Annotation of rpl/lapack/lapack/ztprfs.f, revision 1.16
1.8 bertrand 1: *> \brief \b ZTPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.14 bertrand 5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
1.8 bertrand 7: *
8: *> \htmlonly
1.14 bertrand 9: *> Download ZTPRFS + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztprfs.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztprfs.f">
1.8 bertrand 15: *> [TXT]</a>
1.14 bertrand 16: *> \endhtmlonly
1.8 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
22: * FERR, BERR, WORK, RWORK, INFO )
1.14 bertrand 23: *
1.8 bertrand 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: * ..
1.14 bertrand 32: *
1.8 bertrand 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: *
1.14 bertrand 164: *> \author Univ. of Tennessee
165: *> \author Univ. of California Berkeley
166: *> \author Univ. of Colorado Denver
167: *> \author NAG Ltd.
1.8 bertrand 168: *
1.14 bertrand 169: *> \date December 2016
1.8 bertrand 170: *
171: *> \ingroup complex16OTHERcomputational
172: *
173: * =====================================================================
1.1 bertrand 174: SUBROUTINE ZTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
175: $ FERR, BERR, WORK, RWORK, INFO )
176: *
1.14 bertrand 177: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 178: * -- LAPACK is a software package provided by Univ. of Tennessee, --
179: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.14 bertrand 180: * December 2016
1.1 bertrand 181: *
182: * .. Scalar Arguments ..
183: CHARACTER DIAG, TRANS, UPLO
184: INTEGER INFO, LDB, LDX, N, NRHS
185: * ..
186: * .. Array Arguments ..
187: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
188: COMPLEX*16 AP( * ), B( LDB, * ), WORK( * ), X( LDX, * )
189: * ..
190: *
191: * =====================================================================
192: *
193: * .. Parameters ..
194: DOUBLE PRECISION ZERO
195: PARAMETER ( ZERO = 0.0D+0 )
196: COMPLEX*16 ONE
197: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
198: * ..
199: * .. Local Scalars ..
200: LOGICAL NOTRAN, NOUNIT, UPPER
201: CHARACTER TRANSN, TRANST
202: INTEGER I, J, K, KASE, KC, 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, ZLACN2, ZTPMV, ZTPSV
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: NOTRAN = LSAME( TRANS, 'N' )
233: NOUNIT = LSAME( DIAG, 'N' )
234: *
235: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
236: INFO = -1
237: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
238: $ LSAME( TRANS, 'C' ) ) THEN
239: INFO = -2
240: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
241: INFO = -3
242: ELSE IF( N.LT.0 ) THEN
243: INFO = -4
244: ELSE IF( NRHS.LT.0 ) THEN
245: INFO = -5
246: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
247: INFO = -8
248: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
249: INFO = -10
250: END IF
251: IF( INFO.NE.0 ) THEN
252: CALL XERBLA( 'ZTPRFS', -INFO )
253: RETURN
254: END IF
255: *
256: * Quick return if possible
257: *
258: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
259: DO 10 J = 1, NRHS
260: FERR( J ) = ZERO
261: BERR( J ) = ZERO
262: 10 CONTINUE
263: RETURN
264: END IF
265: *
266: IF( NOTRAN ) THEN
267: TRANSN = 'N'
268: TRANST = 'C'
269: ELSE
270: TRANSN = 'C'
271: TRANST = 'N'
272: END IF
273: *
274: * NZ = maximum number of nonzero elements in each row of A, plus 1
275: *
276: NZ = N + 1
277: EPS = DLAMCH( 'Epsilon' )
278: SAFMIN = DLAMCH( 'Safe minimum' )
279: SAFE1 = NZ*SAFMIN
280: SAFE2 = SAFE1 / EPS
281: *
282: * Do for each right hand side
283: *
284: DO 250 J = 1, NRHS
285: *
286: * Compute residual R = B - op(A) * X,
287: * where op(A) = A, A**T, or A**H, depending on TRANS.
288: *
289: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
290: CALL ZTPMV( UPLO, TRANS, DIAG, N, AP, WORK, 1 )
291: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
292: *
293: * Compute componentwise relative backward error from formula
294: *
295: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
296: *
297: * where abs(Z) is the componentwise absolute value of the matrix
298: * or vector Z. If the i-th component of the denominator is less
299: * than SAFE2, then SAFE1 is added to the i-th components of the
300: * numerator and denominator before dividing.
301: *
302: DO 20 I = 1, N
303: RWORK( I ) = CABS1( B( I, J ) )
304: 20 CONTINUE
305: *
306: IF( NOTRAN ) THEN
307: *
308: * Compute abs(A)*abs(X) + abs(B).
309: *
310: IF( UPPER ) THEN
311: KC = 1
312: IF( NOUNIT ) THEN
313: DO 40 K = 1, N
314: XK = CABS1( X( K, J ) )
315: DO 30 I = 1, K
316: RWORK( I ) = RWORK( I ) +
317: $ CABS1( AP( KC+I-1 ) )*XK
318: 30 CONTINUE
319: KC = KC + K
320: 40 CONTINUE
321: ELSE
322: DO 60 K = 1, N
323: XK = CABS1( X( K, J ) )
324: DO 50 I = 1, K - 1
325: RWORK( I ) = RWORK( I ) +
326: $ CABS1( AP( KC+I-1 ) )*XK
327: 50 CONTINUE
328: RWORK( K ) = RWORK( K ) + XK
329: KC = KC + K
330: 60 CONTINUE
331: END IF
332: ELSE
333: KC = 1
334: IF( NOUNIT ) THEN
335: DO 80 K = 1, N
336: XK = CABS1( X( K, J ) )
337: DO 70 I = K, N
338: RWORK( I ) = RWORK( I ) +
339: $ CABS1( AP( KC+I-K ) )*XK
340: 70 CONTINUE
341: KC = KC + N - K + 1
342: 80 CONTINUE
343: ELSE
344: DO 100 K = 1, N
345: XK = CABS1( X( K, J ) )
346: DO 90 I = K + 1, N
347: RWORK( I ) = RWORK( I ) +
348: $ CABS1( AP( KC+I-K ) )*XK
349: 90 CONTINUE
350: RWORK( K ) = RWORK( K ) + XK
351: KC = KC + N - K + 1
352: 100 CONTINUE
353: END IF
354: END IF
355: ELSE
356: *
357: * Compute abs(A**H)*abs(X) + abs(B).
358: *
359: IF( UPPER ) THEN
360: KC = 1
361: IF( NOUNIT ) THEN
362: DO 120 K = 1, N
363: S = ZERO
364: DO 110 I = 1, K
365: S = S + CABS1( AP( KC+I-1 ) )*CABS1( X( I, J ) )
366: 110 CONTINUE
367: RWORK( K ) = RWORK( K ) + S
368: KC = KC + K
369: 120 CONTINUE
370: ELSE
371: DO 140 K = 1, N
372: S = CABS1( X( K, J ) )
373: DO 130 I = 1, K - 1
374: S = S + CABS1( AP( KC+I-1 ) )*CABS1( X( I, J ) )
375: 130 CONTINUE
376: RWORK( K ) = RWORK( K ) + S
377: KC = KC + K
378: 140 CONTINUE
379: END IF
380: ELSE
381: KC = 1
382: IF( NOUNIT ) THEN
383: DO 160 K = 1, N
384: S = ZERO
385: DO 150 I = K, N
386: S = S + CABS1( AP( KC+I-K ) )*CABS1( X( I, J ) )
387: 150 CONTINUE
388: RWORK( K ) = RWORK( K ) + S
389: KC = KC + N - K + 1
390: 160 CONTINUE
391: ELSE
392: DO 180 K = 1, N
393: S = CABS1( X( K, J ) )
394: DO 170 I = K + 1, N
395: S = S + CABS1( AP( KC+I-K ) )*CABS1( X( I, J ) )
396: 170 CONTINUE
397: RWORK( K ) = RWORK( K ) + S
398: KC = KC + N - K + 1
399: 180 CONTINUE
400: END IF
401: END IF
402: END IF
403: S = ZERO
404: DO 190 I = 1, N
405: IF( RWORK( I ).GT.SAFE2 ) THEN
406: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
407: ELSE
408: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
409: $ ( RWORK( I )+SAFE1 ) )
410: END IF
411: 190 CONTINUE
412: BERR( J ) = S
413: *
414: * Bound error from formula
415: *
416: * norm(X - XTRUE) / norm(X) .le. FERR =
417: * norm( abs(inv(op(A)))*
418: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
419: *
420: * where
421: * norm(Z) is the magnitude of the largest component of Z
422: * inv(op(A)) is the inverse of op(A)
423: * abs(Z) is the componentwise absolute value of the matrix or
424: * vector Z
425: * NZ is the maximum number of nonzeros in any row of A, plus 1
426: * EPS is machine epsilon
427: *
428: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
429: * is incremented by SAFE1 if the i-th component of
430: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
431: *
432: * Use ZLACN2 to estimate the infinity-norm of the matrix
433: * inv(op(A)) * diag(W),
434: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
435: *
436: DO 200 I = 1, N
437: IF( RWORK( I ).GT.SAFE2 ) THEN
438: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
439: ELSE
440: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
441: $ SAFE1
442: END IF
443: 200 CONTINUE
444: *
445: KASE = 0
446: 210 CONTINUE
447: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
448: IF( KASE.NE.0 ) THEN
449: IF( KASE.EQ.1 ) THEN
450: *
451: * Multiply by diag(W)*inv(op(A)**H).
452: *
453: CALL ZTPSV( UPLO, TRANST, DIAG, N, AP, WORK, 1 )
454: DO 220 I = 1, N
455: WORK( I ) = RWORK( I )*WORK( I )
456: 220 CONTINUE
457: ELSE
458: *
459: * Multiply by inv(op(A))*diag(W).
460: *
461: DO 230 I = 1, N
462: WORK( I ) = RWORK( I )*WORK( I )
463: 230 CONTINUE
464: CALL ZTPSV( UPLO, TRANSN, DIAG, N, AP, WORK, 1 )
465: END IF
466: GO TO 210
467: END IF
468: *
469: * Normalize error.
470: *
471: LSTRES = ZERO
472: DO 240 I = 1, N
473: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
474: 240 CONTINUE
475: IF( LSTRES.NE.ZERO )
476: $ FERR( J ) = FERR( J ) / LSTRES
477: *
478: 250 CONTINUE
479: *
480: RETURN
481: *
482: * End of ZTPRFS
483: *
484: END
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