Annotation of rpl/lapack/lapack/dsyrfs.f, revision 1.4
1.1 bertrand 1: SUBROUTINE DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
2: $ X, 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 IPIV( * ), IWORK( * )
17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DSYRFS improves the computed solution to a system of linear
25: * equations when the coefficient matrix is symmetric indefinite, and
26: * provides error bounds and backward error estimates for the solution.
27: *
28: * Arguments
29: * =========
30: *
31: * UPLO (input) CHARACTER*1
32: * = 'U': Upper triangle of A is stored;
33: * = 'L': Lower triangle of A is stored.
34: *
35: * N (input) INTEGER
36: * The order of the matrix A. N >= 0.
37: *
38: * NRHS (input) INTEGER
39: * The number of right hand sides, i.e., the number of columns
40: * of the matrices B and X. NRHS >= 0.
41: *
42: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
43: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
44: * upper triangular part of A contains the upper triangular part
45: * of the matrix A, and the strictly lower triangular part of A
46: * is not referenced. If UPLO = 'L', the leading N-by-N lower
47: * triangular part of A contains the lower triangular part of
48: * the matrix A, and the strictly upper triangular part of A is
49: * not referenced.
50: *
51: * LDA (input) INTEGER
52: * The leading dimension of the array A. LDA >= max(1,N).
53: *
54: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
55: * The factored form of the matrix A. AF contains the block
56: * diagonal matrix D and the multipliers used to obtain the
57: * factor U or L from the factorization A = U*D*U**T or
58: * A = L*D*L**T as computed by DSYTRF.
59: *
60: * LDAF (input) INTEGER
61: * The leading dimension of the array AF. LDAF >= max(1,N).
62: *
63: * IPIV (input) INTEGER array, dimension (N)
64: * Details of the interchanges and the block structure of D
65: * as determined by DSYTRF.
66: *
67: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
68: * The right hand side matrix B.
69: *
70: * LDB (input) INTEGER
71: * The leading dimension of the array B. LDB >= max(1,N).
72: *
73: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
74: * On entry, the solution matrix X, as computed by DSYTRS.
75: * On exit, the improved solution matrix X.
76: *
77: * LDX (input) INTEGER
78: * The leading dimension of the array X. LDX >= max(1,N).
79: *
80: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
81: * The estimated forward error bound for each solution vector
82: * X(j) (the j-th column of the solution matrix X).
83: * If XTRUE is the true solution corresponding to X(j), FERR(j)
84: * is an estimated upper bound for the magnitude of the largest
85: * element in (X(j) - XTRUE) divided by the magnitude of the
86: * largest element in X(j). The estimate is as reliable as
87: * the estimate for RCOND, and is almost always a slight
88: * overestimate of the true error.
89: *
90: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
91: * The componentwise relative backward error of each solution
92: * vector X(j) (i.e., the smallest relative change in
93: * any element of A or B that makes X(j) an exact solution).
94: *
95: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
96: *
97: * IWORK (workspace) INTEGER array, dimension (N)
98: *
99: * INFO (output) INTEGER
100: * = 0: successful exit
101: * < 0: if INFO = -i, the i-th argument had an illegal value
102: *
103: * Internal Parameters
104: * ===================
105: *
106: * ITMAX is the maximum number of steps of iterative refinement.
107: *
108: * =====================================================================
109: *
110: * .. Parameters ..
111: INTEGER ITMAX
112: PARAMETER ( ITMAX = 5 )
113: DOUBLE PRECISION ZERO
114: PARAMETER ( ZERO = 0.0D+0 )
115: DOUBLE PRECISION ONE
116: PARAMETER ( ONE = 1.0D+0 )
117: DOUBLE PRECISION TWO
118: PARAMETER ( TWO = 2.0D+0 )
119: DOUBLE PRECISION THREE
120: PARAMETER ( THREE = 3.0D+0 )
121: * ..
122: * .. Local Scalars ..
123: LOGICAL UPPER
124: INTEGER COUNT, I, J, K, KASE, NZ
125: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
126: * ..
127: * .. Local Arrays ..
128: INTEGER ISAVE( 3 )
129: * ..
130: * .. External Subroutines ..
131: EXTERNAL DAXPY, DCOPY, DLACN2, DSYMV, DSYTRS, XERBLA
132: * ..
133: * .. Intrinsic Functions ..
134: INTRINSIC ABS, MAX
135: * ..
136: * .. External Functions ..
137: LOGICAL LSAME
138: DOUBLE PRECISION DLAMCH
139: EXTERNAL LSAME, DLAMCH
140: * ..
141: * .. Executable Statements ..
142: *
143: * Test the input parameters.
144: *
145: INFO = 0
146: UPPER = LSAME( UPLO, 'U' )
147: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
148: INFO = -1
149: ELSE IF( N.LT.0 ) THEN
150: INFO = -2
151: ELSE IF( NRHS.LT.0 ) THEN
152: INFO = -3
153: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
154: INFO = -5
155: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
156: INFO = -7
157: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
158: INFO = -10
159: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
160: INFO = -12
161: END IF
162: IF( INFO.NE.0 ) THEN
163: CALL XERBLA( 'DSYRFS', -INFO )
164: RETURN
165: END IF
166: *
167: * Quick return if possible
168: *
169: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
170: DO 10 J = 1, NRHS
171: FERR( J ) = ZERO
172: BERR( J ) = ZERO
173: 10 CONTINUE
174: RETURN
175: END IF
176: *
177: * NZ = maximum number of nonzero elements in each row of A, plus 1
178: *
179: NZ = N + 1
180: EPS = DLAMCH( 'Epsilon' )
181: SAFMIN = DLAMCH( 'Safe minimum' )
182: SAFE1 = NZ*SAFMIN
183: SAFE2 = SAFE1 / EPS
184: *
185: * Do for each right hand side
186: *
187: DO 140 J = 1, NRHS
188: *
189: COUNT = 1
190: LSTRES = THREE
191: 20 CONTINUE
192: *
193: * Loop until stopping criterion is satisfied.
194: *
195: * Compute residual R = B - A * X
196: *
197: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
198: CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
199: $ WORK( N+1 ), 1 )
200: *
201: * Compute componentwise relative backward error from formula
202: *
203: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
204: *
205: * where abs(Z) is the componentwise absolute value of the matrix
206: * or vector Z. If the i-th component of the denominator is less
207: * than SAFE2, then SAFE1 is added to the i-th components of the
208: * numerator and denominator before dividing.
209: *
210: DO 30 I = 1, N
211: WORK( I ) = ABS( B( I, J ) )
212: 30 CONTINUE
213: *
214: * Compute abs(A)*abs(X) + abs(B).
215: *
216: IF( UPPER ) THEN
217: DO 50 K = 1, N
218: S = ZERO
219: XK = ABS( X( K, J ) )
220: DO 40 I = 1, K - 1
221: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
222: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
223: 40 CONTINUE
224: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
225: 50 CONTINUE
226: ELSE
227: DO 70 K = 1, N
228: S = ZERO
229: XK = ABS( X( K, J ) )
230: WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
231: DO 60 I = K + 1, N
232: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
233: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
234: 60 CONTINUE
235: WORK( K ) = WORK( K ) + S
236: 70 CONTINUE
237: END IF
238: S = ZERO
239: DO 80 I = 1, N
240: IF( WORK( I ).GT.SAFE2 ) THEN
241: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
242: ELSE
243: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
244: $ ( WORK( I )+SAFE1 ) )
245: END IF
246: 80 CONTINUE
247: BERR( J ) = S
248: *
249: * Test stopping criterion. Continue iterating if
250: * 1) The residual BERR(J) is larger than machine epsilon, and
251: * 2) BERR(J) decreased by at least a factor of 2 during the
252: * last iteration, and
253: * 3) At most ITMAX iterations tried.
254: *
255: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
256: $ COUNT.LE.ITMAX ) THEN
257: *
258: * Update solution and try again.
259: *
260: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
261: $ INFO )
262: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
263: LSTRES = BERR( J )
264: COUNT = COUNT + 1
265: GO TO 20
266: END IF
267: *
268: * Bound error from formula
269: *
270: * norm(X - XTRUE) / norm(X) .le. FERR =
271: * norm( abs(inv(A))*
272: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
273: *
274: * where
275: * norm(Z) is the magnitude of the largest component of Z
276: * inv(A) is the inverse of A
277: * abs(Z) is the componentwise absolute value of the matrix or
278: * vector Z
279: * NZ is the maximum number of nonzeros in any row of A, plus 1
280: * EPS is machine epsilon
281: *
282: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
283: * is incremented by SAFE1 if the i-th component of
284: * abs(A)*abs(X) + abs(B) is less than SAFE2.
285: *
286: * Use DLACN2 to estimate the infinity-norm of the matrix
287: * inv(A) * diag(W),
288: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
289: *
290: DO 90 I = 1, N
291: IF( WORK( I ).GT.SAFE2 ) THEN
292: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
293: ELSE
294: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
295: END IF
296: 90 CONTINUE
297: *
298: KASE = 0
299: 100 CONTINUE
300: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
301: $ KASE, ISAVE )
302: IF( KASE.NE.0 ) THEN
303: IF( KASE.EQ.1 ) THEN
304: *
305: * Multiply by diag(W)*inv(A').
306: *
307: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
308: $ INFO )
309: DO 110 I = 1, N
310: WORK( N+I ) = WORK( I )*WORK( N+I )
311: 110 CONTINUE
312: ELSE IF( KASE.EQ.2 ) THEN
313: *
314: * Multiply by inv(A)*diag(W).
315: *
316: DO 120 I = 1, N
317: WORK( N+I ) = WORK( I )*WORK( N+I )
318: 120 CONTINUE
319: CALL DSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
320: $ INFO )
321: END IF
322: GO TO 100
323: END IF
324: *
325: * Normalize error.
326: *
327: LSTRES = ZERO
328: DO 130 I = 1, N
329: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
330: 130 CONTINUE
331: IF( LSTRES.NE.ZERO )
332: $ FERR( J ) = FERR( J ) / LSTRES
333: *
334: 140 CONTINUE
335: *
336: RETURN
337: *
338: * End of DSYRFS
339: *
340: END
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to dsyrfs.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack