1: SUBROUTINE DTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
2: $ LDX, FERR, BERR, WORK, IWORK, INFO )
3: *
4: * -- LAPACK routine (version 3.3.1) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * -- April 2011 --
8: *
9: * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
10: *
11: * .. Scalar Arguments ..
12: CHARACTER DIAG, TRANS, UPLO
13: INTEGER INFO, LDA, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: INTEGER IWORK( * )
17: DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
18: $ WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DTRRFS provides error bounds and backward error estimates for the
25: * solution to a system of linear equations with a triangular
26: * coefficient matrix.
27: *
28: * The solution matrix X must be computed by DTRTRS or some other
29: * means before entering this routine. DTRRFS does not do iterative
30: * refinement because doing so cannot improve the backward error.
31: *
32: * Arguments
33: * =========
34: *
35: * UPLO (input) CHARACTER*1
36: * = 'U': A is upper triangular;
37: * = 'L': A is lower triangular.
38: *
39: * TRANS (input) CHARACTER*1
40: * Specifies the form of the system of equations:
41: * = 'N': A * X = B (No transpose)
42: * = 'T': A**T * X = B (Transpose)
43: * = 'C': A**H * X = B (Conjugate transpose = Transpose)
44: *
45: * DIAG (input) CHARACTER*1
46: * = 'N': A is non-unit triangular;
47: * = 'U': A is unit triangular.
48: *
49: * N (input) INTEGER
50: * The order of the matrix A. N >= 0.
51: *
52: * NRHS (input) INTEGER
53: * The number of right hand sides, i.e., the number of columns
54: * of the matrices B and X. NRHS >= 0.
55: *
56: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
57: * The triangular matrix A. If UPLO = 'U', the leading N-by-N
58: * upper triangular part of the array A contains the upper
59: * triangular matrix, and the strictly lower triangular part of
60: * A is not referenced. If UPLO = 'L', the leading N-by-N lower
61: * triangular part of the array A contains the lower triangular
62: * matrix, and the strictly upper triangular part of A is not
63: * referenced. If DIAG = 'U', the diagonal elements of A are
64: * also not referenced and are assumed to be 1.
65: *
66: * LDA (input) INTEGER
67: * The leading dimension of the array A. LDA >= max(1,N).
68: *
69: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
70: * The right hand side matrix B.
71: *
72: * LDB (input) INTEGER
73: * The leading dimension of the array B. LDB >= max(1,N).
74: *
75: * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
76: * The solution matrix X.
77: *
78: * LDX (input) INTEGER
79: * The leading dimension of the array X. LDX >= max(1,N).
80: *
81: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
82: * The estimated forward error bound for each solution vector
83: * X(j) (the j-th column of the solution matrix X).
84: * If XTRUE is the true solution corresponding to X(j), FERR(j)
85: * is an estimated upper bound for the magnitude of the largest
86: * element in (X(j) - XTRUE) divided by the magnitude of the
87: * largest element in X(j). The estimate is as reliable as
88: * the estimate for RCOND, and is almost always a slight
89: * overestimate of the true error.
90: *
91: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
92: * The componentwise relative backward error of each solution
93: * vector X(j) (i.e., the smallest relative change in
94: * any element of A or B that makes X(j) an exact solution).
95: *
96: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
97: *
98: * IWORK (workspace) INTEGER array, dimension (N)
99: *
100: * INFO (output) INTEGER
101: * = 0: successful exit
102: * < 0: if INFO = -i, the i-th argument had an illegal value
103: *
104: * =====================================================================
105: *
106: * .. Parameters ..
107: DOUBLE PRECISION ZERO
108: PARAMETER ( ZERO = 0.0D+0 )
109: DOUBLE PRECISION ONE
110: PARAMETER ( ONE = 1.0D+0 )
111: * ..
112: * .. Local Scalars ..
113: LOGICAL NOTRAN, NOUNIT, UPPER
114: CHARACTER TRANST
115: INTEGER I, J, K, KASE, NZ
116: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
117: * ..
118: * .. Local Arrays ..
119: INTEGER ISAVE( 3 )
120: * ..
121: * .. External Subroutines ..
122: EXTERNAL DAXPY, DCOPY, DLACN2, DTRMV, DTRSV, XERBLA
123: * ..
124: * .. Intrinsic Functions ..
125: INTRINSIC ABS, MAX
126: * ..
127: * .. External Functions ..
128: LOGICAL LSAME
129: DOUBLE PRECISION DLAMCH
130: EXTERNAL LSAME, DLAMCH
131: * ..
132: * .. Executable Statements ..
133: *
134: * Test the input parameters.
135: *
136: INFO = 0
137: UPPER = LSAME( UPLO, 'U' )
138: NOTRAN = LSAME( TRANS, 'N' )
139: NOUNIT = LSAME( DIAG, 'N' )
140: *
141: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
142: INFO = -1
143: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
144: $ LSAME( TRANS, 'C' ) ) THEN
145: INFO = -2
146: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
147: INFO = -3
148: ELSE IF( N.LT.0 ) THEN
149: INFO = -4
150: ELSE IF( NRHS.LT.0 ) THEN
151: INFO = -5
152: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
153: INFO = -7
154: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
155: INFO = -9
156: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
157: INFO = -11
158: END IF
159: IF( INFO.NE.0 ) THEN
160: CALL XERBLA( 'DTRRFS', -INFO )
161: RETURN
162: END IF
163: *
164: * Quick return if possible
165: *
166: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
167: DO 10 J = 1, NRHS
168: FERR( J ) = ZERO
169: BERR( J ) = ZERO
170: 10 CONTINUE
171: RETURN
172: END IF
173: *
174: IF( NOTRAN ) THEN
175: TRANST = 'T'
176: ELSE
177: TRANST = 'N'
178: END IF
179: *
180: * NZ = maximum number of nonzero elements in each row of A, plus 1
181: *
182: NZ = N + 1
183: EPS = DLAMCH( 'Epsilon' )
184: SAFMIN = DLAMCH( 'Safe minimum' )
185: SAFE1 = NZ*SAFMIN
186: SAFE2 = SAFE1 / EPS
187: *
188: * Do for each right hand side
189: *
190: DO 250 J = 1, NRHS
191: *
192: * Compute residual R = B - op(A) * X,
193: * where op(A) = A or A**T, depending on TRANS.
194: *
195: CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
196: CALL DTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ), 1 )
197: CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
198: *
199: * Compute componentwise relative backward error from formula
200: *
201: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
202: *
203: * where abs(Z) is the componentwise absolute value of the matrix
204: * or vector Z. If the i-th component of the denominator is less
205: * than SAFE2, then SAFE1 is added to the i-th components of the
206: * numerator and denominator before dividing.
207: *
208: DO 20 I = 1, N
209: WORK( I ) = ABS( B( I, J ) )
210: 20 CONTINUE
211: *
212: IF( NOTRAN ) THEN
213: *
214: * Compute abs(A)*abs(X) + abs(B).
215: *
216: IF( UPPER ) THEN
217: IF( NOUNIT ) THEN
218: DO 40 K = 1, N
219: XK = ABS( X( K, J ) )
220: DO 30 I = 1, K
221: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
222: 30 CONTINUE
223: 40 CONTINUE
224: ELSE
225: DO 60 K = 1, N
226: XK = ABS( X( K, J ) )
227: DO 50 I = 1, K - 1
228: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
229: 50 CONTINUE
230: WORK( K ) = WORK( K ) + XK
231: 60 CONTINUE
232: END IF
233: ELSE
234: IF( NOUNIT ) THEN
235: DO 80 K = 1, N
236: XK = ABS( X( K, J ) )
237: DO 70 I = K, N
238: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
239: 70 CONTINUE
240: 80 CONTINUE
241: ELSE
242: DO 100 K = 1, N
243: XK = ABS( X( K, J ) )
244: DO 90 I = K + 1, N
245: WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
246: 90 CONTINUE
247: WORK( K ) = WORK( K ) + XK
248: 100 CONTINUE
249: END IF
250: END IF
251: ELSE
252: *
253: * Compute abs(A**T)*abs(X) + abs(B).
254: *
255: IF( UPPER ) THEN
256: IF( NOUNIT ) THEN
257: DO 120 K = 1, N
258: S = ZERO
259: DO 110 I = 1, K
260: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
261: 110 CONTINUE
262: WORK( K ) = WORK( K ) + S
263: 120 CONTINUE
264: ELSE
265: DO 140 K = 1, N
266: S = ABS( X( K, J ) )
267: DO 130 I = 1, K - 1
268: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
269: 130 CONTINUE
270: WORK( K ) = WORK( K ) + S
271: 140 CONTINUE
272: END IF
273: ELSE
274: IF( NOUNIT ) THEN
275: DO 160 K = 1, N
276: S = ZERO
277: DO 150 I = K, N
278: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
279: 150 CONTINUE
280: WORK( K ) = WORK( K ) + S
281: 160 CONTINUE
282: ELSE
283: DO 180 K = 1, N
284: S = ABS( X( K, J ) )
285: DO 170 I = K + 1, N
286: S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
287: 170 CONTINUE
288: WORK( K ) = WORK( K ) + S
289: 180 CONTINUE
290: END IF
291: END IF
292: END IF
293: S = ZERO
294: DO 190 I = 1, N
295: IF( WORK( I ).GT.SAFE2 ) THEN
296: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
297: ELSE
298: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
299: $ ( WORK( I )+SAFE1 ) )
300: END IF
301: 190 CONTINUE
302: BERR( J ) = S
303: *
304: * Bound error from formula
305: *
306: * norm(X - XTRUE) / norm(X) .le. FERR =
307: * norm( abs(inv(op(A)))*
308: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
309: *
310: * where
311: * norm(Z) is the magnitude of the largest component of Z
312: * inv(op(A)) is the inverse of op(A)
313: * abs(Z) is the componentwise absolute value of the matrix or
314: * vector Z
315: * NZ is the maximum number of nonzeros in any row of A, plus 1
316: * EPS is machine epsilon
317: *
318: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
319: * is incremented by SAFE1 if the i-th component of
320: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
321: *
322: * Use DLACN2 to estimate the infinity-norm of the matrix
323: * inv(op(A)) * diag(W),
324: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
325: *
326: DO 200 I = 1, N
327: IF( WORK( I ).GT.SAFE2 ) THEN
328: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
329: ELSE
330: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
331: END IF
332: 200 CONTINUE
333: *
334: KASE = 0
335: 210 CONTINUE
336: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
337: $ KASE, ISAVE )
338: IF( KASE.NE.0 ) THEN
339: IF( KASE.EQ.1 ) THEN
340: *
341: * Multiply by diag(W)*inv(op(A)**T).
342: *
343: CALL DTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK( N+1 ),
344: $ 1 )
345: DO 220 I = 1, N
346: WORK( N+I ) = WORK( I )*WORK( N+I )
347: 220 CONTINUE
348: ELSE
349: *
350: * Multiply by inv(op(A))*diag(W).
351: *
352: DO 230 I = 1, N
353: WORK( N+I ) = WORK( I )*WORK( N+I )
354: 230 CONTINUE
355: CALL DTRSV( UPLO, TRANS, DIAG, N, A, LDA, WORK( N+1 ),
356: $ 1 )
357: END IF
358: GO TO 210
359: END IF
360: *
361: * Normalize error.
362: *
363: LSTRES = ZERO
364: DO 240 I = 1, N
365: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
366: 240 CONTINUE
367: IF( LSTRES.NE.ZERO )
368: $ FERR( J ) = FERR( J ) / LSTRES
369: *
370: 250 CONTINUE
371: *
372: RETURN
373: *
374: * End of DTRRFS
375: *
376: END
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