1: SUBROUTINE DPTRFS( N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR,
2: $ BERR, WORK, INFO )
3: *
4: * -- LAPACK routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * .. Scalar Arguments ..
10: INTEGER INFO, LDB, LDX, N, NRHS
11: * ..
12: * .. Array Arguments ..
13: DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DF( * ),
14: $ E( * ), EF( * ), FERR( * ), WORK( * ),
15: $ X( LDX, * )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * DPTRFS improves the computed solution to a system of linear
22: * equations when the coefficient matrix is symmetric positive definite
23: * and tridiagonal, and provides error bounds and backward error
24: * estimates for the solution.
25: *
26: * Arguments
27: * =========
28: *
29: * N (input) INTEGER
30: * The order of the matrix A. N >= 0.
31: *
32: * NRHS (input) INTEGER
33: * The number of right hand sides, i.e., the number of columns
34: * of the matrix B. NRHS >= 0.
35: *
36: * D (input) DOUBLE PRECISION array, dimension (N)
37: * The n diagonal elements of the tridiagonal matrix A.
38: *
39: * E (input) DOUBLE PRECISION array, dimension (N-1)
40: * The (n-1) subdiagonal elements of the tridiagonal matrix A.
41: *
42: * DF (input) DOUBLE PRECISION array, dimension (N)
43: * The n diagonal elements of the diagonal matrix D from the
44: * factorization computed by DPTTRF.
45: *
46: * EF (input) DOUBLE PRECISION array, dimension (N-1)
47: * The (n-1) subdiagonal elements of the unit bidiagonal factor
48: * L from the factorization computed by DPTTRF.
49: *
50: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
51: * The right hand side matrix B.
52: *
53: * LDB (input) INTEGER
54: * The leading dimension of the array B. LDB >= max(1,N).
55: *
56: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
57: * On entry, the solution matrix X, as computed by DPTTRS.
58: * On exit, the improved solution matrix X.
59: *
60: * LDX (input) INTEGER
61: * The leading dimension of the array X. LDX >= max(1,N).
62: *
63: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
64: * The forward error bound for each solution vector
65: * X(j) (the j-th column of the solution matrix X).
66: * If XTRUE is the true solution corresponding to X(j), FERR(j)
67: * is an estimated upper bound for the magnitude of the largest
68: * element in (X(j) - XTRUE) divided by the magnitude of the
69: * largest element in X(j).
70: *
71: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
72: * The componentwise relative backward error of each solution
73: * vector X(j) (i.e., the smallest relative change in
74: * any element of A or B that makes X(j) an exact solution).
75: *
76: * WORK (workspace) DOUBLE PRECISION array, dimension (2*N)
77: *
78: * INFO (output) INTEGER
79: * = 0: successful exit
80: * < 0: if INFO = -i, the i-th argument had an illegal value
81: *
82: * Internal Parameters
83: * ===================
84: *
85: * ITMAX is the maximum number of steps of iterative refinement.
86: *
87: * =====================================================================
88: *
89: * .. Parameters ..
90: INTEGER ITMAX
91: PARAMETER ( ITMAX = 5 )
92: DOUBLE PRECISION ZERO
93: PARAMETER ( ZERO = 0.0D+0 )
94: DOUBLE PRECISION ONE
95: PARAMETER ( ONE = 1.0D+0 )
96: DOUBLE PRECISION TWO
97: PARAMETER ( TWO = 2.0D+0 )
98: DOUBLE PRECISION THREE
99: PARAMETER ( THREE = 3.0D+0 )
100: * ..
101: * .. Local Scalars ..
102: INTEGER COUNT, I, IX, J, NZ
103: DOUBLE PRECISION BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
104: $ SAFMIN
105: * ..
106: * .. External Subroutines ..
107: EXTERNAL DAXPY, DPTTRS, XERBLA
108: * ..
109: * .. Intrinsic Functions ..
110: INTRINSIC ABS, MAX
111: * ..
112: * .. External Functions ..
113: INTEGER IDAMAX
114: DOUBLE PRECISION DLAMCH
115: EXTERNAL IDAMAX, DLAMCH
116: * ..
117: * .. Executable Statements ..
118: *
119: * Test the input parameters.
120: *
121: INFO = 0
122: IF( N.LT.0 ) THEN
123: INFO = -1
124: ELSE IF( NRHS.LT.0 ) THEN
125: INFO = -2
126: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
127: INFO = -8
128: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
129: INFO = -10
130: END IF
131: IF( INFO.NE.0 ) THEN
132: CALL XERBLA( 'DPTRFS', -INFO )
133: RETURN
134: END IF
135: *
136: * Quick return if possible
137: *
138: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
139: DO 10 J = 1, NRHS
140: FERR( J ) = ZERO
141: BERR( J ) = ZERO
142: 10 CONTINUE
143: RETURN
144: END IF
145: *
146: * NZ = maximum number of nonzero elements in each row of A, plus 1
147: *
148: NZ = 4
149: EPS = DLAMCH( 'Epsilon' )
150: SAFMIN = DLAMCH( 'Safe minimum' )
151: SAFE1 = NZ*SAFMIN
152: SAFE2 = SAFE1 / EPS
153: *
154: * Do for each right hand side
155: *
156: DO 90 J = 1, NRHS
157: *
158: COUNT = 1
159: LSTRES = THREE
160: 20 CONTINUE
161: *
162: * Loop until stopping criterion is satisfied.
163: *
164: * Compute residual R = B - A * X. Also compute
165: * abs(A)*abs(x) + abs(b) for use in the backward error bound.
166: *
167: IF( N.EQ.1 ) THEN
168: BI = B( 1, J )
169: DX = D( 1 )*X( 1, J )
170: WORK( N+1 ) = BI - DX
171: WORK( 1 ) = ABS( BI ) + ABS( DX )
172: ELSE
173: BI = B( 1, J )
174: DX = D( 1 )*X( 1, J )
175: EX = E( 1 )*X( 2, J )
176: WORK( N+1 ) = BI - DX - EX
177: WORK( 1 ) = ABS( BI ) + ABS( DX ) + ABS( EX )
178: DO 30 I = 2, N - 1
179: BI = B( I, J )
180: CX = E( I-1 )*X( I-1, J )
181: DX = D( I )*X( I, J )
182: EX = E( I )*X( I+1, J )
183: WORK( N+I ) = BI - CX - DX - EX
184: WORK( I ) = ABS( BI ) + ABS( CX ) + ABS( DX ) + ABS( EX )
185: 30 CONTINUE
186: BI = B( N, J )
187: CX = E( N-1 )*X( N-1, J )
188: DX = D( N )*X( N, J )
189: WORK( N+N ) = BI - CX - DX
190: WORK( N ) = ABS( BI ) + ABS( CX ) + ABS( DX )
191: END IF
192: *
193: * Compute componentwise relative backward error from formula
194: *
195: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
196: *
197: * where abs(Z) is the componentwise absolute value of the matrix
198: * or vector Z. If the i-th component of the denominator is less
199: * than SAFE2, then SAFE1 is added to the i-th components of the
200: * numerator and denominator before dividing.
201: *
202: S = ZERO
203: DO 40 I = 1, N
204: IF( WORK( I ).GT.SAFE2 ) THEN
205: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
206: ELSE
207: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
208: $ ( WORK( I )+SAFE1 ) )
209: END IF
210: 40 CONTINUE
211: BERR( J ) = S
212: *
213: * Test stopping criterion. Continue iterating if
214: * 1) The residual BERR(J) is larger than machine epsilon, and
215: * 2) BERR(J) decreased by at least a factor of 2 during the
216: * last iteration, and
217: * 3) At most ITMAX iterations tried.
218: *
219: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
220: $ COUNT.LE.ITMAX ) THEN
221: *
222: * Update solution and try again.
223: *
224: CALL DPTTRS( N, 1, DF, EF, WORK( N+1 ), N, INFO )
225: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
226: LSTRES = BERR( J )
227: COUNT = COUNT + 1
228: GO TO 20
229: END IF
230: *
231: * Bound error from formula
232: *
233: * norm(X - XTRUE) / norm(X) .le. FERR =
234: * norm( abs(inv(A))*
235: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
236: *
237: * where
238: * norm(Z) is the magnitude of the largest component of Z
239: * inv(A) is the inverse of A
240: * abs(Z) is the componentwise absolute value of the matrix or
241: * vector Z
242: * NZ is the maximum number of nonzeros in any row of A, plus 1
243: * EPS is machine epsilon
244: *
245: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
246: * is incremented by SAFE1 if the i-th component of
247: * abs(A)*abs(X) + abs(B) is less than SAFE2.
248: *
249: DO 50 I = 1, N
250: IF( WORK( I ).GT.SAFE2 ) THEN
251: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
252: ELSE
253: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
254: END IF
255: 50 CONTINUE
256: IX = IDAMAX( N, WORK, 1 )
257: FERR( J ) = WORK( IX )
258: *
259: * Estimate the norm of inv(A).
260: *
261: * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
262: *
263: * m(i,j) = abs(A(i,j)), i = j,
264: * m(i,j) = -abs(A(i,j)), i .ne. j,
265: *
266: * and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'.
267: *
268: * Solve M(L) * x = e.
269: *
270: WORK( 1 ) = ONE
271: DO 60 I = 2, N
272: WORK( I ) = ONE + WORK( I-1 )*ABS( EF( I-1 ) )
273: 60 CONTINUE
274: *
275: * Solve D * M(L)' * x = b.
276: *
277: WORK( N ) = WORK( N ) / DF( N )
278: DO 70 I = N - 1, 1, -1
279: WORK( I ) = WORK( I ) / DF( I ) + WORK( I+1 )*ABS( EF( I ) )
280: 70 CONTINUE
281: *
282: * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
283: *
284: IX = IDAMAX( N, WORK, 1 )
285: FERR( J ) = FERR( J )*ABS( WORK( IX ) )
286: *
287: * Normalize error.
288: *
289: LSTRES = ZERO
290: DO 80 I = 1, N
291: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
292: 80 CONTINUE
293: IF( LSTRES.NE.ZERO )
294: $ FERR( J ) = FERR( J ) / LSTRES
295: *
296: 90 CONTINUE
297: *
298: RETURN
299: *
300: * End of DPTRFS
301: *
302: END
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