1: *> \brief \b ZPTRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZPTRFS + 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/zptrfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPTRFS( UPLO, N, NRHS, D, E, DF, EF, 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: * DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
30: * $ RWORK( * )
31: * COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
32: * $ X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZPTRFS improves the computed solution to a system of linear
42: *> equations when the coefficient matrix is Hermitian positive definite
43: *> and tridiagonal, and provides error bounds and backward error
44: *> estimates for the solution.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] UPLO
51: *> \verbatim
52: *> UPLO is CHARACTER*1
53: *> Specifies whether the superdiagonal or the subdiagonal of the
54: *> tridiagonal matrix A is stored and the form of the
55: *> factorization:
56: *> = 'U': E is the superdiagonal of A, and A = U**H*D*U;
57: *> = 'L': E is the subdiagonal of A, and A = L*D*L**H.
58: *> (The two forms are equivalent if A is real.)
59: *> \endverbatim
60: *>
61: *> \param[in] N
62: *> \verbatim
63: *> N is INTEGER
64: *> The order of the matrix A. N >= 0.
65: *> \endverbatim
66: *>
67: *> \param[in] NRHS
68: *> \verbatim
69: *> NRHS is INTEGER
70: *> The number of right hand sides, i.e., the number of columns
71: *> of the matrix B. NRHS >= 0.
72: *> \endverbatim
73: *>
74: *> \param[in] D
75: *> \verbatim
76: *> D is DOUBLE PRECISION array, dimension (N)
77: *> The n real diagonal elements of the tridiagonal matrix A.
78: *> \endverbatim
79: *>
80: *> \param[in] E
81: *> \verbatim
82: *> E is COMPLEX*16 array, dimension (N-1)
83: *> The (n-1) off-diagonal elements of the tridiagonal matrix A
84: *> (see UPLO).
85: *> \endverbatim
86: *>
87: *> \param[in] DF
88: *> \verbatim
89: *> DF is DOUBLE PRECISION array, dimension (N)
90: *> The n diagonal elements of the diagonal matrix D from
91: *> the factorization computed by ZPTTRF.
92: *> \endverbatim
93: *>
94: *> \param[in] EF
95: *> \verbatim
96: *> EF is COMPLEX*16 array, dimension (N-1)
97: *> The (n-1) off-diagonal elements of the unit bidiagonal
98: *> factor U or L from the factorization computed by ZPTTRF
99: *> (see UPLO).
100: *> \endverbatim
101: *>
102: *> \param[in] B
103: *> \verbatim
104: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
105: *> The right hand side matrix B.
106: *> \endverbatim
107: *>
108: *> \param[in] LDB
109: *> \verbatim
110: *> LDB is INTEGER
111: *> The leading dimension of the array B. LDB >= max(1,N).
112: *> \endverbatim
113: *>
114: *> \param[in,out] X
115: *> \verbatim
116: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
117: *> On entry, the solution matrix X, as computed by ZPTTRS.
118: *> On exit, the improved solution matrix X.
119: *> \endverbatim
120: *>
121: *> \param[in] LDX
122: *> \verbatim
123: *> LDX is INTEGER
124: *> The leading dimension of the array X. LDX >= max(1,N).
125: *> \endverbatim
126: *>
127: *> \param[out] FERR
128: *> \verbatim
129: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
130: *> The forward error bound for each solution vector
131: *> X(j) (the j-th column of the solution matrix X).
132: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
133: *> is an estimated upper bound for the magnitude of the largest
134: *> element in (X(j) - XTRUE) divided by the magnitude of the
135: *> largest element in X(j).
136: *> \endverbatim
137: *>
138: *> \param[out] BERR
139: *> \verbatim
140: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
141: *> The componentwise relative backward error of each solution
142: *> vector X(j) (i.e., the smallest relative change in
143: *> any element of A or B that makes X(j) an exact solution).
144: *> \endverbatim
145: *>
146: *> \param[out] WORK
147: *> \verbatim
148: *> WORK is COMPLEX*16 array, dimension (N)
149: *> \endverbatim
150: *>
151: *> \param[out] RWORK
152: *> \verbatim
153: *> RWORK is DOUBLE PRECISION array, dimension (N)
154: *> \endverbatim
155: *>
156: *> \param[out] INFO
157: *> \verbatim
158: *> INFO is INTEGER
159: *> = 0: successful exit
160: *> < 0: if INFO = -i, the i-th argument had an illegal value
161: *> \endverbatim
162: *
163: *> \par Internal Parameters:
164: * =========================
165: *>
166: *> \verbatim
167: *> ITMAX is the maximum number of steps of iterative refinement.
168: *> \endverbatim
169: *
170: * Authors:
171: * ========
172: *
173: *> \author Univ. of Tennessee
174: *> \author Univ. of California Berkeley
175: *> \author Univ. of Colorado Denver
176: *> \author NAG Ltd.
177: *
178: *> \date November 2011
179: *
180: *> \ingroup complex16OTHERcomputational
181: *
182: * =====================================================================
183: SUBROUTINE ZPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
184: $ FERR, BERR, WORK, RWORK, INFO )
185: *
186: * -- LAPACK computational routine (version 3.4.0) --
187: * -- LAPACK is a software package provided by Univ. of Tennessee, --
188: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189: * November 2011
190: *
191: * .. Scalar Arguments ..
192: CHARACTER UPLO
193: INTEGER INFO, LDB, LDX, N, NRHS
194: * ..
195: * .. Array Arguments ..
196: DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
197: $ RWORK( * )
198: COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
199: $ X( LDX, * )
200: * ..
201: *
202: * =====================================================================
203: *
204: * .. Parameters ..
205: INTEGER ITMAX
206: PARAMETER ( ITMAX = 5 )
207: DOUBLE PRECISION ZERO
208: PARAMETER ( ZERO = 0.0D+0 )
209: DOUBLE PRECISION ONE
210: PARAMETER ( ONE = 1.0D+0 )
211: DOUBLE PRECISION TWO
212: PARAMETER ( TWO = 2.0D+0 )
213: DOUBLE PRECISION THREE
214: PARAMETER ( THREE = 3.0D+0 )
215: * ..
216: * .. Local Scalars ..
217: LOGICAL UPPER
218: INTEGER COUNT, I, IX, J, NZ
219: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
220: COMPLEX*16 BI, CX, DX, EX, ZDUM
221: * ..
222: * .. External Functions ..
223: LOGICAL LSAME
224: INTEGER IDAMAX
225: DOUBLE PRECISION DLAMCH
226: EXTERNAL LSAME, IDAMAX, DLAMCH
227: * ..
228: * .. External Subroutines ..
229: EXTERNAL XERBLA, ZAXPY, ZPTTRS
230: * ..
231: * .. Intrinsic Functions ..
232: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
233: * ..
234: * .. Statement Functions ..
235: DOUBLE PRECISION CABS1
236: * ..
237: * .. Statement Function definitions ..
238: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
239: * ..
240: * .. Executable Statements ..
241: *
242: * Test the input parameters.
243: *
244: INFO = 0
245: UPPER = LSAME( UPLO, 'U' )
246: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
247: INFO = -1
248: ELSE IF( N.LT.0 ) THEN
249: INFO = -2
250: ELSE IF( NRHS.LT.0 ) THEN
251: INFO = -3
252: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
253: INFO = -9
254: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
255: INFO = -11
256: END IF
257: IF( INFO.NE.0 ) THEN
258: CALL XERBLA( 'ZPTRFS', -INFO )
259: RETURN
260: END IF
261: *
262: * Quick return if possible
263: *
264: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
265: DO 10 J = 1, NRHS
266: FERR( J ) = ZERO
267: BERR( J ) = ZERO
268: 10 CONTINUE
269: RETURN
270: END IF
271: *
272: * NZ = maximum number of nonzero elements in each row of A, plus 1
273: *
274: NZ = 4
275: EPS = DLAMCH( 'Epsilon' )
276: SAFMIN = DLAMCH( 'Safe minimum' )
277: SAFE1 = NZ*SAFMIN
278: SAFE2 = SAFE1 / EPS
279: *
280: * Do for each right hand side
281: *
282: DO 100 J = 1, NRHS
283: *
284: COUNT = 1
285: LSTRES = THREE
286: 20 CONTINUE
287: *
288: * Loop until stopping criterion is satisfied.
289: *
290: * Compute residual R = B - A * X. Also compute
291: * abs(A)*abs(x) + abs(b) for use in the backward error bound.
292: *
293: IF( UPPER ) THEN
294: IF( N.EQ.1 ) THEN
295: BI = B( 1, J )
296: DX = D( 1 )*X( 1, J )
297: WORK( 1 ) = BI - DX
298: RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
299: ELSE
300: BI = B( 1, J )
301: DX = D( 1 )*X( 1, J )
302: EX = E( 1 )*X( 2, J )
303: WORK( 1 ) = BI - DX - EX
304: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
305: $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
306: DO 30 I = 2, N - 1
307: BI = B( I, J )
308: CX = DCONJG( E( I-1 ) )*X( I-1, J )
309: DX = D( I )*X( I, J )
310: EX = E( I )*X( I+1, J )
311: WORK( I ) = BI - CX - DX - EX
312: RWORK( I ) = CABS1( BI ) +
313: $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
314: $ CABS1( DX ) + CABS1( E( I ) )*
315: $ CABS1( X( I+1, J ) )
316: 30 CONTINUE
317: BI = B( N, J )
318: CX = DCONJG( E( N-1 ) )*X( N-1, J )
319: DX = D( N )*X( N, J )
320: WORK( N ) = BI - CX - DX
321: RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
322: $ CABS1( X( N-1, J ) ) + CABS1( DX )
323: END IF
324: ELSE
325: IF( N.EQ.1 ) THEN
326: BI = B( 1, J )
327: DX = D( 1 )*X( 1, J )
328: WORK( 1 ) = BI - DX
329: RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
330: ELSE
331: BI = B( 1, J )
332: DX = D( 1 )*X( 1, J )
333: EX = DCONJG( E( 1 ) )*X( 2, J )
334: WORK( 1 ) = BI - DX - EX
335: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
336: $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
337: DO 40 I = 2, N - 1
338: BI = B( I, J )
339: CX = E( I-1 )*X( I-1, J )
340: DX = D( I )*X( I, J )
341: EX = DCONJG( E( I ) )*X( I+1, J )
342: WORK( I ) = BI - CX - DX - EX
343: RWORK( I ) = CABS1( BI ) +
344: $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
345: $ CABS1( DX ) + CABS1( E( I ) )*
346: $ CABS1( X( I+1, J ) )
347: 40 CONTINUE
348: BI = B( N, J )
349: CX = E( N-1 )*X( N-1, J )
350: DX = D( N )*X( N, J )
351: WORK( N ) = BI - CX - DX
352: RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
353: $ CABS1( X( N-1, J ) ) + CABS1( DX )
354: END IF
355: END IF
356: *
357: * Compute componentwise relative backward error from formula
358: *
359: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
360: *
361: * where abs(Z) is the componentwise absolute value of the matrix
362: * or vector Z. If the i-th component of the denominator is less
363: * than SAFE2, then SAFE1 is added to the i-th components of the
364: * numerator and denominator before dividing.
365: *
366: S = ZERO
367: DO 50 I = 1, N
368: IF( RWORK( I ).GT.SAFE2 ) THEN
369: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
370: ELSE
371: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
372: $ ( RWORK( I )+SAFE1 ) )
373: END IF
374: 50 CONTINUE
375: BERR( J ) = S
376: *
377: * Test stopping criterion. Continue iterating if
378: * 1) The residual BERR(J) is larger than machine epsilon, and
379: * 2) BERR(J) decreased by at least a factor of 2 during the
380: * last iteration, and
381: * 3) At most ITMAX iterations tried.
382: *
383: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
384: $ COUNT.LE.ITMAX ) THEN
385: *
386: * Update solution and try again.
387: *
388: CALL ZPTTRS( UPLO, N, 1, DF, EF, WORK, N, INFO )
389: CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
390: LSTRES = BERR( J )
391: COUNT = COUNT + 1
392: GO TO 20
393: END IF
394: *
395: * Bound error from formula
396: *
397: * norm(X - XTRUE) / norm(X) .le. FERR =
398: * norm( abs(inv(A))*
399: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
400: *
401: * where
402: * norm(Z) is the magnitude of the largest component of Z
403: * inv(A) is the inverse of A
404: * abs(Z) is the componentwise absolute value of the matrix or
405: * vector Z
406: * NZ is the maximum number of nonzeros in any row of A, plus 1
407: * EPS is machine epsilon
408: *
409: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
410: * is incremented by SAFE1 if the i-th component of
411: * abs(A)*abs(X) + abs(B) is less than SAFE2.
412: *
413: DO 60 I = 1, N
414: IF( RWORK( I ).GT.SAFE2 ) THEN
415: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
416: ELSE
417: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
418: $ SAFE1
419: END IF
420: 60 CONTINUE
421: IX = IDAMAX( N, RWORK, 1 )
422: FERR( J ) = RWORK( IX )
423: *
424: * Estimate the norm of inv(A).
425: *
426: * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
427: *
428: * m(i,j) = abs(A(i,j)), i = j,
429: * m(i,j) = -abs(A(i,j)), i .ne. j,
430: *
431: * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H.
432: *
433: * Solve M(L) * x = e.
434: *
435: RWORK( 1 ) = ONE
436: DO 70 I = 2, N
437: RWORK( I ) = ONE + RWORK( I-1 )*ABS( EF( I-1 ) )
438: 70 CONTINUE
439: *
440: * Solve D * M(L)**H * x = b.
441: *
442: RWORK( N ) = RWORK( N ) / DF( N )
443: DO 80 I = N - 1, 1, -1
444: RWORK( I ) = RWORK( I ) / DF( I ) +
445: $ RWORK( I+1 )*ABS( EF( I ) )
446: 80 CONTINUE
447: *
448: * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
449: *
450: IX = IDAMAX( N, RWORK, 1 )
451: FERR( J ) = FERR( J )*ABS( RWORK( IX ) )
452: *
453: * Normalize error.
454: *
455: LSTRES = ZERO
456: DO 90 I = 1, N
457: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
458: 90 CONTINUE
459: IF( LSTRES.NE.ZERO )
460: $ FERR( J ) = FERR( J ) / LSTRES
461: *
462: 100 CONTINUE
463: *
464: RETURN
465: *
466: * End of ZPTRFS
467: *
468: END
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