1: *> \brief \b ZTPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZTPRFS + 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/ztprfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
22: * FERR, BERR, WORK, RWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER DIAG, TRANS, UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
30: * COMPLEX*16 AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZTPRFS provides error bounds and backward error estimates for the
40: *> solution to a system of linear equations with a triangular packed
41: *> coefficient matrix.
42: *>
43: *> The solution matrix X must be computed by ZTPTRS or some other
44: *> means before entering this routine. ZTPRFS does not do iterative
45: *> refinement because doing so cannot improve the backward error.
46: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] UPLO
52: *> \verbatim
53: *> UPLO is CHARACTER*1
54: *> = 'U': A is upper triangular;
55: *> = 'L': A is lower triangular.
56: *> \endverbatim
57: *>
58: *> \param[in] TRANS
59: *> \verbatim
60: *> TRANS is CHARACTER*1
61: *> Specifies the form of the system of equations:
62: *> = 'N': A * X = B (No transpose)
63: *> = 'T': A**T * X = B (Transpose)
64: *> = 'C': A**H * X = B (Conjugate transpose)
65: *> \endverbatim
66: *>
67: *> \param[in] DIAG
68: *> \verbatim
69: *> DIAG is CHARACTER*1
70: *> = 'N': A is non-unit triangular;
71: *> = 'U': A is unit triangular.
72: *> \endverbatim
73: *>
74: *> \param[in] N
75: *> \verbatim
76: *> N is INTEGER
77: *> The order of the matrix A. N >= 0.
78: *> \endverbatim
79: *>
80: *> \param[in] NRHS
81: *> \verbatim
82: *> NRHS is INTEGER
83: *> The number of right hand sides, i.e., the number of columns
84: *> of the matrices B and X. NRHS >= 0.
85: *> \endverbatim
86: *>
87: *> \param[in] AP
88: *> \verbatim
89: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
90: *> The upper or lower triangular matrix A, packed columnwise in
91: *> a linear array. The j-th column of A is stored in the array
92: *> AP as follows:
93: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
94: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
95: *> If DIAG = 'U', the diagonal elements of A are not referenced
96: *> and are assumed to be 1.
97: *> \endverbatim
98: *>
99: *> \param[in] B
100: *> \verbatim
101: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
102: *> The right hand side matrix B.
103: *> \endverbatim
104: *>
105: *> \param[in] LDB
106: *> \verbatim
107: *> LDB is INTEGER
108: *> The leading dimension of the array B. LDB >= max(1,N).
109: *> \endverbatim
110: *>
111: *> \param[in] X
112: *> \verbatim
113: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
114: *> The solution matrix X.
115: *> \endverbatim
116: *>
117: *> \param[in] LDX
118: *> \verbatim
119: *> LDX is INTEGER
120: *> The leading dimension of the array X. LDX >= max(1,N).
121: *> \endverbatim
122: *>
123: *> \param[out] FERR
124: *> \verbatim
125: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
126: *> The estimated forward error bound for each solution vector
127: *> X(j) (the j-th column of the solution matrix X).
128: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
129: *> is an estimated upper bound for the magnitude of the largest
130: *> element in (X(j) - XTRUE) divided by the magnitude of the
131: *> largest element in X(j). The estimate is as reliable as
132: *> the estimate for RCOND, and is almost always a slight
133: *> overestimate of the true error.
134: *> \endverbatim
135: *>
136: *> \param[out] BERR
137: *> \verbatim
138: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
139: *> The componentwise relative backward error of each solution
140: *> vector X(j) (i.e., the smallest relative change in
141: *> any element of A or B that makes X(j) an exact solution).
142: *> \endverbatim
143: *>
144: *> \param[out] WORK
145: *> \verbatim
146: *> WORK is COMPLEX*16 array, dimension (2*N)
147: *> \endverbatim
148: *>
149: *> \param[out] RWORK
150: *> \verbatim
151: *> RWORK is DOUBLE PRECISION array, dimension (N)
152: *> \endverbatim
153: *>
154: *> \param[out] INFO
155: *> \verbatim
156: *> INFO is INTEGER
157: *> = 0: successful exit
158: *> < 0: if INFO = -i, the i-th argument had an illegal value
159: *> \endverbatim
160: *
161: * Authors:
162: * ========
163: *
164: *> \author Univ. of Tennessee
165: *> \author Univ. of California Berkeley
166: *> \author Univ. of Colorado Denver
167: *> \author NAG Ltd.
168: *
169: *> \ingroup complex16OTHERcomputational
170: *
171: * =====================================================================
172: SUBROUTINE ZTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
173: $ FERR, BERR, WORK, RWORK, INFO )
174: *
175: * -- LAPACK computational routine --
176: * -- LAPACK is a software package provided by Univ. of Tennessee, --
177: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
178: *
179: * .. Scalar Arguments ..
180: CHARACTER DIAG, TRANS, UPLO
181: INTEGER INFO, LDB, LDX, N, NRHS
182: * ..
183: * .. Array Arguments ..
184: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
185: COMPLEX*16 AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
186: * ..
187: *
188: * =====================================================================
189: *
190: * .. Parameters ..
191: DOUBLE PRECISION ZERO
192: PARAMETER ( ZERO = 0.0D+0 )
193: COMPLEX*16 ONE
194: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
195: * ..
196: * .. Local Scalars ..
197: LOGICAL NOTRAN, NOUNIT, UPPER
198: CHARACTER TRANSN, TRANST
199: INTEGER I, J, K, KASE, KC, NZ
200: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
201: COMPLEX*16 ZDUM
202: * ..
203: * .. Local Arrays ..
204: INTEGER ISAVE( 3 )
205: * ..
206: * .. External Subroutines ..
207: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTPMV, ZTPSV
208: * ..
209: * .. Intrinsic Functions ..
210: INTRINSIC ABS, DBLE, DIMAG, MAX
211: * ..
212: * .. External Functions ..
213: LOGICAL LSAME
214: DOUBLE PRECISION DLAMCH
215: EXTERNAL LSAME, DLAMCH
216: * ..
217: * .. Statement Functions ..
218: DOUBLE PRECISION CABS1
219: * ..
220: * .. Statement Function definitions ..
221: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
222: * ..
223: * .. Executable Statements ..
224: *
225: * Test the input parameters.
226: *
227: INFO = 0
228: UPPER = LSAME( UPLO, 'U' )
229: NOTRAN = LSAME( TRANS, 'N' )
230: NOUNIT = LSAME( DIAG, 'N' )
231: *
232: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
233: INFO = -1
234: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
235: $ LSAME( TRANS, 'C' ) ) THEN
236: INFO = -2
237: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
238: INFO = -3
239: ELSE IF( N.LT.0 ) THEN
240: INFO = -4
241: ELSE IF( NRHS.LT.0 ) THEN
242: INFO = -5
243: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
244: INFO = -8
245: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
246: INFO = -10
247: END IF
248: IF( INFO.NE.0 ) THEN
249: CALL XERBLA( 'ZTPRFS', -INFO )
250: RETURN
251: END IF
252: *
253: * Quick return if possible
254: *
255: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
256: DO 10 J = 1, NRHS
257: FERR( J ) = ZERO
258: BERR( J ) = ZERO
259: 10 CONTINUE
260: RETURN
261: END IF
262: *
263: IF( NOTRAN ) THEN
264: TRANSN = 'N'
265: TRANST = 'C'
266: ELSE
267: TRANSN = 'C'
268: TRANST = 'N'
269: END IF
270: *
271: * NZ = maximum number of nonzero elements in each row of A, plus 1
272: *
273: NZ = N + 1
274: EPS = DLAMCH( 'Epsilon' )
275: SAFMIN = DLAMCH( 'Safe minimum' )
276: SAFE1 = NZ*SAFMIN
277: SAFE2 = SAFE1 / EPS
278: *
279: * Do for each right hand side
280: *
281: DO 250 J = 1, NRHS
282: *
283: * Compute residual R = B - op(A) * X,
284: * where op(A) = A, A**T, or A**H, depending on TRANS.
285: *
286: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
287: CALL ZTPMV( UPLO, TRANS, DIAG, N, AP, WORK, 1 )
288: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
289: *
290: * Compute componentwise relative backward error from formula
291: *
292: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
293: *
294: * where abs(Z) is the componentwise absolute value of the matrix
295: * or vector Z. If the i-th component of the denominator is less
296: * than SAFE2, then SAFE1 is added to the i-th components of the
297: * numerator and denominator before dividing.
298: *
299: DO 20 I = 1, N
300: RWORK( I ) = CABS1( B( I, J ) )
301: 20 CONTINUE
302: *
303: IF( NOTRAN ) THEN
304: *
305: * Compute abs(A)*abs(X) + abs(B).
306: *
307: IF( UPPER ) THEN
308: KC = 1
309: IF( NOUNIT ) THEN
310: DO 40 K = 1, N
311: XK = CABS1( X( K, J ) )
312: DO 30 I = 1, K
313: RWORK( I ) = RWORK( I ) +
314: $ CABS1( AP( KC+I-1 ) )*XK
315: 30 CONTINUE
316: KC = KC + K
317: 40 CONTINUE
318: ELSE
319: DO 60 K = 1, N
320: XK = CABS1( X( K, J ) )
321: DO 50 I = 1, K - 1
322: RWORK( I ) = RWORK( I ) +
323: $ CABS1( AP( KC+I-1 ) )*XK
324: 50 CONTINUE
325: RWORK( K ) = RWORK( K ) + XK
326: KC = KC + K
327: 60 CONTINUE
328: END IF
329: ELSE
330: KC = 1
331: IF( NOUNIT ) THEN
332: DO 80 K = 1, N
333: XK = CABS1( X( K, J ) )
334: DO 70 I = K, N
335: RWORK( I ) = RWORK( I ) +
336: $ CABS1( AP( KC+I-K ) )*XK
337: 70 CONTINUE
338: KC = KC + N - K + 1
339: 80 CONTINUE
340: ELSE
341: DO 100 K = 1, N
342: XK = CABS1( X( K, J ) )
343: DO 90 I = K + 1, N
344: RWORK( I ) = RWORK( I ) +
345: $ CABS1( AP( KC+I-K ) )*XK
346: 90 CONTINUE
347: RWORK( K ) = RWORK( K ) + XK
348: KC = KC + N - K + 1
349: 100 CONTINUE
350: END IF
351: END IF
352: ELSE
353: *
354: * Compute abs(A**H)*abs(X) + abs(B).
355: *
356: IF( UPPER ) THEN
357: KC = 1
358: IF( NOUNIT ) THEN
359: DO 120 K = 1, N
360: S = ZERO
361: DO 110 I = 1, K
362: S = S + CABS1( AP( KC+I-1 ) )*CABS1( X( I, J ) )
363: 110 CONTINUE
364: RWORK( K ) = RWORK( K ) + S
365: KC = KC + K
366: 120 CONTINUE
367: ELSE
368: DO 140 K = 1, N
369: S = CABS1( X( K, J ) )
370: DO 130 I = 1, K - 1
371: S = S + CABS1( AP( KC+I-1 ) )*CABS1( X( I, J ) )
372: 130 CONTINUE
373: RWORK( K ) = RWORK( K ) + S
374: KC = KC + K
375: 140 CONTINUE
376: END IF
377: ELSE
378: KC = 1
379: IF( NOUNIT ) THEN
380: DO 160 K = 1, N
381: S = ZERO
382: DO 150 I = K, N
383: S = S + CABS1( AP( KC+I-K ) )*CABS1( X( I, J ) )
384: 150 CONTINUE
385: RWORK( K ) = RWORK( K ) + S
386: KC = KC + N - K + 1
387: 160 CONTINUE
388: ELSE
389: DO 180 K = 1, N
390: S = CABS1( X( K, J ) )
391: DO 170 I = K + 1, N
392: S = S + CABS1( AP( KC+I-K ) )*CABS1( X( I, J ) )
393: 170 CONTINUE
394: RWORK( K ) = RWORK( K ) + S
395: KC = KC + N - K + 1
396: 180 CONTINUE
397: END IF
398: END IF
399: END IF
400: S = ZERO
401: DO 190 I = 1, N
402: IF( RWORK( I ).GT.SAFE2 ) THEN
403: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
404: ELSE
405: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
406: $ ( RWORK( I )+SAFE1 ) )
407: END IF
408: 190 CONTINUE
409: BERR( J ) = S
410: *
411: * Bound error from formula
412: *
413: * norm(X - XTRUE) / norm(X) .le. FERR =
414: * norm( abs(inv(op(A)))*
415: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
416: *
417: * where
418: * norm(Z) is the magnitude of the largest component of Z
419: * inv(op(A)) is the inverse of op(A)
420: * abs(Z) is the componentwise absolute value of the matrix or
421: * vector Z
422: * NZ is the maximum number of nonzeros in any row of A, plus 1
423: * EPS is machine epsilon
424: *
425: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
426: * is incremented by SAFE1 if the i-th component of
427: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
428: *
429: * Use ZLACN2 to estimate the infinity-norm of the matrix
430: * inv(op(A)) * diag(W),
431: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
432: *
433: DO 200 I = 1, N
434: IF( RWORK( I ).GT.SAFE2 ) THEN
435: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
436: ELSE
437: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
438: $ SAFE1
439: END IF
440: 200 CONTINUE
441: *
442: KASE = 0
443: 210 CONTINUE
444: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
445: IF( KASE.NE.0 ) THEN
446: IF( KASE.EQ.1 ) THEN
447: *
448: * Multiply by diag(W)*inv(op(A)**H).
449: *
450: CALL ZTPSV( UPLO, TRANST, DIAG, N, AP, WORK, 1 )
451: DO 220 I = 1, N
452: WORK( I ) = RWORK( I )*WORK( I )
453: 220 CONTINUE
454: ELSE
455: *
456: * Multiply by inv(op(A))*diag(W).
457: *
458: DO 230 I = 1, N
459: WORK( I ) = RWORK( I )*WORK( I )
460: 230 CONTINUE
461: CALL ZTPSV( UPLO, TRANSN, DIAG, N, AP, WORK, 1 )
462: END IF
463: GO TO 210
464: END IF
465: *
466: * Normalize error.
467: *
468: LSTRES = ZERO
469: DO 240 I = 1, N
470: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
471: 240 CONTINUE
472: IF( LSTRES.NE.ZERO )
473: $ FERR( J ) = FERR( J ) / LSTRES
474: *
475: 250 CONTINUE
476: *
477: RETURN
478: *
479: * End of ZTPRFS
480: *
481: END
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