1: SUBROUTINE ZHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
2: $ X, LDX, FERR, BERR, WORK, RWORK, 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 ZLACN2 in place of ZLACON, 10 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( * )
17: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
18: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
19: $ WORK( * ), X( LDX, * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * ZHERFS improves the computed solution to a system of linear
26: * equations when the coefficient matrix is Hermitian indefinite, and
27: * provides error bounds and backward error estimates 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: * 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) COMPLEX*16 array, dimension (LDA,N)
44: * The Hermitian 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) COMPLEX*16 array, dimension (LDAF,N)
56: * The factored form of the matrix A. AF contains the block
57: * diagonal matrix D and the multipliers used to obtain the
58: * factor U or L from the factorization A = U*D*U**H or
59: * A = L*D*L**H as computed by ZHETRF.
60: *
61: * LDAF (input) INTEGER
62: * The leading dimension of the array AF. LDAF >= max(1,N).
63: *
64: * IPIV (input) INTEGER array, dimension (N)
65: * Details of the interchanges and the block structure of D
66: * as determined by ZHETRF.
67: *
68: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
69: * The right hand side matrix B.
70: *
71: * LDB (input) INTEGER
72: * The leading dimension of the array B. LDB >= max(1,N).
73: *
74: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
75: * On entry, the solution matrix X, as computed by ZHETRS.
76: * On exit, the improved solution matrix X.
77: *
78: * LDX (input) INTEGER
79: * The leading dimension of the array X. LDX >= max(1,N).
80: *
81: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
82: * The estimated forward error bound for each solution vector
83: * X(j) (the j-th column of the solution matrix X).
84: * If XTRUE is the true solution corresponding to X(j), FERR(j)
85: * is an estimated upper bound for the magnitude of the largest
86: * element in (X(j) - XTRUE) divided by the magnitude of the
87: * largest element in X(j). The estimate is as reliable as
88: * the estimate for RCOND, and is almost always a slight
89: * overestimate of the true error.
90: *
91: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
92: * The componentwise relative backward error of each solution
93: * vector X(j) (i.e., the smallest relative change in
94: * any element of A or B that makes X(j) an exact solution).
95: *
96: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
97: *
98: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
99: *
100: * INFO (output) INTEGER
101: * = 0: successful exit
102: * < 0: if INFO = -i, the i-th argument had an illegal value
103: *
104: * Internal Parameters
105: * ===================
106: *
107: * ITMAX is the maximum number of steps of iterative refinement.
108: *
109: * =====================================================================
110: *
111: * .. Parameters ..
112: INTEGER ITMAX
113: PARAMETER ( ITMAX = 5 )
114: DOUBLE PRECISION ZERO
115: PARAMETER ( ZERO = 0.0D+0 )
116: COMPLEX*16 ONE
117: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
118: DOUBLE PRECISION TWO
119: PARAMETER ( TWO = 2.0D+0 )
120: DOUBLE PRECISION THREE
121: PARAMETER ( THREE = 3.0D+0 )
122: * ..
123: * .. Local Scalars ..
124: LOGICAL UPPER
125: INTEGER COUNT, I, J, K, KASE, NZ
126: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
127: COMPLEX*16 ZDUM
128: * ..
129: * .. Local Arrays ..
130: INTEGER ISAVE( 3 )
131: * ..
132: * .. External Subroutines ..
133: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZHETRS, ZLACN2
134: * ..
135: * .. Intrinsic Functions ..
136: INTRINSIC ABS, DBLE, DIMAG, MAX
137: * ..
138: * .. External Functions ..
139: LOGICAL LSAME
140: DOUBLE PRECISION DLAMCH
141: EXTERNAL LSAME, DLAMCH
142: * ..
143: * .. Statement Functions ..
144: DOUBLE PRECISION CABS1
145: * ..
146: * .. Statement Function definitions ..
147: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
148: * ..
149: * .. Executable Statements ..
150: *
151: * Test the input parameters.
152: *
153: INFO = 0
154: UPPER = LSAME( UPLO, 'U' )
155: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
156: INFO = -1
157: ELSE IF( N.LT.0 ) THEN
158: INFO = -2
159: ELSE IF( NRHS.LT.0 ) THEN
160: INFO = -3
161: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
162: INFO = -5
163: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
164: INFO = -7
165: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
166: INFO = -10
167: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
168: INFO = -12
169: END IF
170: IF( INFO.NE.0 ) THEN
171: CALL XERBLA( 'ZHERFS', -INFO )
172: RETURN
173: END IF
174: *
175: * Quick return if possible
176: *
177: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
178: DO 10 J = 1, NRHS
179: FERR( J ) = ZERO
180: BERR( J ) = ZERO
181: 10 CONTINUE
182: RETURN
183: END IF
184: *
185: * NZ = maximum number of nonzero elements in each row of A, plus 1
186: *
187: NZ = N + 1
188: EPS = DLAMCH( 'Epsilon' )
189: SAFMIN = DLAMCH( 'Safe minimum' )
190: SAFE1 = NZ*SAFMIN
191: SAFE2 = SAFE1 / EPS
192: *
193: * Do for each right hand side
194: *
195: DO 140 J = 1, NRHS
196: *
197: COUNT = 1
198: LSTRES = THREE
199: 20 CONTINUE
200: *
201: * Loop until stopping criterion is satisfied.
202: *
203: * Compute residual R = B - A * X
204: *
205: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
206: CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
207: *
208: * Compute componentwise relative backward error from formula
209: *
210: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
211: *
212: * where abs(Z) is the componentwise absolute value of the matrix
213: * or vector Z. If the i-th component of the denominator is less
214: * than SAFE2, then SAFE1 is added to the i-th components of the
215: * numerator and denominator before dividing.
216: *
217: DO 30 I = 1, N
218: RWORK( I ) = CABS1( B( I, J ) )
219: 30 CONTINUE
220: *
221: * Compute abs(A)*abs(X) + abs(B).
222: *
223: IF( UPPER ) THEN
224: DO 50 K = 1, N
225: S = ZERO
226: XK = CABS1( X( K, J ) )
227: DO 40 I = 1, K - 1
228: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
229: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
230: 40 CONTINUE
231: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
232: 50 CONTINUE
233: ELSE
234: DO 70 K = 1, N
235: S = ZERO
236: XK = CABS1( X( K, J ) )
237: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
238: DO 60 I = K + 1, N
239: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
240: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
241: 60 CONTINUE
242: RWORK( K ) = RWORK( K ) + S
243: 70 CONTINUE
244: END IF
245: S = ZERO
246: DO 80 I = 1, N
247: IF( RWORK( I ).GT.SAFE2 ) THEN
248: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
249: ELSE
250: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
251: $ ( RWORK( I )+SAFE1 ) )
252: END IF
253: 80 CONTINUE
254: BERR( J ) = S
255: *
256: * Test stopping criterion. Continue iterating if
257: * 1) The residual BERR(J) is larger than machine epsilon, and
258: * 2) BERR(J) decreased by at least a factor of 2 during the
259: * last iteration, and
260: * 3) At most ITMAX iterations tried.
261: *
262: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
263: $ COUNT.LE.ITMAX ) THEN
264: *
265: * Update solution and try again.
266: *
267: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
268: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
269: LSTRES = BERR( J )
270: COUNT = COUNT + 1
271: GO TO 20
272: END IF
273: *
274: * Bound error from formula
275: *
276: * norm(X - XTRUE) / norm(X) .le. FERR =
277: * norm( abs(inv(A))*
278: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
279: *
280: * where
281: * norm(Z) is the magnitude of the largest component of Z
282: * inv(A) is the inverse of A
283: * abs(Z) is the componentwise absolute value of the matrix or
284: * vector Z
285: * NZ is the maximum number of nonzeros in any row of A, plus 1
286: * EPS is machine epsilon
287: *
288: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
289: * is incremented by SAFE1 if the i-th component of
290: * abs(A)*abs(X) + abs(B) is less than SAFE2.
291: *
292: * Use ZLACN2 to estimate the infinity-norm of the matrix
293: * inv(A) * diag(W),
294: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
295: *
296: DO 90 I = 1, N
297: IF( RWORK( I ).GT.SAFE2 ) THEN
298: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
299: ELSE
300: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
301: $ SAFE1
302: END IF
303: 90 CONTINUE
304: *
305: KASE = 0
306: 100 CONTINUE
307: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
308: IF( KASE.NE.0 ) THEN
309: IF( KASE.EQ.1 ) THEN
310: *
311: * Multiply by diag(W)*inv(A').
312: *
313: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
314: DO 110 I = 1, N
315: WORK( I ) = RWORK( I )*WORK( I )
316: 110 CONTINUE
317: ELSE IF( KASE.EQ.2 ) THEN
318: *
319: * Multiply by inv(A)*diag(W).
320: *
321: DO 120 I = 1, N
322: WORK( I ) = RWORK( I )*WORK( I )
323: 120 CONTINUE
324: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
325: END IF
326: GO TO 100
327: END IF
328: *
329: * Normalize error.
330: *
331: LSTRES = ZERO
332: DO 130 I = 1, N
333: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
334: 130 CONTINUE
335: IF( LSTRES.NE.ZERO )
336: $ FERR( J ) = FERR( J ) / LSTRES
337: *
338: 140 CONTINUE
339: *
340: RETURN
341: *
342: * End of ZHERFS
343: *
344: END
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