Annotation of rpl/lapack/lapack/zgtrfs.f, revision 1.5
1.1 bertrand 1: SUBROUTINE ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
2: $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
3: $ INFO )
4: *
5: * -- LAPACK routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
11: *
12: * .. Scalar Arguments ..
13: CHARACTER TRANS
14: INTEGER INFO, LDB, LDX, N, NRHS
15: * ..
16: * .. Array Arguments ..
17: INTEGER IPIV( * )
18: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
19: COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
20: $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
21: $ WORK( * ), X( LDX, * )
22: * ..
23: *
24: * Purpose
25: * =======
26: *
27: * ZGTRFS improves the computed solution to a system of linear
28: * equations when the coefficient matrix is tridiagonal, and provides
29: * error bounds and backward error estimates for the solution.
30: *
31: * Arguments
32: * =========
33: *
34: * TRANS (input) CHARACTER*1
35: * Specifies the form of the system of equations:
36: * = 'N': A * X = B (No transpose)
37: * = 'T': A**T * X = B (Transpose)
38: * = 'C': A**H * X = B (Conjugate transpose)
39: *
40: * N (input) INTEGER
41: * The order of the matrix A. N >= 0.
42: *
43: * NRHS (input) INTEGER
44: * The number of right hand sides, i.e., the number of columns
45: * of the matrix B. NRHS >= 0.
46: *
47: * DL (input) COMPLEX*16 array, dimension (N-1)
48: * The (n-1) subdiagonal elements of A.
49: *
50: * D (input) COMPLEX*16 array, dimension (N)
51: * The diagonal elements of A.
52: *
53: * DU (input) COMPLEX*16 array, dimension (N-1)
54: * The (n-1) superdiagonal elements of A.
55: *
56: * DLF (input) COMPLEX*16 array, dimension (N-1)
57: * The (n-1) multipliers that define the matrix L from the
58: * LU factorization of A as computed by ZGTTRF.
59: *
60: * DF (input) COMPLEX*16 array, dimension (N)
61: * The n diagonal elements of the upper triangular matrix U from
62: * the LU factorization of A.
63: *
64: * DUF (input) COMPLEX*16 array, dimension (N-1)
65: * The (n-1) elements of the first superdiagonal of U.
66: *
67: * DU2 (input) COMPLEX*16 array, dimension (N-2)
68: * The (n-2) elements of the second superdiagonal of U.
69: *
70: * IPIV (input) INTEGER array, dimension (N)
71: * The pivot indices; for 1 <= i <= n, row i of the matrix was
72: * interchanged with row IPIV(i). IPIV(i) will always be either
73: * i or i+1; IPIV(i) = i indicates a row interchange was not
74: * required.
75: *
76: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
77: * The right hand side matrix B.
78: *
79: * LDB (input) INTEGER
80: * The leading dimension of the array B. LDB >= max(1,N).
81: *
82: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
83: * On entry, the solution matrix X, as computed by ZGTTRS.
84: * On exit, the improved solution matrix X.
85: *
86: * LDX (input) INTEGER
87: * The leading dimension of the array X. LDX >= max(1,N).
88: *
89: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
90: * The estimated forward error bound for each solution vector
91: * X(j) (the j-th column of the solution matrix X).
92: * If XTRUE is the true solution corresponding to X(j), FERR(j)
93: * is an estimated upper bound for the magnitude of the largest
94: * element in (X(j) - XTRUE) divided by the magnitude of the
95: * largest element in X(j). The estimate is as reliable as
96: * the estimate for RCOND, and is almost always a slight
97: * overestimate of the true error.
98: *
99: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
100: * The componentwise relative backward error of each solution
101: * vector X(j) (i.e., the smallest relative change in
102: * any element of A or B that makes X(j) an exact solution).
103: *
104: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
105: *
106: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
107: *
108: * INFO (output) INTEGER
109: * = 0: successful exit
110: * < 0: if INFO = -i, the i-th argument had an illegal value
111: *
112: * Internal Parameters
113: * ===================
114: *
115: * ITMAX is the maximum number of steps of iterative refinement.
116: *
117: * =====================================================================
118: *
119: * .. Parameters ..
120: INTEGER ITMAX
121: PARAMETER ( ITMAX = 5 )
122: DOUBLE PRECISION ZERO, ONE
123: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
124: DOUBLE PRECISION TWO
125: PARAMETER ( TWO = 2.0D+0 )
126: DOUBLE PRECISION THREE
127: PARAMETER ( THREE = 3.0D+0 )
128: * ..
129: * .. Local Scalars ..
130: LOGICAL NOTRAN
131: CHARACTER TRANSN, TRANST
132: INTEGER COUNT, I, J, KASE, NZ
133: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
134: COMPLEX*16 ZDUM
135: * ..
136: * .. Local Arrays ..
137: INTEGER ISAVE( 3 )
138: * ..
139: * .. External Subroutines ..
140: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGTTRS, ZLACN2, ZLAGTM
141: * ..
142: * .. Intrinsic Functions ..
143: INTRINSIC ABS, DBLE, DCMPLX, DIMAG, MAX
144: * ..
145: * .. External Functions ..
146: LOGICAL LSAME
147: DOUBLE PRECISION DLAMCH
148: EXTERNAL LSAME, DLAMCH
149: * ..
150: * .. Statement Functions ..
151: DOUBLE PRECISION CABS1
152: * ..
153: * .. Statement Function definitions ..
154: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
155: * ..
156: * .. Executable Statements ..
157: *
158: * Test the input parameters.
159: *
160: INFO = 0
161: NOTRAN = LSAME( TRANS, 'N' )
162: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
163: $ LSAME( TRANS, 'C' ) ) THEN
164: INFO = -1
165: ELSE IF( N.LT.0 ) THEN
166: INFO = -2
167: ELSE IF( NRHS.LT.0 ) THEN
168: INFO = -3
169: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
170: INFO = -13
171: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
172: INFO = -15
173: END IF
174: IF( INFO.NE.0 ) THEN
175: CALL XERBLA( 'ZGTRFS', -INFO )
176: RETURN
177: END IF
178: *
179: * Quick return if possible
180: *
181: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
182: DO 10 J = 1, NRHS
183: FERR( J ) = ZERO
184: BERR( J ) = ZERO
185: 10 CONTINUE
186: RETURN
187: END IF
188: *
189: IF( NOTRAN ) THEN
190: TRANSN = 'N'
191: TRANST = 'C'
192: ELSE
193: TRANSN = 'C'
194: TRANST = 'N'
195: END IF
196: *
197: * NZ = maximum number of nonzero elements in each row of A, plus 1
198: *
199: NZ = 4
200: EPS = DLAMCH( 'Epsilon' )
201: SAFMIN = DLAMCH( 'Safe minimum' )
202: SAFE1 = NZ*SAFMIN
203: SAFE2 = SAFE1 / EPS
204: *
205: * Do for each right hand side
206: *
207: DO 110 J = 1, NRHS
208: *
209: COUNT = 1
210: LSTRES = THREE
211: 20 CONTINUE
212: *
213: * Loop until stopping criterion is satisfied.
214: *
215: * Compute residual R = B - op(A) * X,
216: * where op(A) = A, A**T, or A**H, depending on TRANS.
217: *
218: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
219: CALL ZLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE,
220: $ WORK, N )
221: *
222: * Compute abs(op(A))*abs(x) + abs(b) for use in the backward
223: * error bound.
224: *
225: IF( NOTRAN ) THEN
226: IF( N.EQ.1 ) THEN
227: RWORK( 1 ) = CABS1( B( 1, J ) ) +
228: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
229: ELSE
230: RWORK( 1 ) = CABS1( B( 1, J ) ) +
231: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
232: $ CABS1( DU( 1 ) )*CABS1( X( 2, J ) )
233: DO 30 I = 2, N - 1
234: RWORK( I ) = CABS1( B( I, J ) ) +
235: $ CABS1( DL( I-1 ) )*CABS1( X( I-1, J ) ) +
236: $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
237: $ CABS1( DU( I ) )*CABS1( X( I+1, J ) )
238: 30 CONTINUE
239: RWORK( N ) = CABS1( B( N, J ) ) +
240: $ CABS1( DL( N-1 ) )*CABS1( X( N-1, J ) ) +
241: $ CABS1( D( N ) )*CABS1( X( N, J ) )
242: END IF
243: ELSE
244: IF( N.EQ.1 ) THEN
245: RWORK( 1 ) = CABS1( B( 1, J ) ) +
246: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
247: ELSE
248: RWORK( 1 ) = CABS1( B( 1, J ) ) +
249: $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
250: $ CABS1( DL( 1 ) )*CABS1( X( 2, J ) )
251: DO 40 I = 2, N - 1
252: RWORK( I ) = CABS1( B( I, J ) ) +
253: $ CABS1( DU( I-1 ) )*CABS1( X( I-1, J ) ) +
254: $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
255: $ CABS1( DL( I ) )*CABS1( X( I+1, J ) )
256: 40 CONTINUE
257: RWORK( N ) = CABS1( B( N, J ) ) +
258: $ CABS1( DU( N-1 ) )*CABS1( X( N-1, J ) ) +
259: $ CABS1( D( N ) )*CABS1( X( N, J ) )
260: END IF
261: END IF
262: *
263: * Compute componentwise relative backward error from formula
264: *
265: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
266: *
267: * where abs(Z) is the componentwise absolute value of the matrix
268: * or vector Z. If the i-th component of the denominator is less
269: * than SAFE2, then SAFE1 is added to the i-th components of the
270: * numerator and denominator before dividing.
271: *
272: S = ZERO
273: DO 50 I = 1, N
274: IF( RWORK( I ).GT.SAFE2 ) THEN
275: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
276: ELSE
277: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
278: $ ( RWORK( I )+SAFE1 ) )
279: END IF
280: 50 CONTINUE
281: BERR( J ) = S
282: *
283: * Test stopping criterion. Continue iterating if
284: * 1) The residual BERR(J) is larger than machine epsilon, and
285: * 2) BERR(J) decreased by at least a factor of 2 during the
286: * last iteration, and
287: * 3) At most ITMAX iterations tried.
288: *
289: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
290: $ COUNT.LE.ITMAX ) THEN
291: *
292: * Update solution and try again.
293: *
294: CALL ZGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, WORK, N,
295: $ INFO )
296: CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
297: LSTRES = BERR( J )
298: COUNT = COUNT + 1
299: GO TO 20
300: END IF
301: *
302: * Bound error from formula
303: *
304: * norm(X - XTRUE) / norm(X) .le. FERR =
305: * norm( abs(inv(op(A)))*
306: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
307: *
308: * where
309: * norm(Z) is the magnitude of the largest component of Z
310: * inv(op(A)) is the inverse of op(A)
311: * abs(Z) is the componentwise absolute value of the matrix or
312: * vector Z
313: * NZ is the maximum number of nonzeros in any row of A, plus 1
314: * EPS is machine epsilon
315: *
316: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
317: * is incremented by SAFE1 if the i-th component of
318: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
319: *
320: * Use ZLACN2 to estimate the infinity-norm of the matrix
321: * inv(op(A)) * diag(W),
322: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
323: *
324: DO 60 I = 1, N
325: IF( RWORK( I ).GT.SAFE2 ) THEN
326: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
327: ELSE
328: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
329: $ SAFE1
330: END IF
331: 60 CONTINUE
332: *
333: KASE = 0
334: 70 CONTINUE
335: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
336: IF( KASE.NE.0 ) THEN
337: IF( KASE.EQ.1 ) THEN
338: *
339: * Multiply by diag(W)*inv(op(A)**H).
340: *
341: CALL ZGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
342: $ N, INFO )
343: DO 80 I = 1, N
344: WORK( I ) = RWORK( I )*WORK( I )
345: 80 CONTINUE
346: ELSE
347: *
348: * Multiply by inv(op(A))*diag(W).
349: *
350: DO 90 I = 1, N
351: WORK( I ) = RWORK( I )*WORK( I )
352: 90 CONTINUE
353: CALL ZGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
354: $ N, INFO )
355: END IF
356: GO TO 70
357: END IF
358: *
359: * Normalize error.
360: *
361: LSTRES = ZERO
362: DO 100 I = 1, N
363: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
364: 100 CONTINUE
365: IF( LSTRES.NE.ZERO )
366: $ FERR( J ) = FERR( J ) / LSTRES
367: *
368: 110 CONTINUE
369: *
370: RETURN
371: *
372: * End of ZGTRFS
373: *
374: END
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