1: SUBROUTINE ZTRRFS( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
2: $ LDX, FERR, BERR, WORK, RWORK, 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 ZLACN2 in place of ZLACON, 10 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: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
18: $ X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * ZTRRFS 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 ZTRTRS or some other
29: * means before entering this routine. ZTRRFS 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)
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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 array, dimension (2*N)
97: *
98: * RWORK (workspace) DOUBLE PRECISION 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: COMPLEX*16 ONE
110: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
111: * ..
112: * .. Local Scalars ..
113: LOGICAL NOTRAN, NOUNIT, UPPER
114: CHARACTER TRANSN, TRANST
115: INTEGER I, J, K, KASE, NZ
116: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
117: COMPLEX*16 ZDUM
118: * ..
119: * .. Local Arrays ..
120: INTEGER ISAVE( 3 )
121: * ..
122: * .. External Subroutines ..
123: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTRMV, ZTRSV
124: * ..
125: * .. Intrinsic Functions ..
126: INTRINSIC ABS, DBLE, DIMAG, MAX
127: * ..
128: * .. External Functions ..
129: LOGICAL LSAME
130: DOUBLE PRECISION DLAMCH
131: EXTERNAL LSAME, DLAMCH
132: * ..
133: * .. Statement Functions ..
134: DOUBLE PRECISION CABS1
135: * ..
136: * .. Statement Function definitions ..
137: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
138: * ..
139: * .. Executable Statements ..
140: *
141: * Test the input parameters.
142: *
143: INFO = 0
144: UPPER = LSAME( UPLO, 'U' )
145: NOTRAN = LSAME( TRANS, 'N' )
146: NOUNIT = LSAME( DIAG, 'N' )
147: *
148: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
149: INFO = -1
150: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
151: $ LSAME( TRANS, 'C' ) ) THEN
152: INFO = -2
153: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
154: INFO = -3
155: ELSE IF( N.LT.0 ) THEN
156: INFO = -4
157: ELSE IF( NRHS.LT.0 ) THEN
158: INFO = -5
159: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
160: INFO = -7
161: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
162: INFO = -9
163: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
164: INFO = -11
165: END IF
166: IF( INFO.NE.0 ) THEN
167: CALL XERBLA( 'ZTRRFS', -INFO )
168: RETURN
169: END IF
170: *
171: * Quick return if possible
172: *
173: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
174: DO 10 J = 1, NRHS
175: FERR( J ) = ZERO
176: BERR( J ) = ZERO
177: 10 CONTINUE
178: RETURN
179: END IF
180: *
181: IF( NOTRAN ) THEN
182: TRANSN = 'N'
183: TRANST = 'C'
184: ELSE
185: TRANSN = 'C'
186: TRANST = 'N'
187: END IF
188: *
189: * NZ = maximum number of nonzero elements in each row of A, plus 1
190: *
191: NZ = N + 1
192: EPS = DLAMCH( 'Epsilon' )
193: SAFMIN = DLAMCH( 'Safe minimum' )
194: SAFE1 = NZ*SAFMIN
195: SAFE2 = SAFE1 / EPS
196: *
197: * Do for each right hand side
198: *
199: DO 250 J = 1, NRHS
200: *
201: * Compute residual R = B - op(A) * X,
202: * where op(A) = A, A**T, or A**H, depending on TRANS.
203: *
204: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
205: CALL ZTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 )
206: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
207: *
208: * Compute componentwise relative backward error from formula
209: *
210: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
211: *
212: * where abs(Z) is the componentwise absolute value of the matrix
213: * or vector Z. If the i-th component of the denominator is less
214: * than SAFE2, then SAFE1 is added to the i-th components of the
215: * numerator and denominator before dividing.
216: *
217: DO 20 I = 1, N
218: RWORK( I ) = CABS1( B( I, J ) )
219: 20 CONTINUE
220: *
221: IF( NOTRAN ) THEN
222: *
223: * Compute abs(A)*abs(X) + abs(B).
224: *
225: IF( UPPER ) THEN
226: IF( NOUNIT ) THEN
227: DO 40 K = 1, N
228: XK = CABS1( X( K, J ) )
229: DO 30 I = 1, K
230: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
231: 30 CONTINUE
232: 40 CONTINUE
233: ELSE
234: DO 60 K = 1, N
235: XK = CABS1( X( K, J ) )
236: DO 50 I = 1, K - 1
237: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
238: 50 CONTINUE
239: RWORK( K ) = RWORK( K ) + XK
240: 60 CONTINUE
241: END IF
242: ELSE
243: IF( NOUNIT ) THEN
244: DO 80 K = 1, N
245: XK = CABS1( X( K, J ) )
246: DO 70 I = K, N
247: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
248: 70 CONTINUE
249: 80 CONTINUE
250: ELSE
251: DO 100 K = 1, N
252: XK = CABS1( X( K, J ) )
253: DO 90 I = K + 1, N
254: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
255: 90 CONTINUE
256: RWORK( K ) = RWORK( K ) + XK
257: 100 CONTINUE
258: END IF
259: END IF
260: ELSE
261: *
262: * Compute abs(A**H)*abs(X) + abs(B).
263: *
264: IF( UPPER ) THEN
265: IF( NOUNIT ) THEN
266: DO 120 K = 1, N
267: S = ZERO
268: DO 110 I = 1, K
269: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
270: 110 CONTINUE
271: RWORK( K ) = RWORK( K ) + S
272: 120 CONTINUE
273: ELSE
274: DO 140 K = 1, N
275: S = CABS1( X( K, J ) )
276: DO 130 I = 1, K - 1
277: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
278: 130 CONTINUE
279: RWORK( K ) = RWORK( K ) + S
280: 140 CONTINUE
281: END IF
282: ELSE
283: IF( NOUNIT ) THEN
284: DO 160 K = 1, N
285: S = ZERO
286: DO 150 I = K, N
287: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
288: 150 CONTINUE
289: RWORK( K ) = RWORK( K ) + S
290: 160 CONTINUE
291: ELSE
292: DO 180 K = 1, N
293: S = CABS1( X( K, J ) )
294: DO 170 I = K + 1, N
295: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
296: 170 CONTINUE
297: RWORK( K ) = RWORK( K ) + S
298: 180 CONTINUE
299: END IF
300: END IF
301: END IF
302: S = ZERO
303: DO 190 I = 1, N
304: IF( RWORK( I ).GT.SAFE2 ) THEN
305: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
306: ELSE
307: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
308: $ ( RWORK( I )+SAFE1 ) )
309: END IF
310: 190 CONTINUE
311: BERR( J ) = S
312: *
313: * Bound error from formula
314: *
315: * norm(X - XTRUE) / norm(X) .le. FERR =
316: * norm( abs(inv(op(A)))*
317: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
318: *
319: * where
320: * norm(Z) is the magnitude of the largest component of Z
321: * inv(op(A)) is the inverse of op(A)
322: * abs(Z) is the componentwise absolute value of the matrix or
323: * vector Z
324: * NZ is the maximum number of nonzeros in any row of A, plus 1
325: * EPS is machine epsilon
326: *
327: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
328: * is incremented by SAFE1 if the i-th component of
329: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
330: *
331: * Use ZLACN2 to estimate the infinity-norm of the matrix
332: * inv(op(A)) * diag(W),
333: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
334: *
335: DO 200 I = 1, N
336: IF( RWORK( I ).GT.SAFE2 ) THEN
337: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
338: ELSE
339: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
340: $ SAFE1
341: END IF
342: 200 CONTINUE
343: *
344: KASE = 0
345: 210 CONTINUE
346: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
347: IF( KASE.NE.0 ) THEN
348: IF( KASE.EQ.1 ) THEN
349: *
350: * Multiply by diag(W)*inv(op(A)**H).
351: *
352: CALL ZTRSV( UPLO, TRANST, DIAG, N, A, LDA, WORK, 1 )
353: DO 220 I = 1, N
354: WORK( I ) = RWORK( I )*WORK( I )
355: 220 CONTINUE
356: ELSE
357: *
358: * Multiply by inv(op(A))*diag(W).
359: *
360: DO 230 I = 1, N
361: WORK( I ) = RWORK( I )*WORK( I )
362: 230 CONTINUE
363: CALL ZTRSV( UPLO, TRANSN, DIAG, N, A, LDA, WORK, 1 )
364: END IF
365: GO TO 210
366: END IF
367: *
368: * Normalize error.
369: *
370: LSTRES = ZERO
371: DO 240 I = 1, N
372: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
373: 240 CONTINUE
374: IF( LSTRES.NE.ZERO )
375: $ FERR( J ) = FERR( J ) / LSTRES
376: *
377: 250 CONTINUE
378: *
379: RETURN
380: *
381: * End of ZTRRFS
382: *
383: END
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