Annotation of rpl/lapack/lapack/dgerfs.f, revision 1.5
1.1 bertrand 1: SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
2: $ X, LDX, FERR, BERR, WORK, IWORK, 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: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
10: *
11: * .. Scalar Arguments ..
12: CHARACTER TRANS
13: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: INTEGER IPIV( * ), IWORK( * )
17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DGERFS improves the computed solution to a system of linear
25: * equations and provides error bounds and backward error estimates for
26: * the solution.
27: *
28: * Arguments
29: * =========
30: *
31: * TRANS (input) CHARACTER*1
32: * Specifies the form of the system of equations:
33: * = 'N': A * X = B (No transpose)
34: * = 'T': A**T * X = B (Transpose)
35: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
36: *
37: * N (input) INTEGER
38: * The order of the matrix A. N >= 0.
39: *
40: * NRHS (input) INTEGER
41: * The number of right hand sides, i.e., the number of columns
42: * of the matrices B and X. NRHS >= 0.
43: *
44: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
45: * The original N-by-N matrix A.
46: *
47: * LDA (input) INTEGER
48: * The leading dimension of the array A. LDA >= max(1,N).
49: *
50: * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
51: * The factors L and U from the factorization A = P*L*U
52: * as computed by DGETRF.
53: *
54: * LDAF (input) INTEGER
55: * The leading dimension of the array AF. LDAF >= max(1,N).
56: *
57: * IPIV (input) INTEGER array, dimension (N)
58: * The pivot indices from DGETRF; for 1<=i<=N, row i of the
59: * matrix was interchanged with row IPIV(i).
60: *
61: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
62: * The right hand side matrix B.
63: *
64: * LDB (input) INTEGER
65: * The leading dimension of the array B. LDB >= max(1,N).
66: *
67: * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
68: * On entry, the solution matrix X, as computed by DGETRS.
69: * On exit, the improved solution matrix X.
70: *
71: * LDX (input) INTEGER
72: * The leading dimension of the array X. LDX >= max(1,N).
73: *
74: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
75: * The estimated forward error bound for each solution vector
76: * X(j) (the j-th column of the solution matrix X).
77: * If XTRUE is the true solution corresponding to X(j), FERR(j)
78: * is an estimated upper bound for the magnitude of the largest
79: * element in (X(j) - XTRUE) divided by the magnitude of the
80: * largest element in X(j). The estimate is as reliable as
81: * the estimate for RCOND, and is almost always a slight
82: * overestimate of the true error.
83: *
84: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
85: * The componentwise relative backward error of each solution
86: * vector X(j) (i.e., the smallest relative change in
87: * any element of A or B that makes X(j) an exact solution).
88: *
89: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
90: *
91: * IWORK (workspace) INTEGER array, dimension (N)
92: *
93: * INFO (output) INTEGER
94: * = 0: successful exit
95: * < 0: if INFO = -i, the i-th argument had an illegal value
96: *
97: * Internal Parameters
98: * ===================
99: *
100: * ITMAX is the maximum number of steps of iterative refinement.
101: *
102: * =====================================================================
103: *
104: * .. Parameters ..
105: INTEGER ITMAX
106: PARAMETER ( ITMAX = 5 )
107: DOUBLE PRECISION ZERO
108: PARAMETER ( ZERO = 0.0D+0 )
109: DOUBLE PRECISION ONE
110: PARAMETER ( ONE = 1.0D+0 )
111: DOUBLE PRECISION TWO
112: PARAMETER ( TWO = 2.0D+0 )
113: DOUBLE PRECISION THREE
114: PARAMETER ( THREE = 3.0D+0 )
115: * ..
116: * .. Local Scalars ..
117: LOGICAL NOTRAN
118: CHARACTER TRANST
119: INTEGER COUNT, I, J, K, KASE, NZ
120: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
121: * ..
122: * .. Local Arrays ..
123: INTEGER ISAVE( 3 )
124: * ..
125: * .. External Subroutines ..
126: EXTERNAL DAXPY, DCOPY, DGEMV, DGETRS, DLACN2, XERBLA
127: * ..
128: * .. Intrinsic Functions ..
129: INTRINSIC ABS, MAX
130: * ..
131: * .. External Functions ..
132: LOGICAL LSAME
133: DOUBLE PRECISION DLAMCH
134: EXTERNAL LSAME, DLAMCH
135: * ..
136: * .. Executable Statements ..
137: *
138: * Test the input parameters.
139: *
140: INFO = 0
141: NOTRAN = LSAME( TRANS, 'N' )
142: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
143: $ LSAME( TRANS, 'C' ) ) THEN
144: INFO = -1
145: ELSE IF( N.LT.0 ) THEN
146: INFO = -2
147: ELSE IF( NRHS.LT.0 ) THEN
148: INFO = -3
149: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
150: INFO = -5
151: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
152: INFO = -7
153: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
154: INFO = -10
155: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
156: INFO = -12
157: END IF
158: IF( INFO.NE.0 ) THEN
159: CALL XERBLA( 'DGERFS', -INFO )
160: RETURN
161: END IF
162: *
163: * Quick return if possible
164: *
165: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
166: DO 10 J = 1, NRHS
167: FERR( J ) = ZERO
168: BERR( J ) = ZERO
169: 10 CONTINUE
170: RETURN
171: END IF
172: *
173: IF( NOTRAN ) THEN
174: TRANST = 'T'
175: ELSE
176: TRANST = 'N'
177: END IF
178: *
179: * NZ = maximum number of nonzero elements in each row of A, plus 1
180: *
181: NZ = N + 1
182: EPS = DLAMCH( 'Epsilon' )
183: SAFMIN = DLAMCH( 'Safe minimum' )
184: SAFE1 = NZ*SAFMIN
185: SAFE2 = SAFE1 / EPS
186: *
187: * Do for each right hand side
188: *
189: DO 140 J = 1, NRHS
190: *
191: COUNT = 1
192: LSTRES = THREE
193: 20 CONTINUE
194: *
195: * Loop until stopping criterion is satisfied.
196: *
197: * Compute residual R = B - op(A) * X,
198: * where op(A) = A, A**T, or A**H, depending on TRANS.
199: *
200: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
201: CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
202: $ WORK( N+1 ), 1 )
203: *
204: * Compute componentwise relative backward error from formula
205: *
206: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
207: *
208: * where abs(Z) is the componentwise absolute value of the matrix
209: * or vector Z. If the i-th component of the denominator is less
210: * than SAFE2, then SAFE1 is added to the i-th components of the
211: * numerator and denominator before dividing.
212: *
213: DO 30 I = 1, N
214: WORK( I ) = ABS( B( I, J ) )
215: 30 CONTINUE
216: *
217: * Compute abs(op(A))*abs(X) + abs(B).
218: *
219: IF( NOTRAN ) THEN
220: DO 50 K = 1, N
221: XK = ABS( X( K, J ) )
222: DO 40 I = 1, N
223: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
224: 40 CONTINUE
225: 50 CONTINUE
226: ELSE
227: DO 70 K = 1, N
228: S = ZERO
229: DO 60 I = 1, N
230: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
231: 60 CONTINUE
232: WORK( K ) = WORK( K ) + S
233: 70 CONTINUE
234: END IF
235: S = ZERO
236: DO 80 I = 1, N
237: IF( WORK( I ).GT.SAFE2 ) THEN
238: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
239: ELSE
240: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
241: $ ( WORK( I )+SAFE1 ) )
242: END IF
243: 80 CONTINUE
244: BERR( J ) = S
245: *
246: * Test stopping criterion. Continue iterating if
247: * 1) The residual BERR(J) is larger than machine epsilon, and
248: * 2) BERR(J) decreased by at least a factor of 2 during the
249: * last iteration, and
250: * 3) At most ITMAX iterations tried.
251: *
252: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
253: $ COUNT.LE.ITMAX ) THEN
254: *
255: * Update solution and try again.
256: *
257: CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
258: $ INFO )
259: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
260: LSTRES = BERR( J )
261: COUNT = COUNT + 1
262: GO TO 20
263: END IF
264: *
265: * Bound error from formula
266: *
267: * norm(X - XTRUE) / norm(X) .le. FERR =
268: * norm( abs(inv(op(A)))*
269: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
270: *
271: * where
272: * norm(Z) is the magnitude of the largest component of Z
273: * inv(op(A)) is the inverse of op(A)
274: * abs(Z) is the componentwise absolute value of the matrix or
275: * vector Z
276: * NZ is the maximum number of nonzeros in any row of A, plus 1
277: * EPS is machine epsilon
278: *
279: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
280: * is incremented by SAFE1 if the i-th component of
281: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
282: *
283: * Use DLACN2 to estimate the infinity-norm of the matrix
284: * inv(op(A)) * diag(W),
285: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
286: *
287: DO 90 I = 1, N
288: IF( WORK( I ).GT.SAFE2 ) THEN
289: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
290: ELSE
291: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
292: END IF
293: 90 CONTINUE
294: *
295: KASE = 0
296: 100 CONTINUE
297: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
298: $ KASE, ISAVE )
299: IF( KASE.NE.0 ) THEN
300: IF( KASE.EQ.1 ) THEN
301: *
302: * Multiply by diag(W)*inv(op(A)**T).
303: *
304: CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
305: $ N, INFO )
306: DO 110 I = 1, N
307: WORK( N+I ) = WORK( I )*WORK( N+I )
308: 110 CONTINUE
309: ELSE
310: *
311: * Multiply by inv(op(A))*diag(W).
312: *
313: DO 120 I = 1, N
314: WORK( N+I ) = WORK( I )*WORK( N+I )
315: 120 CONTINUE
316: CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
317: $ INFO )
318: END IF
319: GO TO 100
320: END IF
321: *
322: * Normalize error.
323: *
324: LSTRES = ZERO
325: DO 130 I = 1, N
326: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
327: 130 CONTINUE
328: IF( LSTRES.NE.ZERO )
329: $ FERR( J ) = FERR( J ) / LSTRES
330: *
331: 140 CONTINUE
332: *
333: RETURN
334: *
335: * End of DGERFS
336: *
337: END
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