1: SUBROUTINE DTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
2: $ LDB, X, 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, KD, LDAB, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: INTEGER IWORK( * )
17: DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
18: $ FERR( * ), WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DTBRFS provides error bounds and backward error estimates for the
25: * solution to a system of linear equations with a triangular band
26: * coefficient matrix.
27: *
28: * The solution matrix X must be computed by DTBTRS or some other
29: * means before entering this routine. DTBRFS 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: * KD (input) INTEGER
53: * The number of superdiagonals or subdiagonals of the
54: * triangular band matrix A. KD >= 0.
55: *
56: * NRHS (input) INTEGER
57: * The number of right hand sides, i.e., the number of columns
58: * of the matrices B and X. NRHS >= 0.
59: *
60: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
61: * The upper or lower triangular band matrix A, stored in the
62: * first kd+1 rows of the array. The j-th column of A is stored
63: * in the j-th column of the array AB as follows:
64: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
65: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
66: * If DIAG = 'U', the diagonal elements of A are not referenced
67: * and are assumed to be 1.
68: *
69: * LDAB (input) INTEGER
70: * The leading dimension of the array AB. LDAB >= KD+1.
71: *
72: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
73: * The right hand side matrix B.
74: *
75: * LDB (input) INTEGER
76: * The leading dimension of the array B. LDB >= max(1,N).
77: *
78: * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
79: * The solution matrix X.
80: *
81: * LDX (input) INTEGER
82: * The leading dimension of the array X. LDX >= max(1,N).
83: *
84: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
85: * The estimated forward error bound for each solution vector
86: * X(j) (the j-th column of the solution matrix X).
87: * If XTRUE is the true solution corresponding to X(j), FERR(j)
88: * is an estimated upper bound for the magnitude of the largest
89: * element in (X(j) - XTRUE) divided by the magnitude of the
90: * largest element in X(j). The estimate is as reliable as
91: * the estimate for RCOND, and is almost always a slight
92: * overestimate of the true error.
93: *
94: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
95: * The componentwise relative backward error of each solution
96: * vector X(j) (i.e., the smallest relative change in
97: * any element of A or B that makes X(j) an exact solution).
98: *
99: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
100: *
101: * IWORK (workspace) INTEGER array, dimension (N)
102: *
103: * INFO (output) INTEGER
104: * = 0: successful exit
105: * < 0: if INFO = -i, the i-th argument had an illegal value
106: *
107: * =====================================================================
108: *
109: * .. Parameters ..
110: DOUBLE PRECISION ZERO
111: PARAMETER ( ZERO = 0.0D+0 )
112: DOUBLE PRECISION ONE
113: PARAMETER ( ONE = 1.0D+0 )
114: * ..
115: * .. Local Scalars ..
116: LOGICAL NOTRAN, NOUNIT, UPPER
117: CHARACTER TRANST
118: INTEGER I, J, K, KASE, NZ
119: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
120: * ..
121: * .. Local Arrays ..
122: INTEGER ISAVE( 3 )
123: * ..
124: * .. External Subroutines ..
125: EXTERNAL DAXPY, DCOPY, DLACN2, DTBMV, DTBSV, XERBLA
126: * ..
127: * .. Intrinsic Functions ..
128: INTRINSIC ABS, MAX, MIN
129: * ..
130: * .. External Functions ..
131: LOGICAL LSAME
132: DOUBLE PRECISION DLAMCH
133: EXTERNAL LSAME, DLAMCH
134: * ..
135: * .. Executable Statements ..
136: *
137: * Test the input parameters.
138: *
139: INFO = 0
140: UPPER = LSAME( UPLO, 'U' )
141: NOTRAN = LSAME( TRANS, 'N' )
142: NOUNIT = LSAME( DIAG, 'N' )
143: *
144: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
145: INFO = -1
146: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
147: $ LSAME( TRANS, 'C' ) ) THEN
148: INFO = -2
149: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
150: INFO = -3
151: ELSE IF( N.LT.0 ) THEN
152: INFO = -4
153: ELSE IF( KD.LT.0 ) THEN
154: INFO = -5
155: ELSE IF( NRHS.LT.0 ) THEN
156: INFO = -6
157: ELSE IF( LDAB.LT.KD+1 ) THEN
158: INFO = -8
159: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
160: INFO = -10
161: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
162: INFO = -12
163: END IF
164: IF( INFO.NE.0 ) THEN
165: CALL XERBLA( 'DTBRFS', -INFO )
166: RETURN
167: END IF
168: *
169: * Quick return if possible
170: *
171: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
172: DO 10 J = 1, NRHS
173: FERR( J ) = ZERO
174: BERR( J ) = ZERO
175: 10 CONTINUE
176: RETURN
177: END IF
178: *
179: IF( NOTRAN ) THEN
180: TRANST = 'T'
181: ELSE
182: TRANST = 'N'
183: END IF
184: *
185: * NZ = maximum number of nonzero elements in each row of A, plus 1
186: *
187: NZ = KD + 2
188: EPS = DLAMCH( 'Epsilon' )
189: SAFMIN = DLAMCH( 'Safe minimum' )
190: SAFE1 = NZ*SAFMIN
191: SAFE2 = SAFE1 / EPS
192: *
193: * Do for each right hand side
194: *
195: DO 250 J = 1, NRHS
196: *
197: * Compute residual R = B - op(A) * X,
198: * where op(A) = A or A**T, depending on TRANS.
199: *
200: CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
201: CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK( N+1 ),
202: $ 1 )
203: CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
204: *
205: * Compute componentwise relative backward error from formula
206: *
207: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
208: *
209: * where abs(Z) is the componentwise absolute value of the matrix
210: * or vector Z. If the i-th component of the denominator is less
211: * than SAFE2, then SAFE1 is added to the i-th components of the
212: * numerator and denominator before dividing.
213: *
214: DO 20 I = 1, N
215: WORK( I ) = ABS( B( I, J ) )
216: 20 CONTINUE
217: *
218: IF( NOTRAN ) THEN
219: *
220: * Compute abs(A)*abs(X) + abs(B).
221: *
222: IF( UPPER ) THEN
223: IF( NOUNIT ) THEN
224: DO 40 K = 1, N
225: XK = ABS( X( K, J ) )
226: DO 30 I = MAX( 1, K-KD ), K
227: WORK( I ) = WORK( I ) +
228: $ ABS( AB( KD+1+I-K, K ) )*XK
229: 30 CONTINUE
230: 40 CONTINUE
231: ELSE
232: DO 60 K = 1, N
233: XK = ABS( X( K, J ) )
234: DO 50 I = MAX( 1, K-KD ), K - 1
235: WORK( I ) = WORK( I ) +
236: $ ABS( AB( KD+1+I-K, K ) )*XK
237: 50 CONTINUE
238: WORK( K ) = WORK( K ) + XK
239: 60 CONTINUE
240: END IF
241: ELSE
242: IF( NOUNIT ) THEN
243: DO 80 K = 1, N
244: XK = ABS( X( K, J ) )
245: DO 70 I = K, MIN( N, K+KD )
246: WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
247: 70 CONTINUE
248: 80 CONTINUE
249: ELSE
250: DO 100 K = 1, N
251: XK = ABS( X( K, J ) )
252: DO 90 I = K + 1, MIN( N, K+KD )
253: WORK( I ) = WORK( I ) + ABS( AB( 1+I-K, K ) )*XK
254: 90 CONTINUE
255: WORK( K ) = WORK( K ) + XK
256: 100 CONTINUE
257: END IF
258: END IF
259: ELSE
260: *
261: * Compute abs(A**T)*abs(X) + abs(B).
262: *
263: IF( UPPER ) THEN
264: IF( NOUNIT ) THEN
265: DO 120 K = 1, N
266: S = ZERO
267: DO 110 I = MAX( 1, K-KD ), K
268: S = S + ABS( AB( KD+1+I-K, K ) )*
269: $ ABS( X( I, J ) )
270: 110 CONTINUE
271: WORK( K ) = WORK( K ) + S
272: 120 CONTINUE
273: ELSE
274: DO 140 K = 1, N
275: S = ABS( X( K, J ) )
276: DO 130 I = MAX( 1, K-KD ), K - 1
277: S = S + ABS( AB( KD+1+I-K, K ) )*
278: $ ABS( X( I, J ) )
279: 130 CONTINUE
280: WORK( K ) = WORK( K ) + S
281: 140 CONTINUE
282: END IF
283: ELSE
284: IF( NOUNIT ) THEN
285: DO 160 K = 1, N
286: S = ZERO
287: DO 150 I = K, MIN( N, K+KD )
288: S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
289: 150 CONTINUE
290: WORK( K ) = WORK( K ) + S
291: 160 CONTINUE
292: ELSE
293: DO 180 K = 1, N
294: S = ABS( X( K, J ) )
295: DO 170 I = K + 1, MIN( N, K+KD )
296: S = S + ABS( AB( 1+I-K, K ) )*ABS( X( I, J ) )
297: 170 CONTINUE
298: WORK( K ) = WORK( K ) + S
299: 180 CONTINUE
300: END IF
301: END IF
302: END IF
303: S = ZERO
304: DO 190 I = 1, N
305: IF( WORK( I ).GT.SAFE2 ) THEN
306: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
307: ELSE
308: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
309: $ ( WORK( I )+SAFE1 ) )
310: END IF
311: 190 CONTINUE
312: BERR( J ) = S
313: *
314: * Bound error from formula
315: *
316: * norm(X - XTRUE) / norm(X) .le. FERR =
317: * norm( abs(inv(op(A)))*
318: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
319: *
320: * where
321: * norm(Z) is the magnitude of the largest component of Z
322: * inv(op(A)) is the inverse of op(A)
323: * abs(Z) is the componentwise absolute value of the matrix or
324: * vector Z
325: * NZ is the maximum number of nonzeros in any row of A, plus 1
326: * EPS is machine epsilon
327: *
328: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
329: * is incremented by SAFE1 if the i-th component of
330: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
331: *
332: * Use DLACN2 to estimate the infinity-norm of the matrix
333: * inv(op(A)) * diag(W),
334: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
335: *
336: DO 200 I = 1, N
337: IF( WORK( I ).GT.SAFE2 ) THEN
338: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
339: ELSE
340: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
341: END IF
342: 200 CONTINUE
343: *
344: KASE = 0
345: 210 CONTINUE
346: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
347: $ KASE, ISAVE )
348: IF( KASE.NE.0 ) THEN
349: IF( KASE.EQ.1 ) THEN
350: *
351: * Multiply by diag(W)*inv(op(A)**T).
352: *
353: CALL DTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB,
354: $ WORK( N+1 ), 1 )
355: DO 220 I = 1, N
356: WORK( N+I ) = WORK( I )*WORK( N+I )
357: 220 CONTINUE
358: ELSE
359: *
360: * Multiply by inv(op(A))*diag(W).
361: *
362: DO 230 I = 1, N
363: WORK( N+I ) = WORK( I )*WORK( N+I )
364: 230 CONTINUE
365: CALL DTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB,
366: $ WORK( N+1 ), 1 )
367: END IF
368: GO TO 210
369: END IF
370: *
371: * Normalize error.
372: *
373: LSTRES = ZERO
374: DO 240 I = 1, N
375: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
376: 240 CONTINUE
377: IF( LSTRES.NE.ZERO )
378: $ FERR( J ) = FERR( J ) / LSTRES
379: *
380: 250 CONTINUE
381: *
382: RETURN
383: *
384: * End of DTBRFS
385: *
386: END
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