1: *> \brief \b DPTRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPTRFS + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dptrfs.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dptrfs.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dptrfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
22: * BERR, WORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, LDB, LDX, N, NRHS
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
29: * $ E( * ), EF( * ), FERR( * ), WORK( * ),
30: * $ X( LDX, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> DPTRFS improves the computed solution to a system of linear
40: *> equations when the coefficient matrix is symmetric positive definite
41: *> and tridiagonal, and provides error bounds and backward error
42: *> estimates for the solution.
43: *> \endverbatim
44: *
45: * Arguments:
46: * ==========
47: *
48: *> \param[in] N
49: *> \verbatim
50: *> N is INTEGER
51: *> The order of the matrix A. N >= 0.
52: *> \endverbatim
53: *>
54: *> \param[in] NRHS
55: *> \verbatim
56: *> NRHS is INTEGER
57: *> The number of right hand sides, i.e., the number of columns
58: *> of the matrix B. NRHS >= 0.
59: *> \endverbatim
60: *>
61: *> \param[in] D
62: *> \verbatim
63: *> D is DOUBLE PRECISION array, dimension (N)
64: *> The n diagonal elements of the tridiagonal matrix A.
65: *> \endverbatim
66: *>
67: *> \param[in] E
68: *> \verbatim
69: *> E is DOUBLE PRECISION array, dimension (N-1)
70: *> The (n-1) subdiagonal elements of the tridiagonal matrix A.
71: *> \endverbatim
72: *>
73: *> \param[in] DF
74: *> \verbatim
75: *> DF is DOUBLE PRECISION array, dimension (N)
76: *> The n diagonal elements of the diagonal matrix D from the
77: *> factorization computed by DPTTRF.
78: *> \endverbatim
79: *>
80: *> \param[in] EF
81: *> \verbatim
82: *> EF is DOUBLE PRECISION array, dimension (N-1)
83: *> The (n-1) subdiagonal elements of the unit bidiagonal factor
84: *> L from the factorization computed by DPTTRF.
85: *> \endverbatim
86: *>
87: *> \param[in] B
88: *> \verbatim
89: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
90: *> The right hand side matrix B.
91: *> \endverbatim
92: *>
93: *> \param[in] LDB
94: *> \verbatim
95: *> LDB is INTEGER
96: *> The leading dimension of the array B. LDB >= max(1,N).
97: *> \endverbatim
98: *>
99: *> \param[in,out] X
100: *> \verbatim
101: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
102: *> On entry, the solution matrix X, as computed by DPTTRS.
103: *> On exit, the improved solution matrix X.
104: *> \endverbatim
105: *>
106: *> \param[in] LDX
107: *> \verbatim
108: *> LDX is INTEGER
109: *> The leading dimension of the array X. LDX >= max(1,N).
110: *> \endverbatim
111: *>
112: *> \param[out] FERR
113: *> \verbatim
114: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
115: *> The forward error bound for each solution vector
116: *> X(j) (the j-th column of the solution matrix X).
117: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
118: *> is an estimated upper bound for the magnitude of the largest
119: *> element in (X(j) - XTRUE) divided by the magnitude of the
120: *> largest element in X(j).
121: *> \endverbatim
122: *>
123: *> \param[out] BERR
124: *> \verbatim
125: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
126: *> The componentwise relative backward error of each solution
127: *> vector X(j) (i.e., the smallest relative change in
128: *> any element of A or B that makes X(j) an exact solution).
129: *> \endverbatim
130: *>
131: *> \param[out] WORK
132: *> \verbatim
133: *> WORK is DOUBLE PRECISION array, dimension (2*N)
134: *> \endverbatim
135: *>
136: *> \param[out] INFO
137: *> \verbatim
138: *> INFO is INTEGER
139: *> = 0: successful exit
140: *> < 0: if INFO = -i, the i-th argument had an illegal value
141: *> \endverbatim
142: *
143: *> \par Internal Parameters:
144: * =========================
145: *>
146: *> \verbatim
147: *> ITMAX is the maximum number of steps of iterative refinement.
148: *> \endverbatim
149: *
150: * Authors:
151: * ========
152: *
153: *> \author Univ. of Tennessee
154: *> \author Univ. of California Berkeley
155: *> \author Univ. of Colorado Denver
156: *> \author NAG Ltd.
157: *
158: *> \date September 2012
159: *
160: *> \ingroup doublePTcomputational
161: *
162: * =====================================================================
163: SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
164: $ BERR, WORK, INFO )
165: *
166: * -- LAPACK computational routine (version 3.4.2) --
167: * -- LAPACK is a software package provided by Univ. of Tennessee, --
168: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
169: * September 2012
170: *
171: * .. Scalar Arguments ..
172: INTEGER INFO, LDB, LDX, N, NRHS
173: * ..
174: * .. Array Arguments ..
175: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
176: $ E( * ), EF( * ), FERR( * ), WORK( * ),
177: $ X( LDX, * )
178: * ..
179: *
180: * =====================================================================
181: *
182: * .. Parameters ..
183: INTEGER ITMAX
184: PARAMETER ( ITMAX = 5 )
185: DOUBLE PRECISION ZERO
186: PARAMETER ( ZERO = 0.0D+0 )
187: DOUBLE PRECISION ONE
188: PARAMETER ( ONE = 1.0D+0 )
189: DOUBLE PRECISION TWO
190: PARAMETER ( TWO = 2.0D+0 )
191: DOUBLE PRECISION THREE
192: PARAMETER ( THREE = 3.0D+0 )
193: * ..
194: * .. Local Scalars ..
195: INTEGER COUNT, I, IX, J, NZ
196: DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
197: $ SAFMIN
198: * ..
199: * .. External Subroutines ..
200: EXTERNAL DAXPY, DPTTRS, XERBLA
201: * ..
202: * .. Intrinsic Functions ..
203: INTRINSIC ABS, MAX
204: * ..
205: * .. External Functions ..
206: INTEGER IDAMAX
207: DOUBLE PRECISION DLAMCH
208: EXTERNAL IDAMAX, DLAMCH
209: * ..
210: * .. Executable Statements ..
211: *
212: * Test the input parameters.
213: *
214: INFO = 0
215: IF( N.LT.0 ) THEN
216: INFO = -1
217: ELSE IF( NRHS.LT.0 ) THEN
218: INFO = -2
219: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
220: INFO = -8
221: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
222: INFO = -10
223: END IF
224: IF( INFO.NE.0 ) THEN
225: CALL XERBLA( 'DPTRFS', -INFO )
226: RETURN
227: END IF
228: *
229: * Quick return if possible
230: *
231: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
232: DO 10 J = 1, NRHS
233: FERR( J ) = ZERO
234: BERR( J ) = ZERO
235: 10 CONTINUE
236: RETURN
237: END IF
238: *
239: * NZ = maximum number of nonzero elements in each row of A, plus 1
240: *
241: NZ = 4
242: EPS = DLAMCH( 'Epsilon' )
243: SAFMIN = DLAMCH( 'Safe minimum' )
244: SAFE1 = NZ*SAFMIN
245: SAFE2 = SAFE1 / EPS
246: *
247: * Do for each right hand side
248: *
249: DO 90 J = 1, NRHS
250: *
251: COUNT = 1
252: LSTRES = THREE
253: 20 CONTINUE
254: *
255: * Loop until stopping criterion is satisfied.
256: *
257: * Compute residual R = B - A * X. Also compute
258: * abs(A)*abs(x) + abs(b) for use in the backward error bound.
259: *
260: IF( N.EQ.1 ) THEN
261: BI = B( 1, J )
262: DX = D( 1 )*X( 1, J )
263: WORK( N+1 ) = BI - DX
264: WORK( 1 ) = ABS( BI ) + ABS( DX )
265: ELSE
266: BI = B( 1, J )
267: DX = D( 1 )*X( 1, J )
268: EX = E( 1 )*X( 2, J )
269: WORK( N+1 ) = BI - DX - EX
270: WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
271: DO 30 I = 2, N - 1
272: BI = B( I, J )
273: CX = E( I-1 )*X( I-1, J )
274: DX = D( I )*X( I, J )
275: EX = E( I )*X( I+1, J )
276: WORK( N+I ) = BI - CX - DX - EX
277: WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
278: 30 CONTINUE
279: BI = B( N, J )
280: CX = E( N-1 )*X( N-1, J )
281: DX = D( N )*X( N, J )
282: WORK( N+N ) = BI - CX - DX
283: WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
284: END IF
285: *
286: * Compute componentwise relative backward error from formula
287: *
288: * max(i) ( abs(R(i)) / ( abs(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: S = ZERO
296: DO 40 I = 1, N
297: IF( WORK( I ).GT.SAFE2 ) THEN
298: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
299: ELSE
300: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
301: $ ( WORK( I )+SAFE1 ) )
302: END IF
303: 40 CONTINUE
304: BERR( J ) = S
305: *
306: * Test stopping criterion. Continue iterating if
307: * 1) The residual BERR(J) is larger than machine epsilon, and
308: * 2) BERR(J) decreased by at least a factor of 2 during the
309: * last iteration, and
310: * 3) At most ITMAX iterations tried.
311: *
312: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
313: $ COUNT.LE.ITMAX ) THEN
314: *
315: * Update solution and try again.
316: *
317: CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
318: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
319: LSTRES = BERR( J )
320: COUNT = COUNT + 1
321: GO TO 20
322: END IF
323: *
324: * Bound error from formula
325: *
326: * norm(X - XTRUE) / norm(X) .le. FERR =
327: * norm( abs(inv(A))*
328: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
329: *
330: * where
331: * norm(Z) is the magnitude of the largest component of Z
332: * inv(A) is the inverse of A
333: * abs(Z) is the componentwise absolute value of the matrix or
334: * vector Z
335: * NZ is the maximum number of nonzeros in any row of A, plus 1
336: * EPS is machine epsilon
337: *
338: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
339: * is incremented by SAFE1 if the i-th component of
340: * abs(A)*abs(X) + abs(B) is less than SAFE2.
341: *
342: DO 50 I = 1, N
343: IF( WORK( I ).GT.SAFE2 ) THEN
344: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
345: ELSE
346: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
347: END IF
348: 50 CONTINUE
349: IX = IDAMAX( N, WORK, 1 )
350: FERR( J ) = WORK( IX )
351: *
352: * Estimate the norm of inv(A).
353: *
354: * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
355: *
356: * m(i,j) = abs(A(i,j)), i = j,
357: * m(i,j) = -abs(A(i,j)), i .ne. j,
358: *
359: * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**T.
360: *
361: * Solve M(L) * x = e.
362: *
363: WORK( 1 ) = ONE
364: DO 60 I = 2, N
365: WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
366: 60 CONTINUE
367: *
368: * Solve D * M(L)**T * x = b.
369: *
370: WORK( N ) = WORK( N ) / DF( N )
371: DO 70 I = N - 1, 1, -1
372: WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
373: 70 CONTINUE
374: *
375: * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
376: *
377: IX = IDAMAX( N, WORK, 1 )
378: FERR( J ) = FERR( J )*ABS( WORK( IX ) )
379: *
380: * Normalize error.
381: *
382: LSTRES = ZERO
383: DO 80 I = 1, N
384: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
385: 80 CONTINUE
386: IF( LSTRES.NE.ZERO )
387: $ FERR( J ) = FERR( J ) / LSTRES
388: *
389: 90 CONTINUE
390: *
391: RETURN
392: *
393: * End of DPTRFS
394: *
395: END
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