Annotation of rpl/lapack/lapack/dtprfs.f, revision 1.16
1.9 bertrand 1: *> \brief \b DTPRFS
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 DTPRFS + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtprfs.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtprfs.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtprfs.f">
1.9 bertrand 15: *> [TXT]</a>
1.15 bertrand 16: *> \endhtmlonly
1.9 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
22: * FERR, BERR, WORK, IWORK, INFO )
1.15 bertrand 23: *
1.9 bertrand 24: * .. Scalar Arguments ..
25: * CHARACTER DIAG, TRANS, UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IWORK( * )
30: * DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
31: * $ WORK( * ), X( LDX, * )
32: * ..
1.15 bertrand 33: *
1.9 bertrand 34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> DTPRFS provides error bounds and backward error estimates for the
41: *> solution to a system of linear equations with a triangular packed
42: *> coefficient matrix.
43: *>
44: *> The solution matrix X must be computed by DTPTRS or some other
45: *> means before entering this routine. DTPRFS does not do iterative
46: *> refinement because doing so cannot improve the backward error.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] UPLO
53: *> \verbatim
54: *> UPLO is CHARACTER*1
55: *> = 'U': A is upper triangular;
56: *> = 'L': A is lower triangular.
57: *> \endverbatim
58: *>
59: *> \param[in] TRANS
60: *> \verbatim
61: *> TRANS is CHARACTER*1
62: *> Specifies the form of the system of equations:
63: *> = 'N': A * X = B (No transpose)
64: *> = 'T': A**T * X = B (Transpose)
65: *> = 'C': A**H * X = B (Conjugate transpose = Transpose)
66: *> \endverbatim
67: *>
68: *> \param[in] DIAG
69: *> \verbatim
70: *> DIAG is CHARACTER*1
71: *> = 'N': A is non-unit triangular;
72: *> = 'U': A is unit triangular.
73: *> \endverbatim
74: *>
75: *> \param[in] N
76: *> \verbatim
77: *> N is INTEGER
78: *> The order of the matrix A. N >= 0.
79: *> \endverbatim
80: *>
81: *> \param[in] NRHS
82: *> \verbatim
83: *> NRHS is INTEGER
84: *> The number of right hand sides, i.e., the number of columns
85: *> of the matrices B and X. NRHS >= 0.
86: *> \endverbatim
87: *>
88: *> \param[in] AP
89: *> \verbatim
90: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
91: *> The upper or lower triangular matrix A, packed columnwise in
92: *> a linear array. The j-th column of A is stored in the array
93: *> AP as follows:
94: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
95: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
96: *> If DIAG = 'U', the diagonal elements of A are not referenced
97: *> and are assumed to be 1.
98: *> \endverbatim
99: *>
100: *> \param[in] B
101: *> \verbatim
102: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
103: *> The right hand side matrix B.
104: *> \endverbatim
105: *>
106: *> \param[in] LDB
107: *> \verbatim
108: *> LDB is INTEGER
109: *> The leading dimension of the array B. LDB >= max(1,N).
110: *> \endverbatim
111: *>
112: *> \param[in] X
113: *> \verbatim
114: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
115: *> The solution matrix X.
116: *> \endverbatim
117: *>
118: *> \param[in] LDX
119: *> \verbatim
120: *> LDX is INTEGER
121: *> The leading dimension of the array X. LDX >= max(1,N).
122: *> \endverbatim
123: *>
124: *> \param[out] FERR
125: *> \verbatim
126: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
127: *> The estimated forward error bound for each solution vector
128: *> X(j) (the j-th column of the solution matrix X).
129: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
130: *> is an estimated upper bound for the magnitude of the largest
131: *> element in (X(j) - XTRUE) divided by the magnitude of the
132: *> largest element in X(j). The estimate is as reliable as
133: *> the estimate for RCOND, and is almost always a slight
134: *> overestimate of the true error.
135: *> \endverbatim
136: *>
137: *> \param[out] BERR
138: *> \verbatim
139: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
140: *> The componentwise relative backward error of each solution
141: *> vector X(j) (i.e., the smallest relative change in
142: *> any element of A or B that makes X(j) an exact solution).
143: *> \endverbatim
144: *>
145: *> \param[out] WORK
146: *> \verbatim
147: *> WORK is DOUBLE PRECISION array, dimension (3*N)
148: *> \endverbatim
149: *>
150: *> \param[out] IWORK
151: *> \verbatim
152: *> IWORK is INTEGER array, dimension (N)
153: *> \endverbatim
154: *>
155: *> \param[out] INFO
156: *> \verbatim
157: *> INFO is INTEGER
158: *> = 0: successful exit
159: *> < 0: if INFO = -i, the i-th argument had an illegal value
160: *> \endverbatim
161: *
162: * Authors:
163: * ========
164: *
1.15 bertrand 165: *> \author Univ. of Tennessee
166: *> \author Univ. of California Berkeley
167: *> \author Univ. of Colorado Denver
168: *> \author NAG Ltd.
1.9 bertrand 169: *
1.15 bertrand 170: *> \date December 2016
1.9 bertrand 171: *
172: *> \ingroup doubleOTHERcomputational
173: *
174: * =====================================================================
1.1 bertrand 175: SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
176: $ FERR, BERR, WORK, IWORK, INFO )
177: *
1.15 bertrand 178: * -- LAPACK computational routine (version 3.7.0) --
1.1 bertrand 179: * -- LAPACK is a software package provided by Univ. of Tennessee, --
180: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.15 bertrand 181: * December 2016
1.1 bertrand 182: *
183: * .. Scalar Arguments ..
184: CHARACTER DIAG, TRANS, UPLO
185: INTEGER INFO, LDB, LDX, N, NRHS
186: * ..
187: * .. Array Arguments ..
188: INTEGER IWORK( * )
189: DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
190: $ WORK( * ), X( LDX, * )
191: * ..
192: *
193: * =====================================================================
194: *
195: * .. Parameters ..
196: DOUBLE PRECISION ZERO
197: PARAMETER ( ZERO = 0.0D+0 )
198: DOUBLE PRECISION ONE
199: PARAMETER ( ONE = 1.0D+0 )
200: * ..
201: * .. Local Scalars ..
202: LOGICAL NOTRAN, NOUNIT, UPPER
203: CHARACTER TRANST
204: INTEGER I, J, K, KASE, KC, NZ
205: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
206: * ..
207: * .. Local Arrays ..
208: INTEGER ISAVE( 3 )
209: * ..
210: * .. External Subroutines ..
211: EXTERNAL DAXPY, DCOPY, DLACN2, DTPMV, DTPSV, XERBLA
212: * ..
213: * .. Intrinsic Functions ..
214: INTRINSIC ABS, MAX
215: * ..
216: * .. External Functions ..
217: LOGICAL LSAME
218: DOUBLE PRECISION DLAMCH
219: EXTERNAL LSAME, DLAMCH
220: * ..
221: * .. Executable Statements ..
222: *
223: * Test the input parameters.
224: *
225: INFO = 0
226: UPPER = LSAME( UPLO, 'U' )
227: NOTRAN = LSAME( TRANS, 'N' )
228: NOUNIT = LSAME( DIAG, 'N' )
229: *
230: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
231: INFO = -1
232: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
233: $ LSAME( TRANS, 'C' ) ) THEN
234: INFO = -2
235: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
236: INFO = -3
237: ELSE IF( N.LT.0 ) THEN
238: INFO = -4
239: ELSE IF( NRHS.LT.0 ) THEN
240: INFO = -5
241: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
242: INFO = -8
243: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
244: INFO = -10
245: END IF
246: IF( INFO.NE.0 ) THEN
247: CALL XERBLA( 'DTPRFS', -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: IF( NOTRAN ) THEN
262: TRANST = 'T'
263: ELSE
264: TRANST = 'N'
265: END IF
266: *
267: * NZ = maximum number of nonzero elements in each row of A, plus 1
268: *
269: NZ = N + 1
270: EPS = DLAMCH( 'Epsilon' )
271: SAFMIN = DLAMCH( 'Safe minimum' )
272: SAFE1 = NZ*SAFMIN
273: SAFE2 = SAFE1 / EPS
274: *
275: * Do for each right hand side
276: *
277: DO 250 J = 1, NRHS
278: *
279: * Compute residual R = B - op(A) * X,
1.8 bertrand 280: * where op(A) = A or A**T, depending on TRANS.
1.1 bertrand 281: *
282: CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
283: CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
284: CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
285: *
286: * Compute componentwise relative backward error from formula
287: *
288: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
289: *
290: * where abs(Z) is the componentwise absolute value of the matrix
291: * or vector Z. If the i-th component of the denominator is less
292: * than SAFE2, then SAFE1 is added to the i-th components of the
293: * numerator and denominator before dividing.
294: *
295: DO 20 I = 1, N
296: WORK( I ) = ABS( B( I, J ) )
297: 20 CONTINUE
298: *
299: IF( NOTRAN ) THEN
300: *
301: * Compute abs(A)*abs(X) + abs(B).
302: *
303: IF( UPPER ) THEN
304: KC = 1
305: IF( NOUNIT ) THEN
306: DO 40 K = 1, N
307: XK = ABS( X( K, J ) )
308: DO 30 I = 1, K
309: WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
310: 30 CONTINUE
311: KC = KC + K
312: 40 CONTINUE
313: ELSE
314: DO 60 K = 1, N
315: XK = ABS( X( K, J ) )
316: DO 50 I = 1, K - 1
317: WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
318: 50 CONTINUE
319: WORK( K ) = WORK( K ) + XK
320: KC = KC + K
321: 60 CONTINUE
322: END IF
323: ELSE
324: KC = 1
325: IF( NOUNIT ) THEN
326: DO 80 K = 1, N
327: XK = ABS( X( K, J ) )
328: DO 70 I = K, N
329: WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
330: 70 CONTINUE
331: KC = KC + N - K + 1
332: 80 CONTINUE
333: ELSE
334: DO 100 K = 1, N
335: XK = ABS( X( K, J ) )
336: DO 90 I = K + 1, N
337: WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
338: 90 CONTINUE
339: WORK( K ) = WORK( K ) + XK
340: KC = KC + N - K + 1
341: 100 CONTINUE
342: END IF
343: END IF
344: ELSE
345: *
1.8 bertrand 346: * Compute abs(A**T)*abs(X) + abs(B).
1.1 bertrand 347: *
348: IF( UPPER ) THEN
349: KC = 1
350: IF( NOUNIT ) THEN
351: DO 120 K = 1, N
352: S = ZERO
353: DO 110 I = 1, K
354: S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
355: 110 CONTINUE
356: WORK( K ) = WORK( K ) + S
357: KC = KC + K
358: 120 CONTINUE
359: ELSE
360: DO 140 K = 1, N
361: S = ABS( X( K, J ) )
362: DO 130 I = 1, K - 1
363: S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
364: 130 CONTINUE
365: WORK( K ) = WORK( K ) + S
366: KC = KC + K
367: 140 CONTINUE
368: END IF
369: ELSE
370: KC = 1
371: IF( NOUNIT ) THEN
372: DO 160 K = 1, N
373: S = ZERO
374: DO 150 I = K, N
375: S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
376: 150 CONTINUE
377: WORK( K ) = WORK( K ) + S
378: KC = KC + N - K + 1
379: 160 CONTINUE
380: ELSE
381: DO 180 K = 1, N
382: S = ABS( X( K, J ) )
383: DO 170 I = K + 1, N
384: S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
385: 170 CONTINUE
386: WORK( K ) = WORK( K ) + S
387: KC = KC + N - K + 1
388: 180 CONTINUE
389: END IF
390: END IF
391: END IF
392: S = ZERO
393: DO 190 I = 1, N
394: IF( WORK( I ).GT.SAFE2 ) THEN
395: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
396: ELSE
397: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
398: $ ( WORK( I )+SAFE1 ) )
399: END IF
400: 190 CONTINUE
401: BERR( J ) = S
402: *
403: * Bound error from formula
404: *
405: * norm(X - XTRUE) / norm(X) .le. FERR =
406: * norm( abs(inv(op(A)))*
407: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
408: *
409: * where
410: * norm(Z) is the magnitude of the largest component of Z
411: * inv(op(A)) is the inverse of op(A)
412: * abs(Z) is the componentwise absolute value of the matrix or
413: * vector Z
414: * NZ is the maximum number of nonzeros in any row of A, plus 1
415: * EPS is machine epsilon
416: *
417: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
418: * is incremented by SAFE1 if the i-th component of
419: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
420: *
421: * Use DLACN2 to estimate the infinity-norm of the matrix
422: * inv(op(A)) * diag(W),
423: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
424: *
425: DO 200 I = 1, N
426: IF( WORK( I ).GT.SAFE2 ) THEN
427: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
428: ELSE
429: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
430: END IF
431: 200 CONTINUE
432: *
433: KASE = 0
434: 210 CONTINUE
435: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
436: $ KASE, ISAVE )
437: IF( KASE.NE.0 ) THEN
438: IF( KASE.EQ.1 ) THEN
439: *
1.8 bertrand 440: * Multiply by diag(W)*inv(op(A)**T).
1.1 bertrand 441: *
442: CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ), 1 )
443: DO 220 I = 1, N
444: WORK( N+I ) = WORK( I )*WORK( N+I )
445: 220 CONTINUE
446: ELSE
447: *
448: * Multiply by inv(op(A))*diag(W).
449: *
450: DO 230 I = 1, N
451: WORK( N+I ) = WORK( I )*WORK( N+I )
452: 230 CONTINUE
453: CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
454: END IF
455: GO TO 210
456: END IF
457: *
458: * Normalize error.
459: *
460: LSTRES = ZERO
461: DO 240 I = 1, N
462: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
463: 240 CONTINUE
464: IF( LSTRES.NE.ZERO )
465: $ FERR( J ) = FERR( J ) / LSTRES
466: *
467: 250 CONTINUE
468: *
469: RETURN
470: *
471: * End of DTPRFS
472: *
473: END
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