Annotation of rpl/lapack/lapack/zhprfs.f, revision 1.4
1.1 bertrand 1: SUBROUTINE ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
2: $ 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, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: INTEGER IPIV( * )
17: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
18: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
19: $ X( LDX, * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * ZHPRFS improves the computed solution to a system of linear
26: * equations when the coefficient matrix is Hermitian indefinite
27: * and packed, and provides error bounds and backward error estimates
28: * for the solution.
29: *
30: * Arguments
31: * =========
32: *
33: * UPLO (input) CHARACTER*1
34: * = 'U': Upper triangle of A is stored;
35: * = 'L': Lower triangle of A is stored.
36: *
37: * N (input) INTEGER
38: * The order of the matrix A. N >= 0.
39: *
40: * NRHS (input) INTEGER
41: * The number of right hand sides, i.e., the number of columns
42: * of the matrices B and X. NRHS >= 0.
43: *
44: * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
45: * The upper or lower triangle of the Hermitian matrix A, packed
46: * columnwise in a linear array. The j-th column of A is stored
47: * in the array AP as follows:
48: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
49: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
50: *
51: * AFP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
52: * The factored form of the matrix A. AFP contains the block
53: * diagonal matrix D and the multipliers used to obtain the
54: * factor U or L from the factorization A = U*D*U**H or
55: * A = L*D*L**H as computed by ZHPTRF, stored as a packed
56: * triangular matrix.
57: *
58: * IPIV (input) INTEGER array, dimension (N)
59: * Details of the interchanges and the block structure of D
60: * as determined by ZHPTRF.
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 ZHPTRS.
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, IK, J, K, KASE, KK, 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, ZHPMV, ZHPTRS, ZLACN2
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( LDB.LT.MAX( 1, N ) ) THEN
156: INFO = -8
157: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
158: INFO = -10
159: END IF
160: IF( INFO.NE.0 ) THEN
161: CALL XERBLA( 'ZHPRFS', -INFO )
162: RETURN
163: END IF
164: *
165: * Quick return if possible
166: *
167: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
168: DO 10 J = 1, NRHS
169: FERR( J ) = ZERO
170: BERR( J ) = ZERO
171: 10 CONTINUE
172: RETURN
173: END IF
174: *
175: * NZ = maximum number of nonzero elements in each row of A, plus 1
176: *
177: NZ = N + 1
178: EPS = DLAMCH( 'Epsilon' )
179: SAFMIN = DLAMCH( 'Safe minimum' )
180: SAFE1 = NZ*SAFMIN
181: SAFE2 = SAFE1 / EPS
182: *
183: * Do for each right hand side
184: *
185: DO 140 J = 1, NRHS
186: *
187: COUNT = 1
188: LSTRES = THREE
189: 20 CONTINUE
190: *
191: * Loop until stopping criterion is satisfied.
192: *
193: * Compute residual R = B - A * X
194: *
195: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
196: CALL ZHPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 )
197: *
198: * Compute componentwise relative backward error from formula
199: *
200: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
201: *
202: * where abs(Z) is the componentwise absolute value of the matrix
203: * or vector Z. If the i-th component of the denominator is less
204: * than SAFE2, then SAFE1 is added to the i-th components of the
205: * numerator and denominator before dividing.
206: *
207: DO 30 I = 1, N
208: RWORK( I ) = CABS1( B( I, J ) )
209: 30 CONTINUE
210: *
211: * Compute abs(A)*abs(X) + abs(B).
212: *
213: KK = 1
214: IF( UPPER ) THEN
215: DO 50 K = 1, N
216: S = ZERO
217: XK = CABS1( X( K, J ) )
218: IK = KK
219: DO 40 I = 1, K - 1
220: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
221: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
222: IK = IK + 1
223: 40 CONTINUE
224: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
225: $ XK + S
226: KK = KK + K
227: 50 CONTINUE
228: ELSE
229: DO 70 K = 1, N
230: S = ZERO
231: XK = CABS1( X( K, J ) )
232: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
233: IK = KK + 1
234: DO 60 I = K + 1, N
235: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
236: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
237: IK = IK + 1
238: 60 CONTINUE
239: RWORK( K ) = RWORK( K ) + S
240: KK = KK + ( N-K+1 )
241: 70 CONTINUE
242: END IF
243: S = ZERO
244: DO 80 I = 1, N
245: IF( RWORK( I ).GT.SAFE2 ) THEN
246: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
247: ELSE
248: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
249: $ ( RWORK( I )+SAFE1 ) )
250: END IF
251: 80 CONTINUE
252: BERR( J ) = S
253: *
254: * Test stopping criterion. Continue iterating if
255: * 1) The residual BERR(J) is larger than machine epsilon, and
256: * 2) BERR(J) decreased by at least a factor of 2 during the
257: * last iteration, and
258: * 3) At most ITMAX iterations tried.
259: *
260: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
261: $ COUNT.LE.ITMAX ) THEN
262: *
263: * Update solution and try again.
264: *
265: CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
266: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
267: LSTRES = BERR( J )
268: COUNT = COUNT + 1
269: GO TO 20
270: END IF
271: *
272: * Bound error from formula
273: *
274: * norm(X - XTRUE) / norm(X) .le. FERR =
275: * norm( abs(inv(A))*
276: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
277: *
278: * where
279: * norm(Z) is the magnitude of the largest component of Z
280: * inv(A) is the inverse of A
281: * abs(Z) is the componentwise absolute value of the matrix or
282: * vector Z
283: * NZ is the maximum number of nonzeros in any row of A, plus 1
284: * EPS is machine epsilon
285: *
286: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
287: * is incremented by SAFE1 if the i-th component of
288: * abs(A)*abs(X) + abs(B) is less than SAFE2.
289: *
290: * Use ZLACN2 to estimate the infinity-norm of the matrix
291: * inv(A) * diag(W),
292: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
293: *
294: DO 90 I = 1, N
295: IF( RWORK( I ).GT.SAFE2 ) THEN
296: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
297: ELSE
298: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
299: $ SAFE1
300: END IF
301: 90 CONTINUE
302: *
303: KASE = 0
304: 100 CONTINUE
305: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
306: IF( KASE.NE.0 ) THEN
307: IF( KASE.EQ.1 ) THEN
308: *
309: * Multiply by diag(W)*inv(A').
310: *
311: CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
312: DO 110 I = 1, N
313: WORK( I ) = RWORK( I )*WORK( I )
314: 110 CONTINUE
315: ELSE IF( KASE.EQ.2 ) THEN
316: *
317: * Multiply by inv(A)*diag(W).
318: *
319: DO 120 I = 1, N
320: WORK( I ) = RWORK( I )*WORK( I )
321: 120 CONTINUE
322: CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, 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, CABS1( 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 ZHPRFS
341: *
342: END
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