error from formula
263: *
264: * norm(X - XTRUE) / norm(X) .le. FERR =
265: * norm( abs(inv(A))*
266: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
267: *
268: * where
269: * norm(Z) is the magnitude of the largest component of Z
270: * inv(A) is the inverse of A
271: * abs(Z) is the componentwise absolute value of the matrix or
272: * vector Z
273: * NZ is the maximum number of nonzeros in any row of A, plus 1
274: * EPS is machine epsilon
275: *
276: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
277: * is incremented by SAFE1 if the i-th component of
278: * abs(A)*abs(X) + abs(B) is less than SAFE2.
279: *
280: * Use DLACN2 to estimate the infinity-norm of the matrix
281: * inv(A) * diag(W),
282: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
283: *
284: DO 90 I = 1, N
285: IF( WORK( I ).GT.SAFE2 ) THEN
286: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
287: ELSE
288: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
289: END IF
290: 90 CONTINUE
291: *
292: KASE = 0
293: 100 CONTINUE
294: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
295: $ KASE, ISAVE )
296: IF( KASE.NE.0 ) THEN
297: IF( KASE.EQ.1 ) THEN
298: *
299: * Multiply by diag(W)*inv(A').
300: *
301: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
302: DO 110 I = 1, N
303: WORK( N+I ) = WORK( I )*WORK( N+I )
304: 110 CONTINUE
305: ELSE IF( KASE.EQ.2 ) THEN
306: *
307: * Multiply by inv(A)*diag(W).
308: *
309: DO 120 I = 1, N
310: WORK( N+I ) = WORK( I )*WORK( N+I )
311: 120 CONTINUE
312: CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
313: END IF
314: GO TO 100
315: END IF
316: *
317: * Normalize error.
318: *
319: LSTRES = ZERO
320: DO 130 I = 1, N
321: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
322: 130 CONTINUE
323: IF( LSTRES.NE.ZERO )
324: $ FERR( J ) = FERR( J ) / LSTRES
325: *
326: 140 CONTINUE
327: *
328: RETURN
329: *
330: * End of DPORFS
331: *
332: END
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