Annotation of rpl/lapack/lapack/zpprfs.f, revision 1.18
1.9 bertrand 1: *> \brief \b ZPPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.15 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.9 bertrand 7: *
8: *> \htmlonly
1.15 bertrand 9: *> Download ZPPRFS + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpprfs.f">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
22: * BERR, WORK, RWORK, INFO )
1.15 bertrand 23: *
1.9 bertrand 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: * ..
1.15 bertrand 33: *
1.9 bertrand 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: *
1.15 bertrand 161: *> \author Univ. of Tennessee
162: *> \author Univ. of California Berkeley
163: *> \author Univ. of Colorado Denver
164: *> \author NAG Ltd.
1.9 bertrand 165: *
166: *> \ingroup complex16OTHERcomputational
167: *
168: * =====================================================================
1.1 bertrand 169: SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
170: $ BERR, WORK, RWORK, INFO )
171: *
1.18 ! bertrand 172: * -- LAPACK computational routine --
1.1 bertrand 173: * -- LAPACK is a software package provided by Univ. of Tennessee, --
174: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175: *
176: * .. Scalar Arguments ..
177: CHARACTER UPLO
178: INTEGER INFO, LDB, LDX, N, NRHS
179: * ..
180: * .. Array Arguments ..
181: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
182: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
183: $ X( LDX, * )
184: * ..
185: *
186: * ====================================================================
187: *
188: * .. Parameters ..
189: INTEGER ITMAX
190: PARAMETER ( ITMAX = 5 )
191: DOUBLE PRECISION ZERO
192: PARAMETER ( ZERO = 0.0D+0 )
193: COMPLEX*16 CONE
194: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
195: DOUBLE PRECISION TWO
196: PARAMETER ( TWO = 2.0D+0 )
197: DOUBLE PRECISION THREE
198: PARAMETER ( THREE = 3.0D+0 )
199: * ..
200: * .. Local Scalars ..
201: LOGICAL UPPER
202: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
203: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
204: COMPLEX*16 ZDUM
205: * ..
206: * .. Local Arrays ..
207: INTEGER ISAVE( 3 )
208: * ..
209: * .. External Subroutines ..
210: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS
211: * ..
212: * .. Intrinsic Functions ..
213: INTRINSIC ABS, DBLE, DIMAG, MAX
214: * ..
215: * .. External Functions ..
216: LOGICAL LSAME
217: DOUBLE PRECISION DLAMCH
218: EXTERNAL LSAME, DLAMCH
219: * ..
220: * .. Statement Functions ..
221: DOUBLE PRECISION CABS1
222: * ..
223: * .. Statement Function definitions ..
224: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
225: * ..
226: * .. Executable Statements ..
227: *
228: * Test the input parameters.
229: *
230: INFO = 0
231: UPPER = LSAME( UPLO, 'U' )
232: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
233: INFO = -1
234: ELSE IF( N.LT.0 ) THEN
235: INFO = -2
236: ELSE IF( NRHS.LT.0 ) THEN
237: INFO = -3
238: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
239: INFO = -7
240: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
241: INFO = -9
242: END IF
243: IF( INFO.NE.0 ) THEN
244: CALL XERBLA( 'ZPPRFS', -INFO )
245: RETURN
246: END IF
247: *
248: * Quick return if possible
249: *
250: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
251: DO 10 J = 1, NRHS
252: FERR( J ) = ZERO
253: BERR( J ) = ZERO
254: 10 CONTINUE
255: RETURN
256: END IF
257: *
258: * NZ = maximum number of nonzero elements in each row of A, plus 1
259: *
260: NZ = N + 1
261: EPS = DLAMCH( 'Epsilon' )
262: SAFMIN = DLAMCH( 'Safe minimum' )
263: SAFE1 = NZ*SAFMIN
264: SAFE2 = SAFE1 / EPS
265: *
266: * Do for each right hand side
267: *
268: DO 140 J = 1, NRHS
269: *
270: COUNT = 1
271: LSTRES = THREE
272: 20 CONTINUE
273: *
274: * Loop until stopping criterion is satisfied.
275: *
276: * Compute residual R = B - A * X
277: *
278: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
279: CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 )
280: *
281: * Compute componentwise relative backward error from formula
282: *
283: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
284: *
285: * where abs(Z) is the componentwise absolute value of the matrix
286: * or vector Z. If the i-th component of the denominator is less
287: * than SAFE2, then SAFE1 is added to the i-th components of the
288: * numerator and denominator before dividing.
289: *
290: DO 30 I = 1, N
291: RWORK( I ) = CABS1( B( I, J ) )
292: 30 CONTINUE
293: *
294: * Compute abs(A)*abs(X) + abs(B).
295: *
296: KK = 1
297: IF( UPPER ) THEN
298: DO 50 K = 1, N
299: S = ZERO
300: XK = CABS1( X( K, J ) )
301: IK = KK
302: DO 40 I = 1, K - 1
303: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
304: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
305: IK = IK + 1
306: 40 CONTINUE
307: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
308: $ XK + S
309: KK = KK + K
310: 50 CONTINUE
311: ELSE
312: DO 70 K = 1, N
313: S = ZERO
314: XK = CABS1( X( K, J ) )
315: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
316: IK = KK + 1
317: DO 60 I = K + 1, N
318: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
319: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
320: IK = IK + 1
321: 60 CONTINUE
322: RWORK( K ) = RWORK( K ) + S
323: KK = KK + ( N-K+1 )
324: 70 CONTINUE
325: END IF
326: S = ZERO
327: DO 80 I = 1, N
328: IF( RWORK( I ).GT.SAFE2 ) THEN
329: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
330: ELSE
331: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
332: $ ( RWORK( I )+SAFE1 ) )
333: END IF
334: 80 CONTINUE
335: BERR( J ) = S
336: *
337: * Test stopping criterion. Continue iterating if
338: * 1) The residual BERR(J) is larger than machine epsilon, and
339: * 2) BERR(J) decreased by at least a factor of 2 during the
340: * last iteration, and
341: * 3) At most ITMAX iterations tried.
342: *
343: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
344: $ COUNT.LE.ITMAX ) THEN
345: *
346: * Update solution and try again.
347: *
348: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
349: CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
350: LSTRES = BERR( J )
351: COUNT = COUNT + 1
352: GO TO 20
353: END IF
354: *
355: * Bound error from formula
356: *
357: * norm(X - XTRUE) / norm(X) .le. FERR =
358: * norm( abs(inv(A))*
359: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
360: *
361: * where
362: * norm(Z) is the magnitude of the largest component of Z
363: * inv(A) is the inverse of A
364: * abs(Z) is the componentwise absolute value of the matrix or
365: * vector Z
366: * NZ is the maximum number of nonzeros in any row of A, plus 1
367: * EPS is machine epsilon
368: *
369: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
370: * is incremented by SAFE1 if the i-th component of
371: * abs(A)*abs(X) + abs(B) is less than SAFE2.
372: *
373: * Use ZLACN2 to estimate the infinity-norm of the matrix
374: * inv(A) * diag(W),
375: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
376: *
377: DO 90 I = 1, N
378: IF( RWORK( I ).GT.SAFE2 ) THEN
379: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
380: ELSE
381: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
382: $ SAFE1
383: END IF
384: 90 CONTINUE
385: *
386: KASE = 0
387: 100 CONTINUE
388: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
389: IF( KASE.NE.0 ) THEN
390: IF( KASE.EQ.1 ) THEN
391: *
1.8 bertrand 392: * Multiply by diag(W)*inv(A**H).
1.1 bertrand 393: *
394: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
395: DO 110 I = 1, N
396: WORK( I ) = RWORK( I )*WORK( I )
397: 110 CONTINUE
398: ELSE IF( KASE.EQ.2 ) THEN
399: *
400: * Multiply by inv(A)*diag(W).
401: *
402: DO 120 I = 1, N
403: WORK( I ) = RWORK( I )*WORK( I )
404: 120 CONTINUE
405: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
406: END IF
407: GO TO 100
408: END IF
409: *
410: * Normalize error.
411: *
412: LSTRES = ZERO
413: DO 130 I = 1, N
414: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
415: 130 CONTINUE
416: IF( LSTRES.NE.ZERO )
417: $ FERR( J ) = FERR( J ) / LSTRES
418: *
419: 140 CONTINUE
420: *
421: RETURN
422: *
423: * End of ZPPRFS
424: *
425: END
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