1: *> \brief \b ZSPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZSPRFS + dependencies
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
22: * FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * )
30: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
31: * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
32: * $ X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZSPRFS improves the computed solution to a system of linear
42: *> equations when the coefficient matrix is symmetric indefinite
43: *> and packed, and provides error bounds and backward error estimates
44: *> for the solution.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] UPLO
51: *> \verbatim
52: *> UPLO is CHARACTER*1
53: *> = 'U': Upper triangle of A is stored;
54: *> = 'L': Lower triangle of A is stored.
55: *> \endverbatim
56: *>
57: *> \param[in] N
58: *> \verbatim
59: *> N is INTEGER
60: *> The order of the matrix A. N >= 0.
61: *> \endverbatim
62: *>
63: *> \param[in] NRHS
64: *> \verbatim
65: *> NRHS is INTEGER
66: *> The number of right hand sides, i.e., the number of columns
67: *> of the matrices B and X. NRHS >= 0.
68: *> \endverbatim
69: *>
70: *> \param[in] AP
71: *> \verbatim
72: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
73: *> The upper or lower triangle of the symmetric matrix A, packed
74: *> columnwise in a linear array. The j-th column of A is stored
75: *> in the array AP as follows:
76: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
77: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
78: *> \endverbatim
79: *>
80: *> \param[in] AFP
81: *> \verbatim
82: *> AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
83: *> The factored form of the matrix A. AFP contains the block
84: *> diagonal matrix D and the multipliers used to obtain the
85: *> factor U or L from the factorization A = U*D*U**T or
86: *> A = L*D*L**T as computed by ZSPTRF, stored as a packed
87: *> triangular matrix.
88: *> \endverbatim
89: *>
90: *> \param[in] IPIV
91: *> \verbatim
92: *> IPIV is INTEGER array, dimension (N)
93: *> Details of the interchanges and the block structure of D
94: *> as determined by ZSPTRF.
95: *> \endverbatim
96: *>
97: *> \param[in] B
98: *> \verbatim
99: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
100: *> The right hand side matrix B.
101: *> \endverbatim
102: *>
103: *> \param[in] LDB
104: *> \verbatim
105: *> LDB is INTEGER
106: *> The leading dimension of the array B. LDB >= max(1,N).
107: *> \endverbatim
108: *>
109: *> \param[in,out] X
110: *> \verbatim
111: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
112: *> On entry, the solution matrix X, as computed by ZSPTRS.
113: *> On exit, the improved solution matrix X.
114: *> \endverbatim
115: *>
116: *> \param[in] LDX
117: *> \verbatim
118: *> LDX is INTEGER
119: *> The leading dimension of the array X. LDX >= max(1,N).
120: *> \endverbatim
121: *>
122: *> \param[out] FERR
123: *> \verbatim
124: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
125: *> The estimated forward error bound for each solution vector
126: *> X(j) (the j-th column of the solution matrix X).
127: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
128: *> is an estimated upper bound for the magnitude of the largest
129: *> element in (X(j) - XTRUE) divided by the magnitude of the
130: *> largest element in X(j). The estimate is as reliable as
131: *> the estimate for RCOND, and is almost always a slight
132: *> overestimate of the true error.
133: *> \endverbatim
134: *>
135: *> \param[out] BERR
136: *> \verbatim
137: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
138: *> The componentwise relative backward error of each solution
139: *> vector X(j) (i.e., the smallest relative change in
140: *> any element of A or B that makes X(j) an exact solution).
141: *> \endverbatim
142: *>
143: *> \param[out] WORK
144: *> \verbatim
145: *> WORK is COMPLEX*16 array, dimension (2*N)
146: *> \endverbatim
147: *>
148: *> \param[out] RWORK
149: *> \verbatim
150: *> RWORK is DOUBLE PRECISION array, dimension (N)
151: *> \endverbatim
152: *>
153: *> \param[out] INFO
154: *> \verbatim
155: *> INFO is INTEGER
156: *> = 0: successful exit
157: *> < 0: if INFO = -i, the i-th argument had an illegal value
158: *> \endverbatim
159: *
160: *> \par Internal Parameters:
161: * =========================
162: *>
163: *> \verbatim
164: *> ITMAX is the maximum number of steps of iterative refinement.
165: *> \endverbatim
166: *
167: * Authors:
168: * ========
169: *
170: *> \author Univ. of Tennessee
171: *> \author Univ. of California Berkeley
172: *> \author Univ. of Colorado Denver
173: *> \author NAG Ltd.
174: *
175: *> \ingroup complex16OTHERcomputational
176: *
177: * =====================================================================
178: SUBROUTINE ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
179: $ FERR, BERR, WORK, RWORK, INFO )
180: *
181: * -- LAPACK computational routine --
182: * -- LAPACK is a software package provided by Univ. of Tennessee, --
183: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
184: *
185: * .. Scalar Arguments ..
186: CHARACTER UPLO
187: INTEGER INFO, LDB, LDX, N, NRHS
188: * ..
189: * .. Array Arguments ..
190: INTEGER IPIV( * )
191: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
192: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
193: $ X( LDX, * )
194: * ..
195: *
196: * =====================================================================
197: *
198: * .. Parameters ..
199: INTEGER ITMAX
200: PARAMETER ( ITMAX = 5 )
201: DOUBLE PRECISION ZERO
202: PARAMETER ( ZERO = 0.0D+0 )
203: COMPLEX*16 ONE
204: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
205: DOUBLE PRECISION TWO
206: PARAMETER ( TWO = 2.0D+0 )
207: DOUBLE PRECISION THREE
208: PARAMETER ( THREE = 3.0D+0 )
209: * ..
210: * .. Local Scalars ..
211: LOGICAL UPPER
212: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
213: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
214: COMPLEX*16 ZDUM
215: * ..
216: * .. Local Arrays ..
217: INTEGER ISAVE( 3 )
218: * ..
219: * .. External Subroutines ..
220: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSPMV, ZSPTRS
221: * ..
222: * .. Intrinsic Functions ..
223: INTRINSIC ABS, DBLE, DIMAG, MAX
224: * ..
225: * .. External Functions ..
226: LOGICAL LSAME
227: DOUBLE PRECISION DLAMCH
228: EXTERNAL LSAME, DLAMCH
229: * ..
230: * .. Statement Functions ..
231: DOUBLE PRECISION CABS1
232: * ..
233: * .. Statement Function definitions ..
234: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
235: * ..
236: * .. Executable Statements ..
237: *
238: * Test the input parameters.
239: *
240: INFO = 0
241: UPPER = LSAME( UPLO, 'U' )
242: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
243: INFO = -1
244: ELSE IF( N.LT.0 ) THEN
245: INFO = -2
246: ELSE IF( NRHS.LT.0 ) THEN
247: INFO = -3
248: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
249: INFO = -8
250: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
251: INFO = -10
252: END IF
253: IF( INFO.NE.0 ) THEN
254: CALL XERBLA( 'ZSPRFS', -INFO )
255: RETURN
256: END IF
257: *
258: * Quick return if possible
259: *
260: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
261: DO 10 J = 1, NRHS
262: FERR( J ) = ZERO
263: BERR( J ) = ZERO
264: 10 CONTINUE
265: RETURN
266: END IF
267: *
268: * NZ = maximum number of nonzero elements in each row of A, plus 1
269: *
270: NZ = N + 1
271: EPS = DLAMCH( 'Epsilon' )
272: SAFMIN = DLAMCH( 'Safe minimum' )
273: SAFE1 = NZ*SAFMIN
274: SAFE2 = SAFE1 / EPS
275: *
276: * Do for each right hand side
277: *
278: DO 140 J = 1, NRHS
279: *
280: COUNT = 1
281: LSTRES = THREE
282: 20 CONTINUE
283: *
284: * Loop until stopping criterion is satisfied.
285: *
286: * Compute residual R = B - A * X
287: *
288: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
289: CALL ZSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 )
290: *
291: * Compute componentwise relative backward error from formula
292: *
293: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
294: *
295: * where abs(Z) is the componentwise absolute value of the matrix
296: * or vector Z. If the i-th component of the denominator is less
297: * than SAFE2, then SAFE1 is added to the i-th components of the
298: * numerator and denominator before dividing.
299: *
300: DO 30 I = 1, N
301: RWORK( I ) = CABS1( B( I, J ) )
302: 30 CONTINUE
303: *
304: * Compute abs(A)*abs(X) + abs(B).
305: *
306: KK = 1
307: IF( UPPER ) THEN
308: DO 50 K = 1, N
309: S = ZERO
310: XK = CABS1( X( K, J ) )
311: IK = KK
312: DO 40 I = 1, K - 1
313: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
314: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
315: IK = IK + 1
316: 40 CONTINUE
317: RWORK( K ) = RWORK( K ) + CABS1( AP( KK+K-1 ) )*XK + S
318: KK = KK + K
319: 50 CONTINUE
320: ELSE
321: DO 70 K = 1, N
322: S = ZERO
323: XK = CABS1( X( K, J ) )
324: RWORK( K ) = RWORK( K ) + CABS1( AP( KK ) )*XK
325: IK = KK + 1
326: DO 60 I = K + 1, N
327: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
328: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
329: IK = IK + 1
330: 60 CONTINUE
331: RWORK( K ) = RWORK( K ) + S
332: KK = KK + ( N-K+1 )
333: 70 CONTINUE
334: END IF
335: S = ZERO
336: DO 80 I = 1, N
337: IF( RWORK( I ).GT.SAFE2 ) THEN
338: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
339: ELSE
340: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
341: $ ( RWORK( I )+SAFE1 ) )
342: END IF
343: 80 CONTINUE
344: BERR( J ) = S
345: *
346: * Test stopping criterion. Continue iterating if
347: * 1) The residual BERR(J) is larger than machine epsilon, and
348: * 2) BERR(J) decreased by at least a factor of 2 during the
349: * last iteration, and
350: * 3) At most ITMAX iterations tried.
351: *
352: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
353: $ COUNT.LE.ITMAX ) THEN
354: *
355: * Update solution and try again.
356: *
357: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
358: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
359: LSTRES = BERR( J )
360: COUNT = COUNT + 1
361: GO TO 20
362: END IF
363: *
364: * Bound error from formula
365: *
366: * norm(X - XTRUE) / norm(X) .le. FERR =
367: * norm( abs(inv(A))*
368: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
369: *
370: * where
371: * norm(Z) is the magnitude of the largest component of Z
372: * inv(A) is the inverse of A
373: * abs(Z) is the componentwise absolute value of the matrix or
374: * vector Z
375: * NZ is the maximum number of nonzeros in any row of A, plus 1
376: * EPS is machine epsilon
377: *
378: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
379: * is incremented by SAFE1 if the i-th component of
380: * abs(A)*abs(X) + abs(B) is less than SAFE2.
381: *
382: * Use ZLACN2 to estimate the infinity-norm of the matrix
383: * inv(A) * diag(W),
384: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
385: *
386: DO 90 I = 1, N
387: IF( RWORK( I ).GT.SAFE2 ) THEN
388: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
389: ELSE
390: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
391: $ SAFE1
392: END IF
393: 90 CONTINUE
394: *
395: KASE = 0
396: 100 CONTINUE
397: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
398: IF( KASE.NE.0 ) THEN
399: IF( KASE.EQ.1 ) THEN
400: *
401: * Multiply by diag(W)*inv(A**T).
402: *
403: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
404: DO 110 I = 1, N
405: WORK( I ) = RWORK( I )*WORK( I )
406: 110 CONTINUE
407: ELSE IF( KASE.EQ.2 ) THEN
408: *
409: * Multiply by inv(A)*diag(W).
410: *
411: DO 120 I = 1, N
412: WORK( I ) = RWORK( I )*WORK( I )
413: 120 CONTINUE
414: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
415: END IF
416: GO TO 100
417: END IF
418: *
419: * Normalize error.
420: *
421: LSTRES = ZERO
422: DO 130 I = 1, N
423: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
424: 130 CONTINUE
425: IF( LSTRES.NE.ZERO )
426: $ FERR( J ) = FERR( J ) / LSTRES
427: *
428: 140 CONTINUE
429: *
430: RETURN
431: *
432: * End of ZSPRFS
433: *
434: END
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