1: SUBROUTINE DTPRFS( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
2: $ FERR, BERR, WORK, IWORK, 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 DLACN2 in place of DLACON, 5 Feb 03, SJH.
10: *
11: * .. Scalar Arguments ..
12: CHARACTER DIAG, TRANS, UPLO
13: INTEGER INFO, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: INTEGER IWORK( * )
17: DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
18: $ WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DTPRFS provides error bounds and backward error estimates for the
25: * solution to a system of linear equations with a triangular packed
26: * coefficient matrix.
27: *
28: * The solution matrix X must be computed by DTPTRS or some other
29: * means before entering this routine. DTPRFS 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: * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
57: * The upper or lower triangular matrix A, packed columnwise in
58: * a linear array. The j-th column of A is stored in the array
59: * AP as follows:
60: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
61: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
62: * If DIAG = 'U', the diagonal elements of A are not referenced
63: * and are assumed to be 1.
64: *
65: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
66: * The right hand side matrix B.
67: *
68: * LDB (input) INTEGER
69: * The leading dimension of the array B. LDB >= max(1,N).
70: *
71: * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS)
72: * The solution matrix X.
73: *
74: * LDX (input) INTEGER
75: * The leading dimension of the array X. LDX >= max(1,N).
76: *
77: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
78: * The estimated forward error bound for each solution vector
79: * X(j) (the j-th column of the solution matrix X).
80: * If XTRUE is the true solution corresponding to X(j), FERR(j)
81: * is an estimated upper bound for the magnitude of the largest
82: * element in (X(j) - XTRUE) divided by the magnitude of the
83: * largest element in X(j). The estimate is as reliable as
84: * the estimate for RCOND, and is almost always a slight
85: * overestimate of the true error.
86: *
87: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
88: * The componentwise relative backward error of each solution
89: * vector X(j) (i.e., the smallest relative change in
90: * any element of A or B that makes X(j) an exact solution).
91: *
92: * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
93: *
94: * IWORK (workspace) INTEGER array, dimension (N)
95: *
96: * INFO (output) INTEGER
97: * = 0: successful exit
98: * < 0: if INFO = -i, the i-th argument had an illegal value
99: *
100: * =====================================================================
101: *
102: * .. Parameters ..
103: DOUBLE PRECISION ZERO
104: PARAMETER ( ZERO = 0.0D+0 )
105: DOUBLE PRECISION ONE
106: PARAMETER ( ONE = 1.0D+0 )
107: * ..
108: * .. Local Scalars ..
109: LOGICAL NOTRAN, NOUNIT, UPPER
110: CHARACTER TRANST
111: INTEGER I, J, K, KASE, KC, NZ
112: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
113: * ..
114: * .. Local Arrays ..
115: INTEGER ISAVE( 3 )
116: * ..
117: * .. External Subroutines ..
118: EXTERNAL DAXPY, DCOPY, DLACN2, DTPMV, DTPSV, XERBLA
119: * ..
120: * .. Intrinsic Functions ..
121: INTRINSIC ABS, MAX
122: * ..
123: * .. External Functions ..
124: LOGICAL LSAME
125: DOUBLE PRECISION DLAMCH
126: EXTERNAL LSAME, DLAMCH
127: * ..
128: * .. Executable Statements ..
129: *
130: * Test the input parameters.
131: *
132: INFO = 0
133: UPPER = LSAME( UPLO, 'U' )
134: NOTRAN = LSAME( TRANS, 'N' )
135: NOUNIT = LSAME( DIAG, 'N' )
136: *
137: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
138: INFO = -1
139: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
140: $ LSAME( TRANS, 'C' ) ) THEN
141: INFO = -2
142: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
143: INFO = -3
144: ELSE IF( N.LT.0 ) THEN
145: INFO = -4
146: ELSE IF( NRHS.LT.0 ) THEN
147: INFO = -5
148: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
149: INFO = -8
150: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
151: INFO = -10
152: END IF
153: IF( INFO.NE.0 ) THEN
154: CALL XERBLA( 'DTPRFS', -INFO )
155: RETURN
156: END IF
157: *
158: * Quick return if possible
159: *
160: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
161: DO 10 J = 1, NRHS
162: FERR( J ) = ZERO
163: BERR( J ) = ZERO
164: 10 CONTINUE
165: RETURN
166: END IF
167: *
168: IF( NOTRAN ) THEN
169: TRANST = 'T'
170: ELSE
171: TRANST = 'N'
172: END IF
173: *
174: * NZ = maximum number of nonzero elements in each row of A, plus 1
175: *
176: NZ = N + 1
177: EPS = DLAMCH( 'Epsilon' )
178: SAFMIN = DLAMCH( 'Safe minimum' )
179: SAFE1 = NZ*SAFMIN
180: SAFE2 = SAFE1 / EPS
181: *
182: * Do for each right hand side
183: *
184: DO 250 J = 1, NRHS
185: *
186: * Compute residual R = B - op(A) * X,
187: * where op(A) = A or A', depending on TRANS.
188: *
189: CALL DCOPY( N, X( 1, J ), 1, WORK( N+1 ), 1 )
190: CALL DTPMV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
191: CALL DAXPY( N, -ONE, B( 1, J ), 1, WORK( N+1 ), 1 )
192: *
193: * Compute componentwise relative backward error from formula
194: *
195: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
196: *
197: * where abs(Z) is the componentwise absolute value of the matrix
198: * or vector Z. If the i-th component of the denominator is less
199: * than SAFE2, then SAFE1 is added to the i-th components of the
200: * numerator and denominator before dividing.
201: *
202: DO 20 I = 1, N
203: WORK( I ) = ABS( B( I, J ) )
204: 20 CONTINUE
205: *
206: IF( NOTRAN ) THEN
207: *
208: * Compute abs(A)*abs(X) + abs(B).
209: *
210: IF( UPPER ) THEN
211: KC = 1
212: IF( NOUNIT ) THEN
213: DO 40 K = 1, N
214: XK = ABS( X( K, J ) )
215: DO 30 I = 1, K
216: WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
217: 30 CONTINUE
218: KC = KC + K
219: 40 CONTINUE
220: ELSE
221: DO 60 K = 1, N
222: XK = ABS( X( K, J ) )
223: DO 50 I = 1, K - 1
224: WORK( I ) = WORK( I ) + ABS( AP( KC+I-1 ) )*XK
225: 50 CONTINUE
226: WORK( K ) = WORK( K ) + XK
227: KC = KC + K
228: 60 CONTINUE
229: END IF
230: ELSE
231: KC = 1
232: IF( NOUNIT ) THEN
233: DO 80 K = 1, N
234: XK = ABS( X( K, J ) )
235: DO 70 I = K, N
236: WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
237: 70 CONTINUE
238: KC = KC + N - K + 1
239: 80 CONTINUE
240: ELSE
241: DO 100 K = 1, N
242: XK = ABS( X( K, J ) )
243: DO 90 I = K + 1, N
244: WORK( I ) = WORK( I ) + ABS( AP( KC+I-K ) )*XK
245: 90 CONTINUE
246: WORK( K ) = WORK( K ) + XK
247: KC = KC + N - K + 1
248: 100 CONTINUE
249: END IF
250: END IF
251: ELSE
252: *
253: * Compute abs(A')*abs(X) + abs(B).
254: *
255: IF( UPPER ) THEN
256: KC = 1
257: IF( NOUNIT ) THEN
258: DO 120 K = 1, N
259: S = ZERO
260: DO 110 I = 1, K
261: S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
262: 110 CONTINUE
263: WORK( K ) = WORK( K ) + S
264: KC = KC + K
265: 120 CONTINUE
266: ELSE
267: DO 140 K = 1, N
268: S = ABS( X( K, J ) )
269: DO 130 I = 1, K - 1
270: S = S + ABS( AP( KC+I-1 ) )*ABS( X( I, J ) )
271: 130 CONTINUE
272: WORK( K ) = WORK( K ) + S
273: KC = KC + K
274: 140 CONTINUE
275: END IF
276: ELSE
277: KC = 1
278: IF( NOUNIT ) THEN
279: DO 160 K = 1, N
280: S = ZERO
281: DO 150 I = K, N
282: S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
283: 150 CONTINUE
284: WORK( K ) = WORK( K ) + S
285: KC = KC + N - K + 1
286: 160 CONTINUE
287: ELSE
288: DO 180 K = 1, N
289: S = ABS( X( K, J ) )
290: DO 170 I = K + 1, N
291: S = S + ABS( AP( KC+I-K ) )*ABS( X( I, J ) )
292: 170 CONTINUE
293: WORK( K ) = WORK( K ) + S
294: KC = KC + N - K + 1
295: 180 CONTINUE
296: END IF
297: END IF
298: END IF
299: S = ZERO
300: DO 190 I = 1, N
301: IF( WORK( I ).GT.SAFE2 ) THEN
302: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
303: ELSE
304: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
305: $ ( WORK( I )+SAFE1 ) )
306: END IF
307: 190 CONTINUE
308: BERR( J ) = S
309: *
310: * Bound error from formula
311: *
312: * norm(X - XTRUE) / norm(X) .le. FERR =
313: * norm( abs(inv(op(A)))*
314: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
315: *
316: * where
317: * norm(Z) is the magnitude of the largest component of Z
318: * inv(op(A)) is the inverse of op(A)
319: * abs(Z) is the componentwise absolute value of the matrix or
320: * vector Z
321: * NZ is the maximum number of nonzeros in any row of A, plus 1
322: * EPS is machine epsilon
323: *
324: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
325: * is incremented by SAFE1 if the i-th component of
326: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
327: *
328: * Use DLACN2 to estimate the infinity-norm of the matrix
329: * inv(op(A)) * diag(W),
330: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
331: *
332: DO 200 I = 1, N
333: IF( WORK( I ).GT.SAFE2 ) THEN
334: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
335: ELSE
336: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
337: END IF
338: 200 CONTINUE
339: *
340: KASE = 0
341: 210 CONTINUE
342: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
343: $ KASE, ISAVE )
344: IF( KASE.NE.0 ) THEN
345: IF( KASE.EQ.1 ) THEN
346: *
347: * Multiply by diag(W)*inv(op(A)').
348: *
349: CALL DTPSV( UPLO, TRANST, DIAG, N, AP, WORK( N+1 ), 1 )
350: DO 220 I = 1, N
351: WORK( N+I ) = WORK( I )*WORK( N+I )
352: 220 CONTINUE
353: ELSE
354: *
355: * Multiply by inv(op(A))*diag(W).
356: *
357: DO 230 I = 1, N
358: WORK( N+I ) = WORK( I )*WORK( N+I )
359: 230 CONTINUE
360: CALL DTPSV( UPLO, TRANS, DIAG, N, AP, WORK( N+1 ), 1 )
361: END IF
362: GO TO 210
363: END IF
364: *
365: * Normalize error.
366: *
367: LSTRES = ZERO
368: DO 240 I = 1, N
369: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
370: 240 CONTINUE
371: IF( LSTRES.NE.ZERO )
372: $ FERR( J ) = FERR( J ) / LSTRES
373: *
374: 250 CONTINUE
375: *
376: RETURN
377: *
378: * End of DTPRFS
379: *
380: END
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