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: *> \ingroup complex16PTcomputational
179: *
180: * =====================================================================
181: SUBROUTINE ZPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
182: $ FERR, BERR, WORK, RWORK, INFO )
183: *
184: * -- LAPACK computational routine --
185: * -- LAPACK is a software package provided by Univ. of Tennessee, --
186: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187: *
188: * .. Scalar Arguments ..
189: CHARACTER UPLO
190: INTEGER INFO, LDB, LDX, N, NRHS
191: * ..
192: * .. Array Arguments ..
193: DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
194: $ RWORK( * )
195: COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
196: $ X( LDX, * )
197: * ..
198: *
199: * =====================================================================
200: *
201: * .. Parameters ..
202: INTEGER ITMAX
203: PARAMETER ( ITMAX = 5 )
204: DOUBLE PRECISION ZERO
205: PARAMETER ( ZERO = 0.0D+0 )
206: DOUBLE PRECISION ONE
207: PARAMETER ( ONE = 1.0D+0 )
208: DOUBLE PRECISION TWO
209: PARAMETER ( TWO = 2.0D+0 )
210: DOUBLE PRECISION THREE
211: PARAMETER ( THREE = 3.0D+0 )
212: * ..
213: * .. Local Scalars ..
214: LOGICAL UPPER
215: INTEGER COUNT, I, IX, J, NZ
216: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
217: COMPLEX*16 BI, CX, DX, EX, ZDUM
218: * ..
219: * .. External Functions ..
220: LOGICAL LSAME
221: INTEGER IDAMAX
222: DOUBLE PRECISION DLAMCH
223: EXTERNAL LSAME, IDAMAX, DLAMCH
224: * ..
225: * .. External Subroutines ..
226: EXTERNAL XERBLA, ZAXPY, ZPTTRS
227: * ..
228: * .. Intrinsic Functions ..
229: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
230: * ..
231: * .. Statement Functions ..
232: DOUBLE PRECISION CABS1
233: * ..
234: * .. Statement Function definitions ..
235: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
236: * ..
237: * .. Executable Statements ..
238: *
239: * Test the input parameters.
240: *
241: INFO = 0
242: UPPER = LSAME( UPLO, 'U' )
243: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
244: INFO = -1
245: ELSE IF( N.LT.0 ) THEN
246: INFO = -2
247: ELSE IF( NRHS.LT.0 ) THEN
248: INFO = -3
249: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
250: INFO = -9
251: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
252: INFO = -11
253: END IF
254: IF( INFO.NE.0 ) THEN
255: CALL XERBLA( 'ZPTRFS', -INFO )
256: RETURN
257: END IF
258: *
259: * Quick return if possible
260: *
261: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
262: DO 10 J = 1, NRHS
263: FERR( J ) = ZERO
264: BERR( J ) = ZERO
265: 10 CONTINUE
266: RETURN
267: END IF
268: *
269: * NZ = maximum number of nonzero elements in each row of A, plus 1
270: *
271: NZ = 4
272: EPS = DLAMCH( 'Epsilon' )
273: SAFMIN = DLAMCH( 'Safe minimum' )
274: SAFE1 = NZ*SAFMIN
275: SAFE2 = SAFE1 / EPS
276: *
277: * Do for each right hand side
278: *
279: DO 100 J = 1, NRHS
280: *
281: COUNT = 1
282: LSTRES = THREE
283: 20 CONTINUE
284: *
285: * Loop until stopping criterion is satisfied.
286: *
287: * Compute residual R = B - A * X. Also compute
288: * abs(A)*abs(x) + abs(b) for use in the backward error bound.
289: *
290: IF( UPPER ) THEN
291: IF( N.EQ.1 ) THEN
292: BI = B( 1, J )
293: DX = D( 1 )*X( 1, J )
294: WORK( 1 ) = BI - DX
295: RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
296: ELSE
297: BI = B( 1, J )
298: DX = D( 1 )*X( 1, J )
299: EX = E( 1 )*X( 2, J )
300: WORK( 1 ) = BI - DX - EX
301: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
302: $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
303: DO 30 I = 2, N - 1
304: BI = B( I, J )
305: CX = DCONJG( E( I-1 ) )*X( I-1, J )
306: DX = D( I )*X( I, J )
307: EX = E( I )*X( I+1, J )
308: WORK( I ) = BI - CX - DX - EX
309: RWORK( I ) = CABS1( BI ) +
310: $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
311: $ CABS1( DX ) + CABS1( E( I ) )*
312: $ CABS1( X( I+1, J ) )
313: 30 CONTINUE
314: BI = B( N, J )
315: CX = DCONJG( E( N-1 ) )*X( N-1, J )
316: DX = D( N )*X( N, J )
317: WORK( N ) = BI - CX - DX
318: RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
319: $ CABS1( X( N-1, J ) ) + CABS1( DX )
320: END IF
321: ELSE
322: IF( N.EQ.1 ) THEN
323: BI = B( 1, J )
324: DX = D( 1 )*X( 1, J )
325: WORK( 1 ) = BI - DX
326: RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
327: ELSE
328: BI = B( 1, J )
329: DX = D( 1 )*X( 1, J )
330: EX = DCONJG( E( 1 ) )*X( 2, J )
331: WORK( 1 ) = BI - DX - EX
332: RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
333: $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
334: DO 40 I = 2, N - 1
335: BI = B( I, J )
336: CX = E( I-1 )*X( I-1, J )
337: DX = D( I )*X( I, J )
338: EX = DCONJG( E( I ) )*X( I+1, J )
339: WORK( I ) = BI - CX - DX - EX
340: RWORK( I ) = CABS1( BI ) +
341: $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
342: $ CABS1( DX ) + CABS1( E( I ) )*
343: $ CABS1( X( I+1, J ) )
344: 40 CONTINUE
345: BI = B( N, J )
346: CX = E( N-1 )*X( N-1, J )
347: DX = D( N )*X( N, J )
348: WORK( N ) = BI - CX - DX
349: RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
350: $ CABS1( X( N-1, J ) ) + CABS1( DX )
351: END IF
352: END IF
353: *
354: * Compute componentwise relative backward error from formula
355: *
356: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
357: *
358: * where abs(Z) is the componentwise absolute value of the matrix
359: * or vector Z. If the i-th component of the denominator is less
360: * than SAFE2, then SAFE1 is added to the i-th components of the
361: * numerator and denominator before dividing.
362: *
363: S = ZERO
364: DO 50 I = 1, N
365: IF( RWORK( I ).GT.SAFE2 ) THEN
366: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
367: ELSE
368: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
369: $ ( RWORK( I )+SAFE1 ) )
370: END IF
371: 50 CONTINUE
372: BERR( J ) = S
373: *
374: * Test stopping criterion. Continue iterating if
375: * 1) The residual BERR(J) is larger than machine epsilon, and
376: * 2) BERR(J) decreased by at least a factor of 2 during the
377: * last iteration, and
378: * 3) At most ITMAX iterations tried.
379: *
380: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
381: $ COUNT.LE.ITMAX ) THEN
382: *
383: * Update solution and try again.
384: *
385: CALL ZPTTRS( UPLO, N, 1, DF, EF, WORK, N, INFO )
386: CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
387: LSTRES = BERR( J )
388: COUNT = COUNT + 1
389: GO TO 20
390: END IF
391: *
392: * Bound error from formula
393: *
394: * norm(X - XTRUE) / norm(X) .le. FERR =
395: * norm( abs(inv(A))*
396: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
397: *
398: * where
399: * norm(Z) is the magnitude of the largest component of Z
400: * inv(A) is the inverse of A
401: * abs(Z) is the componentwise absolute value of the matrix or
402: * vector Z
403: * NZ is the maximum number of nonzeros in any row of A, plus 1
404: * EPS is machine epsilon
405: *
406: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
407: * is incremented by SAFE1 if the i-th component of
408: * abs(A)*abs(X) + abs(B) is less than SAFE2.
409: *
410: DO 60 I = 1, N
411: IF( RWORK( I ).GT.SAFE2 ) THEN
412: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
413: ELSE
414: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
415: $ SAFE1
416: END IF
417: 60 CONTINUE
418: IX = IDAMAX( N, RWORK, 1 )
419: FERR( J ) = RWORK( IX )
420: *
421: * Estimate the norm of inv(A).
422: *
423: * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
424: *
425: * m(i,j) = abs(A(i,j)), i = j,
426: * m(i,j) = -abs(A(i,j)), i .ne. j,
427: *
428: * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H.
429: *
430: * Solve M(L) * x = e.
431: *
432: RWORK( 1 ) = ONE
433: DO 70 I = 2, N
434: RWORK( I ) = ONE + RWORK( I-1 )*ABS( EF( I-1 ) )
435: 70 CONTINUE
436: *
437: * Solve D * M(L)**H * x = b.
438: *
439: RWORK( N ) = RWORK( N ) / DF( N )
440: DO 80 I = N - 1, 1, -1
441: RWORK( I ) = RWORK( I ) / DF( I ) +
442: $ RWORK( I+1 )*ABS( EF( I ) )
443: 80 CONTINUE
444: *
445: * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
446: *
447: IX = IDAMAX( N, RWORK, 1 )
448: FERR( J ) = FERR( J )*ABS( RWORK( IX ) )
449: *
450: * Normalize error.
451: *
452: LSTRES = ZERO
453: DO 90 I = 1, N
454: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
455: 90 CONTINUE
456: IF( LSTRES.NE.ZERO )
457: $ FERR( J ) = FERR( J ) / LSTRES
458: *
459: 100 CONTINUE
460: *
461: RETURN
462: *
463: * End of ZPTRFS
464: *
465: END
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