Annotation of rpl/lapack/lapack/dpbrfs.f, revision 1.1.1.1
1.1 bertrand 1: SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
2: $ LDB, 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, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: INTEGER IWORK( * )
17: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DPBRFS improves the computed solution to a system of linear
25: * equations when the coefficient matrix is symmetric positive definite
26: * and banded, and provides error bounds and backward error estimates
27: * for the 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: * KD (input) INTEGER
40: * The number of superdiagonals of the matrix A if UPLO = 'U',
41: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
42: *
43: * NRHS (input) INTEGER
44: * The number of right hand sides, i.e., the number of columns
45: * of the matrices B and X. NRHS >= 0.
46: *
47: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
48: * The upper or lower triangle of the symmetric band matrix A,
49: * stored in the first KD+1 rows of the array. The j-th column
50: * of A is stored in the j-th column of the array AB as follows:
51: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
52: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
53: *
54: * LDAB (input) INTEGER
55: * The leading dimension of the array AB. LDAB >= KD+1.
56: *
57: * AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N)
58: * The triangular factor U or L from the Cholesky factorization
59: * A = U**T*U or A = L*L**T of the band matrix A as computed by
60: * DPBTRF, in the same storage format as A (see AB).
61: *
62: * LDAFB (input) INTEGER
63: * The leading dimension of the array AFB. LDAFB >= KD+1.
64: *
65: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
66: * The right hand side matrix B.
67: *
68: * LDB (input) INTEGER
69: * The leading dimension of the array B. LDB >= max(1,N).
70: *
71: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
72: * On entry, the solution matrix X, as computed by DPBTRS.
73: * On exit, the improved solution matrix X.
74: *
75: * LDX (input) INTEGER
76: * The leading dimension of the array X. LDX >= max(1,N).
77: *
78: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
79: * The estimated forward error bound for each solution vector
80: * X(j) (the j-th column of the solution matrix X).
81: * If XTRUE is the true solution corresponding to X(j), FERR(j)
82: * is an estimated upper bound for the magnitude of the largest
83: * element in (X(j) - XTRUE) divided by the magnitude of the
84: * largest element in X(j). The estimate is as reliable as
85: * the estimate for RCOND, and is almost always a slight
86: * overestimate of the true error.
87: *
88: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
89: * The componentwise relative backward error of each solution
90: * vector X(j) (i.e., the smallest relative change in
91: * any element of A or B that makes X(j) an exact solution).
92: *
93: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
94: *
95: * IWORK (workspace) INTEGER array, dimension (N)
96: *
97: * INFO (output) INTEGER
98: * = 0: successful exit
99: * < 0: if INFO = -i, the i-th argument had an illegal value
100: *
101: * Internal Parameters
102: * ===================
103: *
104: * ITMAX is the maximum number of steps of iterative refinement.
105: *
106: * =====================================================================
107: *
108: * .. Parameters ..
109: INTEGER ITMAX
110: PARAMETER ( ITMAX = 5 )
111: DOUBLE PRECISION ZERO
112: PARAMETER ( ZERO = 0.0D+0 )
113: DOUBLE PRECISION ONE
114: PARAMETER ( ONE = 1.0D+0 )
115: DOUBLE PRECISION TWO
116: PARAMETER ( TWO = 2.0D+0 )
117: DOUBLE PRECISION THREE
118: PARAMETER ( THREE = 3.0D+0 )
119: * ..
120: * .. Local Scalars ..
121: LOGICAL UPPER
122: INTEGER COUNT, I, J, K, KASE, L, NZ
123: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
124: * ..
125: * .. Local Arrays ..
126: INTEGER ISAVE( 3 )
127: * ..
128: * .. External Subroutines ..
129: EXTERNAL DAXPY, DCOPY, DLACN2, DPBTRS, DSBMV, XERBLA
130: * ..
131: * .. Intrinsic Functions ..
132: INTRINSIC ABS, MAX, MIN
133: * ..
134: * .. External Functions ..
135: LOGICAL LSAME
136: DOUBLE PRECISION DLAMCH
137: EXTERNAL LSAME, DLAMCH
138: * ..
139: * .. Executable Statements ..
140: *
141: * Test the input parameters.
142: *
143: INFO = 0
144: UPPER = LSAME( UPLO, 'U' )
145: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
146: INFO = -1
147: ELSE IF( N.LT.0 ) THEN
148: INFO = -2
149: ELSE IF( KD.LT.0 ) THEN
150: INFO = -3
151: ELSE IF( NRHS.LT.0 ) THEN
152: INFO = -4
153: ELSE IF( LDAB.LT.KD+1 ) THEN
154: INFO = -6
155: ELSE IF( LDAFB.LT.KD+1 ) THEN
156: INFO = -8
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( 'DPBRFS', -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 = MIN( N+1, 2*KD+2 )
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 DSBMV( UPLO, N, KD, -ONE, AB, LDAB, 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: L = KD + 1 - K
221: DO 40 I = MAX( 1, K-KD ), K - 1
222: WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
223: S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
224: 40 CONTINUE
225: WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
226: 50 CONTINUE
227: ELSE
228: DO 70 K = 1, N
229: S = ZERO
230: XK = ABS( X( K, J ) )
231: WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
232: L = 1 - K
233: DO 60 I = K + 1, MIN( N, K+KD )
234: WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
235: S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
236: 60 CONTINUE
237: WORK( K ) = WORK( K ) + S
238: 70 CONTINUE
239: END IF
240: S = ZERO
241: DO 80 I = 1, N
242: IF( WORK( I ).GT.SAFE2 ) THEN
243: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
244: ELSE
245: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
246: $ ( WORK( I )+SAFE1 ) )
247: END IF
248: 80 CONTINUE
249: BERR( J ) = S
250: *
251: * Test stopping criterion. Continue iterating if
252: * 1) The residual BERR(J) is larger than machine epsilon, and
253: * 2) BERR(J) decreased by at least a factor of 2 during the
254: * last iteration, and
255: * 3) At most ITMAX iterations tried.
256: *
257: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
258: $ COUNT.LE.ITMAX ) THEN
259: *
260: * Update solution and try again.
261: *
262: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
263: $ INFO )
264: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
265: LSTRES = BERR( J )
266: COUNT = COUNT + 1
267: GO TO 20
268: END IF
269: *
270: * Bound error from formula
271: *
272: * norm(X - XTRUE) / norm(X) .le. FERR =
273: * norm( abs(inv(A))*
274: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
275: *
276: * where
277: * norm(Z) is the magnitude of the largest component of Z
278: * inv(A) is the inverse of A
279: * abs(Z) is the componentwise absolute value of the matrix or
280: * vector Z
281: * NZ is the maximum number of nonzeros in any row of A, plus 1
282: * EPS is machine epsilon
283: *
284: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
285: * is incremented by SAFE1 if the i-th component of
286: * abs(A)*abs(X) + abs(B) is less than SAFE2.
287: *
288: * Use DLACN2 to estimate the infinity-norm of the matrix
289: * inv(A) * diag(W),
290: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
291: *
292: DO 90 I = 1, N
293: IF( WORK( I ).GT.SAFE2 ) THEN
294: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
295: ELSE
296: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
297: END IF
298: 90 CONTINUE
299: *
300: KASE = 0
301: 100 CONTINUE
302: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
303: $ KASE, ISAVE )
304: IF( KASE.NE.0 ) THEN
305: IF( KASE.EQ.1 ) THEN
306: *
307: * Multiply by diag(W)*inv(A').
308: *
309: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
310: $ INFO )
311: DO 110 I = 1, N
312: WORK( N+I ) = WORK( N+I )*WORK( I )
313: 110 CONTINUE
314: ELSE IF( KASE.EQ.2 ) THEN
315: *
316: * Multiply by inv(A)*diag(W).
317: *
318: DO 120 I = 1, N
319: WORK( N+I ) = WORK( N+I )*WORK( I )
320: 120 CONTINUE
321: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
322: $ INFO )
323: END IF
324: GO TO 100
325: END IF
326: *
327: * Normalize error.
328: *
329: LSTRES = ZERO
330: DO 130 I = 1, N
331: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
332: 130 CONTINUE
333: IF( LSTRES.NE.ZERO )
334: $ FERR( J ) = FERR( J ) / LSTRES
335: *
336: 140 CONTINUE
337: *
338: RETURN
339: *
340: * End of DPBRFS
341: *
342: END
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