1: SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
2: $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
3: $ INFO )
4: *
5: * -- LAPACK routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
11: *
12: * .. Scalar Arguments ..
13: CHARACTER TRANS
14: INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
15: * ..
16: * .. Array Arguments ..
17: INTEGER IPIV( * )
18: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
19: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
20: $ WORK( * ), X( LDX, * )
21: * ..
22: *
23: * Purpose
24: * =======
25: *
26: * ZGBRFS improves the computed solution to a system of linear
27: * equations when the coefficient matrix is banded, and provides
28: * error bounds and backward error estimates for the solution.
29: *
30: * Arguments
31: * =========
32: *
33: * TRANS (input) CHARACTER*1
34: * Specifies the form of the system of equations:
35: * = 'N': A * X = B (No transpose)
36: * = 'T': A**T * X = B (Transpose)
37: * = 'C': A**H * X = B (Conjugate transpose)
38: *
39: * N (input) INTEGER
40: * The order of the matrix A. N >= 0.
41: *
42: * KL (input) INTEGER
43: * The number of subdiagonals within the band of A. KL >= 0.
44: *
45: * KU (input) INTEGER
46: * The number of superdiagonals within the band of A. KU >= 0.
47: *
48: * NRHS (input) INTEGER
49: * The number of right hand sides, i.e., the number of columns
50: * of the matrices B and X. NRHS >= 0.
51: *
52: * AB (input) COMPLEX*16 array, dimension (LDAB,N)
53: * The original band matrix A, stored in rows 1 to KL+KU+1.
54: * The j-th column of A is stored in the j-th column of the
55: * array AB as follows:
56: * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
57: *
58: * LDAB (input) INTEGER
59: * The leading dimension of the array AB. LDAB >= KL+KU+1.
60: *
61: * AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
62: * Details of the LU factorization of the band matrix A, as
63: * computed by ZGBTRF. U is stored as an upper triangular band
64: * matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
65: * the multipliers used during the factorization are stored in
66: * rows KL+KU+2 to 2*KL+KU+1.
67: *
68: * LDAFB (input) INTEGER
69: * The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
70: *
71: * IPIV (input) INTEGER array, dimension (N)
72: * The pivot indices from ZGBTRF; for 1<=i<=N, row i of the
73: * matrix was interchanged with row IPIV(i).
74: *
75: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
76: * The right hand side matrix B.
77: *
78: * LDB (input) INTEGER
79: * The leading dimension of the array B. LDB >= max(1,N).
80: *
81: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
82: * On entry, the solution matrix X, as computed by ZGBTRS.
83: * On exit, the improved solution matrix X.
84: *
85: * LDX (input) INTEGER
86: * The leading dimension of the array X. LDX >= max(1,N).
87: *
88: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
89: * The estimated forward error bound for each solution vector
90: * X(j) (the j-th column of the solution matrix X).
91: * If XTRUE is the true solution corresponding to X(j), FERR(j)
92: * is an estimated upper bound for the magnitude of the largest
93: * element in (X(j) - XTRUE) divided by the magnitude of the
94: * largest element in X(j). The estimate is as reliable as
95: * the estimate for RCOND, and is almost always a slight
96: * overestimate of the true error.
97: *
98: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
99: * The componentwise relative backward error of each solution
100: * vector X(j) (i.e., the smallest relative change in
101: * any element of A or B that makes X(j) an exact solution).
102: *
103: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
104: *
105: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
106: *
107: * INFO (output) INTEGER
108: * = 0: successful exit
109: * < 0: if INFO = -i, the i-th argument had an illegal value
110: *
111: * Internal Parameters
112: * ===================
113: *
114: * ITMAX is the maximum number of steps of iterative refinement.
115: *
116: * =====================================================================
117: *
118: * .. Parameters ..
119: INTEGER ITMAX
120: PARAMETER ( ITMAX = 5 )
121: DOUBLE PRECISION ZERO
122: PARAMETER ( ZERO = 0.0D+0 )
123: COMPLEX*16 CONE
124: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
125: DOUBLE PRECISION TWO
126: PARAMETER ( TWO = 2.0D+0 )
127: DOUBLE PRECISION THREE
128: PARAMETER ( THREE = 3.0D+0 )
129: * ..
130: * .. Local Scalars ..
131: LOGICAL NOTRAN
132: CHARACTER TRANSN, TRANST
133: INTEGER COUNT, I, J, K, KASE, KK, NZ
134: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
135: COMPLEX*16 ZDUM
136: * ..
137: * .. Local Arrays ..
138: INTEGER ISAVE( 3 )
139: * ..
140: * .. External Subroutines ..
141: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGBMV, ZGBTRS, ZLACN2
142: * ..
143: * .. Intrinsic Functions ..
144: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
145: * ..
146: * .. External Functions ..
147: LOGICAL LSAME
148: DOUBLE PRECISION DLAMCH
149: EXTERNAL LSAME, DLAMCH
150: * ..
151: * .. Statement Functions ..
152: DOUBLE PRECISION CABS1
153: * ..
154: * .. Statement Function definitions ..
155: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
156: * ..
157: * .. Executable Statements ..
158: *
159: * Test the input parameters.
160: *
161: INFO = 0
162: NOTRAN = LSAME( TRANS, 'N' )
163: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
164: $ LSAME( TRANS, 'C' ) ) THEN
165: INFO = -1
166: ELSE IF( N.LT.0 ) THEN
167: INFO = -2
168: ELSE IF( KL.LT.0 ) THEN
169: INFO = -3
170: ELSE IF( KU.LT.0 ) THEN
171: INFO = -4
172: ELSE IF( NRHS.LT.0 ) THEN
173: INFO = -5
174: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
175: INFO = -7
176: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
177: INFO = -9
178: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
179: INFO = -12
180: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
181: INFO = -14
182: END IF
183: IF( INFO.NE.0 ) THEN
184: CALL XERBLA( 'ZGBRFS', -INFO )
185: RETURN
186: END IF
187: *
188: * Quick return if possible
189: *
190: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
191: DO 10 J = 1, NRHS
192: FERR( J ) = ZERO
193: BERR( J ) = ZERO
194: 10 CONTINUE
195: RETURN
196: END IF
197: *
198: IF( NOTRAN ) THEN
199: TRANSN = 'N'
200: TRANST = 'C'
201: ELSE
202: TRANSN = 'C'
203: TRANST = 'N'
204: END IF
205: *
206: * NZ = maximum number of nonzero elements in each row of A, plus 1
207: *
208: NZ = MIN( KL+KU+2, N+1 )
209: EPS = DLAMCH( 'Epsilon' )
210: SAFMIN = DLAMCH( 'Safe minimum' )
211: SAFE1 = NZ*SAFMIN
212: SAFE2 = SAFE1 / EPS
213: *
214: * Do for each right hand side
215: *
216: DO 140 J = 1, NRHS
217: *
218: COUNT = 1
219: LSTRES = THREE
220: 20 CONTINUE
221: *
222: * Loop until stopping criterion is satisfied.
223: *
224: * Compute residual R = B - op(A) * X,
225: * where op(A) = A, A**T, or A**H, depending on TRANS.
226: *
227: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
228: CALL ZGBMV( TRANS, N, N, KL, KU, -CONE, AB, LDAB, X( 1, J ), 1,
229: $ CONE, WORK, 1 )
230: *
231: * Compute componentwise relative backward error from formula
232: *
233: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
234: *
235: * where abs(Z) is the componentwise absolute value of the matrix
236: * or vector Z. If the i-th component of the denominator is less
237: * than SAFE2, then SAFE1 is added to the i-th components of the
238: * numerator and denominator before dividing.
239: *
240: DO 30 I = 1, N
241: RWORK( I ) = CABS1( B( I, J ) )
242: 30 CONTINUE
243: *
244: * Compute abs(op(A))*abs(X) + abs(B).
245: *
246: IF( NOTRAN ) THEN
247: DO 50 K = 1, N
248: KK = KU + 1 - K
249: XK = CABS1( X( K, J ) )
250: DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
251: RWORK( I ) = RWORK( I ) + CABS1( AB( KK+I, K ) )*XK
252: 40 CONTINUE
253: 50 CONTINUE
254: ELSE
255: DO 70 K = 1, N
256: S = ZERO
257: KK = KU + 1 - K
258: DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
259: S = S + CABS1( AB( KK+I, K ) )*CABS1( X( I, J ) )
260: 60 CONTINUE
261: RWORK( K ) = RWORK( K ) + S
262: 70 CONTINUE
263: END IF
264: S = ZERO
265: DO 80 I = 1, N
266: IF( RWORK( I ).GT.SAFE2 ) THEN
267: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
268: ELSE
269: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
270: $ ( RWORK( I )+SAFE1 ) )
271: END IF
272: 80 CONTINUE
273: BERR( J ) = S
274: *
275: * Test stopping criterion. Continue iterating if
276: * 1) The residual BERR(J) is larger than machine epsilon, and
277: * 2) BERR(J) decreased by at least a factor of 2 during the
278: * last iteration, and
279: * 3) At most ITMAX iterations tried.
280: *
281: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
282: $ COUNT.LE.ITMAX ) THEN
283: *
284: * Update solution and try again.
285: *
286: CALL ZGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, WORK, N,
287: $ INFO )
288: CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
289: LSTRES = BERR( J )
290: COUNT = COUNT + 1
291: GO TO 20
292: END IF
293: *
294: * Bound error from formula
295: *
296: * norm(X - XTRUE) / norm(X) .le. FERR =
297: * norm( abs(inv(op(A)))*
298: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
299: *
300: * where
301: * norm(Z) is the magnitude of the largest component of Z
302: * inv(op(A)) is the inverse of op(A)
303: * abs(Z) is the componentwise absolute value of the matrix or
304: * vector Z
305: * NZ is the maximum number of nonzeros in any row of A, plus 1
306: * EPS is machine epsilon
307: *
308: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
309: * is incremented by SAFE1 if the i-th component of
310: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
311: *
312: * Use ZLACN2 to estimate the infinity-norm of the matrix
313: * inv(op(A)) * diag(W),
314: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
315: *
316: DO 90 I = 1, N
317: IF( RWORK( I ).GT.SAFE2 ) THEN
318: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
319: ELSE
320: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
321: $ SAFE1
322: END IF
323: 90 CONTINUE
324: *
325: KASE = 0
326: 100 CONTINUE
327: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
328: IF( KASE.NE.0 ) THEN
329: IF( KASE.EQ.1 ) THEN
330: *
331: * Multiply by diag(W)*inv(op(A)**H).
332: *
333: CALL ZGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
334: $ WORK, N, INFO )
335: DO 110 I = 1, N
336: WORK( I ) = RWORK( I )*WORK( I )
337: 110 CONTINUE
338: ELSE
339: *
340: * Multiply by inv(op(A))*diag(W).
341: *
342: DO 120 I = 1, N
343: WORK( I ) = RWORK( I )*WORK( I )
344: 120 CONTINUE
345: CALL ZGBTRS( TRANSN, N, KL, KU, 1, AFB, LDAFB, IPIV,
346: $ WORK, N, INFO )
347: END IF
348: GO TO 100
349: END IF
350: *
351: * Normalize error.
352: *
353: LSTRES = ZERO
354: DO 130 I = 1, N
355: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
356: 130 CONTINUE
357: IF( LSTRES.NE.ZERO )
358: $ FERR( J ) = FERR( J ) / LSTRES
359: *
360: 140 CONTINUE
361: *
362: RETURN
363: *
364: * End of ZGBRFS
365: *
366: END
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