Annotation of rpl/lapack/lapack/zporfs.f, revision 1.1.1.1
1.1 bertrand 1: SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
2: $ 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: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18: $ WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * ZPORFS improves the computed solution to a system of linear
25: * equations when the coefficient matrix is Hermitian 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) 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 triangular factor U or L from the Cholesky factorization
57: * A = U**H*U or A = L*L**H, as computed by ZPOTRF.
58: *
59: * LDAF (input) INTEGER
60: * The leading dimension of the array AF. LDAF >= max(1,N).
61: *
62: * B (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
69: * On entry, the solution matrix X, as computed by ZPOTRS.
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) COMPLEX*16 array, dimension (2*N)
91: *
92: * RWORK (workspace) DOUBLE PRECISION 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: COMPLEX*16 ONE
111: PARAMETER ( ONE = ( 1.0D+0, 0.0D+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: COMPLEX*16 ZDUM
122: * ..
123: * .. Local Arrays ..
124: INTEGER ISAVE( 3 )
125: * ..
126: * .. External Subroutines ..
127: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS
128: * ..
129: * .. Intrinsic Functions ..
130: INTRINSIC ABS, DBLE, DIMAG, MAX
131: * ..
132: * .. External Functions ..
133: LOGICAL LSAME
134: DOUBLE PRECISION DLAMCH
135: EXTERNAL LSAME, DLAMCH
136: * ..
137: * .. Statement Functions ..
138: DOUBLE PRECISION CABS1
139: * ..
140: * .. Statement Function definitions ..
141: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
142: * ..
143: * .. Executable Statements ..
144: *
145: * Test the input parameters.
146: *
147: INFO = 0
148: UPPER = LSAME( UPLO, 'U' )
149: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
150: INFO = -1
151: ELSE IF( N.LT.0 ) THEN
152: INFO = -2
153: ELSE IF( NRHS.LT.0 ) THEN
154: INFO = -3
155: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
156: INFO = -5
157: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
158: INFO = -7
159: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
160: INFO = -9
161: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
162: INFO = -11
163: END IF
164: IF( INFO.NE.0 ) THEN
165: CALL XERBLA( 'ZPORFS', -INFO )
166: RETURN
167: END IF
168: *
169: * Quick return if possible
170: *
171: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
172: DO 10 J = 1, NRHS
173: FERR( J ) = ZERO
174: BERR( J ) = ZERO
175: 10 CONTINUE
176: RETURN
177: END IF
178: *
179: * NZ = maximum number of nonzero elements in each row of A, plus 1
180: *
181: NZ = N + 1
182: EPS = DLAMCH( 'Epsilon' )
183: SAFMIN = DLAMCH( 'Safe minimum' )
184: SAFE1 = NZ*SAFMIN
185: SAFE2 = SAFE1 / EPS
186: *
187: * Do for each right hand side
188: *
189: DO 140 J = 1, NRHS
190: *
191: COUNT = 1
192: LSTRES = THREE
193: 20 CONTINUE
194: *
195: * Loop until stopping criterion is satisfied.
196: *
197: * Compute residual R = B - A * X
198: *
199: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
200: CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
201: *
202: * Compute componentwise relative backward error from formula
203: *
204: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
205: *
206: * where abs(Z) is the componentwise absolute value of the matrix
207: * or vector Z. If the i-th component of the denominator is less
208: * than SAFE2, then SAFE1 is added to the i-th components of the
209: * numerator and denominator before dividing.
210: *
211: DO 30 I = 1, N
212: RWORK( I ) = CABS1( B( I, J ) )
213: 30 CONTINUE
214: *
215: * Compute abs(A)*abs(X) + abs(B).
216: *
217: IF( UPPER ) THEN
218: DO 50 K = 1, N
219: S = ZERO
220: XK = CABS1( X( K, J ) )
221: DO 40 I = 1, K - 1
222: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
223: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
224: 40 CONTINUE
225: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
226: 50 CONTINUE
227: ELSE
228: DO 70 K = 1, N
229: S = ZERO
230: XK = CABS1( X( K, J ) )
231: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
232: DO 60 I = K + 1, N
233: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
234: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
235: 60 CONTINUE
236: RWORK( K ) = RWORK( K ) + S
237: 70 CONTINUE
238: END IF
239: S = ZERO
240: DO 80 I = 1, N
241: IF( RWORK( I ).GT.SAFE2 ) THEN
242: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
243: ELSE
244: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
245: $ ( RWORK( I )+SAFE1 ) )
246: END IF
247: 80 CONTINUE
248: BERR( J ) = S
249: *
250: * Test stopping criterion. Continue iterating if
251: * 1) The residual BERR(J) is larger than machine epsilon, and
252: * 2) BERR(J) decreased by at least a factor of 2 during the
253: * last iteration, and
254: * 3) At most ITMAX iterations tried.
255: *
256: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
257: $ COUNT.LE.ITMAX ) THEN
258: *
259: * Update solution and try again.
260: *
261: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
262: CALL ZAXPY( N, ONE, WORK, 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 ZLACN2 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( RWORK( I ).GT.SAFE2 ) THEN
292: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
293: ELSE
294: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
295: $ SAFE1
296: END IF
297: 90 CONTINUE
298: *
299: KASE = 0
300: 100 CONTINUE
301: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), 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 ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
308: DO 110 I = 1, N
309: WORK( I ) = RWORK( I )*WORK( I )
310: 110 CONTINUE
311: ELSE IF( KASE.EQ.2 ) THEN
312: *
313: * Multiply by inv(A)*diag(W).
314: *
315: DO 120 I = 1, N
316: WORK( I ) = RWORK( I )*WORK( I )
317: 120 CONTINUE
318: CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
319: END IF
320: GO TO 100
321: END IF
322: *
323: * Normalize error.
324: *
325: LSTRES = ZERO
326: DO 130 I = 1, N
327: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
328: 130 CONTINUE
329: IF( LSTRES.NE.ZERO )
330: $ FERR( J ) = FERR( J ) / LSTRES
331: *
332: 140 CONTINUE
333: *
334: RETURN
335: *
336: * End of ZPORFS
337: *
338: END
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