1: *> \brief \b ZPPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
22: * BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
30: * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
31: * $ X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZPPRFS improves the computed solution to a system of linear
41: *> equations when the coefficient matrix is Hermitian positive definite
42: *> and packed, and provides error bounds and backward error estimates
43: *> for the solution.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] UPLO
50: *> \verbatim
51: *> UPLO is CHARACTER*1
52: *> = 'U': Upper triangle of A is stored;
53: *> = 'L': Lower triangle of A is stored.
54: *> \endverbatim
55: *>
56: *> \param[in] N
57: *> \verbatim
58: *> N is INTEGER
59: *> The order of the matrix A. N >= 0.
60: *> \endverbatim
61: *>
62: *> \param[in] NRHS
63: *> \verbatim
64: *> NRHS is INTEGER
65: *> The number of right hand sides, i.e., the number of columns
66: *> of the matrices B and X. NRHS >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in] AP
70: *> \verbatim
71: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
72: *> The upper or lower triangle of the Hermitian matrix A, packed
73: *> columnwise in a linear array. The j-th column of A is stored
74: *> in the array AP as follows:
75: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
76: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
77: *> \endverbatim
78: *>
79: *> \param[in] AFP
80: *> \verbatim
81: *> AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
82: *> The triangular factor U or L from the Cholesky factorization
83: *> A = U**H*U or A = L*L**H, as computed by DPPTRF/ZPPTRF,
84: *> packed columnwise in a linear array in the same format as A
85: *> (see AP).
86: *> \endverbatim
87: *>
88: *> \param[in] B
89: *> \verbatim
90: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
91: *> The right hand side matrix B.
92: *> \endverbatim
93: *>
94: *> \param[in] LDB
95: *> \verbatim
96: *> LDB is INTEGER
97: *> The leading dimension of the array B. LDB >= max(1,N).
98: *> \endverbatim
99: *>
100: *> \param[in,out] X
101: *> \verbatim
102: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
103: *> On entry, the solution matrix X, as computed by ZPPTRS.
104: *> On exit, the improved solution matrix X.
105: *> \endverbatim
106: *>
107: *> \param[in] LDX
108: *> \verbatim
109: *> LDX is INTEGER
110: *> The leading dimension of the array X. LDX >= max(1,N).
111: *> \endverbatim
112: *>
113: *> \param[out] FERR
114: *> \verbatim
115: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
116: *> The estimated forward error bound for each solution vector
117: *> X(j) (the j-th column of the solution matrix X).
118: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
119: *> is an estimated upper bound for the magnitude of the largest
120: *> element in (X(j) - XTRUE) divided by the magnitude of the
121: *> largest element in X(j). The estimate is as reliable as
122: *> the estimate for RCOND, and is almost always a slight
123: *> overestimate of the true error.
124: *> \endverbatim
125: *>
126: *> \param[out] BERR
127: *> \verbatim
128: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
129: *> The componentwise relative backward error of each solution
130: *> vector X(j) (i.e., the smallest relative change in
131: *> any element of A or B that makes X(j) an exact solution).
132: *> \endverbatim
133: *>
134: *> \param[out] WORK
135: *> \verbatim
136: *> WORK is COMPLEX*16 array, dimension (2*N)
137: *> \endverbatim
138: *>
139: *> \param[out] RWORK
140: *> \verbatim
141: *> RWORK is DOUBLE PRECISION array, dimension (N)
142: *> \endverbatim
143: *>
144: *> \param[out] INFO
145: *> \verbatim
146: *> INFO is INTEGER
147: *> = 0: successful exit
148: *> < 0: if INFO = -i, the i-th argument had an illegal value
149: *> \endverbatim
150: *
151: *> \par Internal Parameters:
152: * =========================
153: *>
154: *> \verbatim
155: *> ITMAX is the maximum number of steps of iterative refinement.
156: *> \endverbatim
157: *
158: * Authors:
159: * ========
160: *
161: *> \author Univ. of Tennessee
162: *> \author Univ. of California Berkeley
163: *> \author Univ. of Colorado Denver
164: *> \author NAG Ltd.
165: *
166: *> \date November 2011
167: *
168: *> \ingroup complex16OTHERcomputational
169: *
170: * =====================================================================
171: SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
172: $ BERR, WORK, RWORK, INFO )
173: *
174: * -- LAPACK computational routine (version 3.4.0) --
175: * -- LAPACK is a software package provided by Univ. of Tennessee, --
176: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177: * November 2011
178: *
179: * .. Scalar Arguments ..
180: CHARACTER UPLO
181: INTEGER INFO, LDB, LDX, N, NRHS
182: * ..
183: * .. Array Arguments ..
184: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
185: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
186: $ X( LDX, * )
187: * ..
188: *
189: * ====================================================================
190: *
191: * .. Parameters ..
192: INTEGER ITMAX
193: PARAMETER ( ITMAX = 5 )
194: DOUBLE PRECISION ZERO
195: PARAMETER ( ZERO = 0.0D+0 )
196: COMPLEX*16 CONE
197: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
198: DOUBLE PRECISION TWO
199: PARAMETER ( TWO = 2.0D+0 )
200: DOUBLE PRECISION THREE
201: PARAMETER ( THREE = 3.0D+0 )
202: * ..
203: * .. Local Scalars ..
204: LOGICAL UPPER
205: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
206: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
207: COMPLEX*16 ZDUM
208: * ..
209: * .. Local Arrays ..
210: INTEGER ISAVE( 3 )
211: * ..
212: * .. External Subroutines ..
213: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS
214: * ..
215: * .. Intrinsic Functions ..
216: INTRINSIC ABS, DBLE, DIMAG, MAX
217: * ..
218: * .. External Functions ..
219: LOGICAL LSAME
220: DOUBLE PRECISION DLAMCH
221: EXTERNAL LSAME, DLAMCH
222: * ..
223: * .. Statement Functions ..
224: DOUBLE PRECISION CABS1
225: * ..
226: * .. Statement Function definitions ..
227: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
228: * ..
229: * .. Executable Statements ..
230: *
231: * Test the input parameters.
232: *
233: INFO = 0
234: UPPER = LSAME( UPLO, 'U' )
235: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
236: INFO = -1
237: ELSE IF( N.LT.0 ) THEN
238: INFO = -2
239: ELSE IF( NRHS.LT.0 ) THEN
240: INFO = -3
241: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
242: INFO = -7
243: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
244: INFO = -9
245: END IF
246: IF( INFO.NE.0 ) THEN
247: CALL XERBLA( 'ZPPRFS', -INFO )
248: RETURN
249: END IF
250: *
251: * Quick return if possible
252: *
253: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
254: DO 10 J = 1, NRHS
255: FERR( J ) = ZERO
256: BERR( J ) = ZERO
257: 10 CONTINUE
258: RETURN
259: END IF
260: *
261: * NZ = maximum number of nonzero elements in each row of A, plus 1
262: *
263: NZ = N + 1
264: EPS = DLAMCH( 'Epsilon' )
265: SAFMIN = DLAMCH( 'Safe minimum' )
266: SAFE1 = NZ*SAFMIN
267: SAFE2 = SAFE1 / EPS
268: *
269: * Do for each right hand side
270: *
271: DO 140 J = 1, NRHS
272: *
273: COUNT = 1
274: LSTRES = THREE
275: 20 CONTINUE
276: *
277: * Loop until stopping criterion is satisfied.
278: *
279: * Compute residual R = B - A * X
280: *
281: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
282: CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 )
283: *
284: * Compute componentwise relative backward error from formula
285: *
286: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
287: *
288: * where abs(Z) is the componentwise absolute value of the matrix
289: * or vector Z. If the i-th component of the denominator is less
290: * than SAFE2, then SAFE1 is added to the i-th components of the
291: * numerator and denominator before dividing.
292: *
293: DO 30 I = 1, N
294: RWORK( I ) = CABS1( B( I, J ) )
295: 30 CONTINUE
296: *
297: * Compute abs(A)*abs(X) + abs(B).
298: *
299: KK = 1
300: IF( UPPER ) THEN
301: DO 50 K = 1, N
302: S = ZERO
303: XK = CABS1( X( K, J ) )
304: IK = KK
305: DO 40 I = 1, K - 1
306: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
307: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
308: IK = IK + 1
309: 40 CONTINUE
310: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
311: $ XK + S
312: KK = KK + K
313: 50 CONTINUE
314: ELSE
315: DO 70 K = 1, N
316: S = ZERO
317: XK = CABS1( X( K, J ) )
318: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
319: IK = KK + 1
320: DO 60 I = K + 1, N
321: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
322: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
323: IK = IK + 1
324: 60 CONTINUE
325: RWORK( K ) = RWORK( K ) + S
326: KK = KK + ( N-K+1 )
327: 70 CONTINUE
328: END IF
329: S = ZERO
330: DO 80 I = 1, N
331: IF( RWORK( I ).GT.SAFE2 ) THEN
332: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
333: ELSE
334: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
335: $ ( RWORK( I )+SAFE1 ) )
336: END IF
337: 80 CONTINUE
338: BERR( J ) = S
339: *
340: * Test stopping criterion. Continue iterating if
341: * 1) The residual BERR(J) is larger than machine epsilon, and
342: * 2) BERR(J) decreased by at least a factor of 2 during the
343: * last iteration, and
344: * 3) At most ITMAX iterations tried.
345: *
346: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
347: $ COUNT.LE.ITMAX ) THEN
348: *
349: * Update solution and try again.
350: *
351: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
352: CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
353: LSTRES = BERR( J )
354: COUNT = COUNT + 1
355: GO TO 20
356: END IF
357: *
358: * Bound error from formula
359: *
360: * norm(X - XTRUE) / norm(X) .le. FERR =
361: * norm( abs(inv(A))*
362: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
363: *
364: * where
365: * norm(Z) is the magnitude of the largest component of Z
366: * inv(A) is the inverse of A
367: * abs(Z) is the componentwise absolute value of the matrix or
368: * vector Z
369: * NZ is the maximum number of nonzeros in any row of A, plus 1
370: * EPS is machine epsilon
371: *
372: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
373: * is incremented by SAFE1 if the i-th component of
374: * abs(A)*abs(X) + abs(B) is less than SAFE2.
375: *
376: * Use ZLACN2 to estimate the infinity-norm of the matrix
377: * inv(A) * diag(W),
378: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
379: *
380: DO 90 I = 1, N
381: IF( RWORK( I ).GT.SAFE2 ) THEN
382: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
383: ELSE
384: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
385: $ SAFE1
386: END IF
387: 90 CONTINUE
388: *
389: KASE = 0
390: 100 CONTINUE
391: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
392: IF( KASE.NE.0 ) THEN
393: IF( KASE.EQ.1 ) THEN
394: *
395: * Multiply by diag(W)*inv(A**H).
396: *
397: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
398: DO 110 I = 1, N
399: WORK( I ) = RWORK( I )*WORK( I )
400: 110 CONTINUE
401: ELSE IF( KASE.EQ.2 ) THEN
402: *
403: * Multiply by inv(A)*diag(W).
404: *
405: DO 120 I = 1, N
406: WORK( I ) = RWORK( I )*WORK( I )
407: 120 CONTINUE
408: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
409: END IF
410: GO TO 100
411: END IF
412: *
413: * Normalize error.
414: *
415: LSTRES = ZERO
416: DO 130 I = 1, N
417: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
418: 130 CONTINUE
419: IF( LSTRES.NE.ZERO )
420: $ FERR( J ) = FERR( J ) / LSTRES
421: *
422: 140 CONTINUE
423: *
424: RETURN
425: *
426: * End of ZPPRFS
427: *
428: END
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