1: SUBROUTINE ZTBRFS( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B,
2: $ LDB, X, 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, KD, LDAB, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17: COMPLEX*16 AB( LDAB, * ), B( LDB, * ), WORK( * ),
18: $ X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * ZTBRFS 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 ZTBTRS or some other
29: * means before entering this routine. ZTBRFS 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: * 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 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) COMPLEX*16 array, dimension (2*N)
100: *
101: * RWORK (workspace) DOUBLE PRECISION 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: COMPLEX*16 ONE
113: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
114: * ..
115: * .. Local Scalars ..
116: LOGICAL NOTRAN, NOUNIT, UPPER
117: CHARACTER TRANSN, TRANST
118: INTEGER I, J, K, KASE, NZ
119: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
120: COMPLEX*16 ZDUM
121: * ..
122: * .. Local Arrays ..
123: INTEGER ISAVE( 3 )
124: * ..
125: * .. External Subroutines ..
126: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZTBMV, ZTBSV
127: * ..
128: * .. Intrinsic Functions ..
129: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
130: * ..
131: * .. External Functions ..
132: LOGICAL LSAME
133: DOUBLE PRECISION DLAMCH
134: EXTERNAL LSAME, DLAMCH
135: * ..
136: * .. Statement Functions ..
137: DOUBLE PRECISION CABS1
138: * ..
139: * .. Statement Function definitions ..
140: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
141: * ..
142: * .. Executable Statements ..
143: *
144: * Test the input parameters.
145: *
146: INFO = 0
147: UPPER = LSAME( UPLO, 'U' )
148: NOTRAN = LSAME( TRANS, 'N' )
149: NOUNIT = LSAME( DIAG, 'N' )
150: *
151: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
152: INFO = -1
153: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
154: $ LSAME( TRANS, 'C' ) ) THEN
155: INFO = -2
156: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
157: INFO = -3
158: ELSE IF( N.LT.0 ) THEN
159: INFO = -4
160: ELSE IF( KD.LT.0 ) THEN
161: INFO = -5
162: ELSE IF( NRHS.LT.0 ) THEN
163: INFO = -6
164: ELSE IF( LDAB.LT.KD+1 ) THEN
165: INFO = -8
166: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
167: INFO = -10
168: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
169: INFO = -12
170: END IF
171: IF( INFO.NE.0 ) THEN
172: CALL XERBLA( 'ZTBRFS', -INFO )
173: RETURN
174: END IF
175: *
176: * Quick return if possible
177: *
178: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
179: DO 10 J = 1, NRHS
180: FERR( J ) = ZERO
181: BERR( J ) = ZERO
182: 10 CONTINUE
183: RETURN
184: END IF
185: *
186: IF( NOTRAN ) THEN
187: TRANSN = 'N'
188: TRANST = 'C'
189: ELSE
190: TRANSN = 'C'
191: TRANST = 'N'
192: END IF
193: *
194: * NZ = maximum number of nonzero elements in each row of A, plus 1
195: *
196: NZ = KD + 2
197: EPS = DLAMCH( 'Epsilon' )
198: SAFMIN = DLAMCH( 'Safe minimum' )
199: SAFE1 = NZ*SAFMIN
200: SAFE2 = SAFE1 / EPS
201: *
202: * Do for each right hand side
203: *
204: DO 250 J = 1, NRHS
205: *
206: * Compute residual R = B - op(A) * X,
207: * where op(A) = A, A**T, or A**H, depending on TRANS.
208: *
209: CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
210: CALL ZTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 )
211: CALL ZAXPY( N, -ONE, B( 1, J ), 1, WORK, 1 )
212: *
213: * Compute componentwise relative backward error from formula
214: *
215: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
216: *
217: * where abs(Z) is the componentwise absolute value of the matrix
218: * or vector Z. If the i-th component of the denominator is less
219: * than SAFE2, then SAFE1 is added to the i-th components of the
220: * numerator and denominator before dividing.
221: *
222: DO 20 I = 1, N
223: RWORK( I ) = CABS1( B( I, J ) )
224: 20 CONTINUE
225: *
226: IF( NOTRAN ) THEN
227: *
228: * Compute abs(A)*abs(X) + abs(B).
229: *
230: IF( UPPER ) THEN
231: IF( NOUNIT ) THEN
232: DO 40 K = 1, N
233: XK = CABS1( X( K, J ) )
234: DO 30 I = MAX( 1, K-KD ), K
235: RWORK( I ) = RWORK( I ) +
236: $ CABS1( AB( KD+1+I-K, K ) )*XK
237: 30 CONTINUE
238: 40 CONTINUE
239: ELSE
240: DO 60 K = 1, N
241: XK = CABS1( X( K, J ) )
242: DO 50 I = MAX( 1, K-KD ), K - 1
243: RWORK( I ) = RWORK( I ) +
244: $ CABS1( AB( KD+1+I-K, K ) )*XK
245: 50 CONTINUE
246: RWORK( K ) = RWORK( K ) + XK
247: 60 CONTINUE
248: END IF
249: ELSE
250: IF( NOUNIT ) THEN
251: DO 80 K = 1, N
252: XK = CABS1( X( K, J ) )
253: DO 70 I = K, MIN( N, K+KD )
254: RWORK( I ) = RWORK( I ) +
255: $ CABS1( AB( 1+I-K, K ) )*XK
256: 70 CONTINUE
257: 80 CONTINUE
258: ELSE
259: DO 100 K = 1, N
260: XK = CABS1( X( K, J ) )
261: DO 90 I = K + 1, MIN( N, K+KD )
262: RWORK( I ) = RWORK( I ) +
263: $ CABS1( AB( 1+I-K, K ) )*XK
264: 90 CONTINUE
265: RWORK( K ) = RWORK( K ) + XK
266: 100 CONTINUE
267: END IF
268: END IF
269: ELSE
270: *
271: * Compute abs(A**H)*abs(X) + abs(B).
272: *
273: IF( UPPER ) THEN
274: IF( NOUNIT ) THEN
275: DO 120 K = 1, N
276: S = ZERO
277: DO 110 I = MAX( 1, K-KD ), K
278: S = S + CABS1( AB( KD+1+I-K, K ) )*
279: $ CABS1( X( I, J ) )
280: 110 CONTINUE
281: RWORK( K ) = RWORK( K ) + S
282: 120 CONTINUE
283: ELSE
284: DO 140 K = 1, N
285: S = CABS1( X( K, J ) )
286: DO 130 I = MAX( 1, K-KD ), K - 1
287: S = S + CABS1( AB( KD+1+I-K, K ) )*
288: $ CABS1( X( I, J ) )
289: 130 CONTINUE
290: RWORK( K ) = RWORK( K ) + S
291: 140 CONTINUE
292: END IF
293: ELSE
294: IF( NOUNIT ) THEN
295: DO 160 K = 1, N
296: S = ZERO
297: DO 150 I = K, MIN( N, K+KD )
298: S = S + CABS1( AB( 1+I-K, K ) )*
299: $ CABS1( X( I, J ) )
300: 150 CONTINUE
301: RWORK( K ) = RWORK( K ) + S
302: 160 CONTINUE
303: ELSE
304: DO 180 K = 1, N
305: S = CABS1( X( K, J ) )
306: DO 170 I = K + 1, MIN( N, K+KD )
307: S = S + CABS1( AB( 1+I-K, K ) )*
308: $ CABS1( X( I, J ) )
309: 170 CONTINUE
310: RWORK( K ) = RWORK( K ) + S
311: 180 CONTINUE
312: END IF
313: END IF
314: END IF
315: S = ZERO
316: DO 190 I = 1, N
317: IF( RWORK( I ).GT.SAFE2 ) THEN
318: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
319: ELSE
320: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
321: $ ( RWORK( I )+SAFE1 ) )
322: END IF
323: 190 CONTINUE
324: BERR( J ) = S
325: *
326: * Bound error from formula
327: *
328: * norm(X - XTRUE) / norm(X) .le. FERR =
329: * norm( abs(inv(op(A)))*
330: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
331: *
332: * where
333: * norm(Z) is the magnitude of the largest component of Z
334: * inv(op(A)) is the inverse of op(A)
335: * abs(Z) is the componentwise absolute value of the matrix or
336: * vector Z
337: * NZ is the maximum number of nonzeros in any row of A, plus 1
338: * EPS is machine epsilon
339: *
340: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
341: * is incremented by SAFE1 if the i-th component of
342: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
343: *
344: * Use ZLACN2 to estimate the infinity-norm of the matrix
345: * inv(op(A)) * diag(W),
346: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
347: *
348: DO 200 I = 1, N
349: IF( RWORK( I ).GT.SAFE2 ) THEN
350: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
351: ELSE
352: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
353: $ SAFE1
354: END IF
355: 200 CONTINUE
356: *
357: KASE = 0
358: 210 CONTINUE
359: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
360: IF( KASE.NE.0 ) THEN
361: IF( KASE.EQ.1 ) THEN
362: *
363: * Multiply by diag(W)*inv(op(A)**H).
364: *
365: CALL ZTBSV( UPLO, TRANST, DIAG, N, KD, AB, LDAB, WORK,
366: $ 1 )
367: DO 220 I = 1, N
368: WORK( I ) = RWORK( I )*WORK( I )
369: 220 CONTINUE
370: ELSE
371: *
372: * Multiply by inv(op(A))*diag(W).
373: *
374: DO 230 I = 1, N
375: WORK( I ) = RWORK( I )*WORK( I )
376: 230 CONTINUE
377: CALL ZTBSV( UPLO, TRANSN, DIAG, N, KD, AB, LDAB, WORK,
378: $ 1 )
379: END IF
380: GO TO 210
381: END IF
382: *
383: * Normalize error.
384: *
385: LSTRES = ZERO
386: DO 240 I = 1, N
387: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
388: 240 CONTINUE
389: IF( LSTRES.NE.ZERO )
390: $ FERR( J ) = FERR( J ) / LSTRES
391: *
392: 250 CONTINUE
393: *
394: RETURN
395: *
396: * End of ZTBRFS
397: *
398: END
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