Annotation of rpl/lapack/lapack/zpprfs.f, revision 1.3
1.1 bertrand 1: SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
2: $ 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 UPLO
13: INTEGER INFO, LDB, LDX, N, NRHS
14: * ..
15: * .. Array Arguments ..
16: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
18: $ X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * ZPPRFS improves the computed solution to a system of linear
25: * equations when the coefficient matrix is Hermitian positive definite
26: * and packed, and provides error bounds and backward error estimates
27: * for the solution.
28: *
29: * Arguments
30: * =========
31: *
32: * UPLO (input) CHARACTER*1
33: * = 'U': Upper triangle of A is stored;
34: * = 'L': Lower triangle of A is stored.
35: *
36: * N (input) INTEGER
37: * The order of the matrix A. N >= 0.
38: *
39: * NRHS (input) INTEGER
40: * The number of right hand sides, i.e., the number of columns
41: * of the matrices B and X. NRHS >= 0.
42: *
43: * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
44: * The upper or lower triangle of the Hermitian matrix A, packed
45: * columnwise in a linear array. The j-th column of A is stored
46: * in the array AP as follows:
47: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
48: * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
49: *
50: * AFP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
51: * The triangular factor U or L from the Cholesky factorization
52: * A = U**H*U or A = L*L**H, as computed by DPPTRF/ZPPTRF,
53: * packed columnwise in a linear array in the same format as A
54: * (see AP).
55: *
56: * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
57: * The right hand side matrix B.
58: *
59: * LDB (input) INTEGER
60: * The leading dimension of the array B. LDB >= max(1,N).
61: *
62: * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
63: * On entry, the solution matrix X, as computed by ZPPTRS.
64: * On exit, the improved solution matrix X.
65: *
66: * LDX (input) INTEGER
67: * The leading dimension of the array X. LDX >= max(1,N).
68: *
69: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
70: * The estimated forward error bound for each solution vector
71: * X(j) (the j-th column of the solution matrix X).
72: * If XTRUE is the true solution corresponding to X(j), FERR(j)
73: * is an estimated upper bound for the magnitude of the largest
74: * element in (X(j) - XTRUE) divided by the magnitude of the
75: * largest element in X(j). The estimate is as reliable as
76: * the estimate for RCOND, and is almost always a slight
77: * overestimate of the true error.
78: *
79: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
80: * The componentwise relative backward error of each solution
81: * vector X(j) (i.e., the smallest relative change in
82: * any element of A or B that makes X(j) an exact solution).
83: *
84: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
85: *
86: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
87: *
88: * INFO (output) INTEGER
89: * = 0: successful exit
90: * < 0: if INFO = -i, the i-th argument had an illegal value
91: *
92: * Internal Parameters
93: * ===================
94: *
95: * ITMAX is the maximum number of steps of iterative refinement.
96: *
97: * ====================================================================
98: *
99: * .. Parameters ..
100: INTEGER ITMAX
101: PARAMETER ( ITMAX = 5 )
102: DOUBLE PRECISION ZERO
103: PARAMETER ( ZERO = 0.0D+0 )
104: COMPLEX*16 CONE
105: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
106: DOUBLE PRECISION TWO
107: PARAMETER ( TWO = 2.0D+0 )
108: DOUBLE PRECISION THREE
109: PARAMETER ( THREE = 3.0D+0 )
110: * ..
111: * .. Local Scalars ..
112: LOGICAL UPPER
113: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
114: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
115: COMPLEX*16 ZDUM
116: * ..
117: * .. Local Arrays ..
118: INTEGER ISAVE( 3 )
119: * ..
120: * .. External Subroutines ..
121: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS
122: * ..
123: * .. Intrinsic Functions ..
124: INTRINSIC ABS, DBLE, DIMAG, MAX
125: * ..
126: * .. External Functions ..
127: LOGICAL LSAME
128: DOUBLE PRECISION DLAMCH
129: EXTERNAL LSAME, DLAMCH
130: * ..
131: * .. Statement Functions ..
132: DOUBLE PRECISION CABS1
133: * ..
134: * .. Statement Function definitions ..
135: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
136: * ..
137: * .. Executable Statements ..
138: *
139: * Test the input parameters.
140: *
141: INFO = 0
142: UPPER = LSAME( UPLO, 'U' )
143: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
144: INFO = -1
145: ELSE IF( N.LT.0 ) THEN
146: INFO = -2
147: ELSE IF( NRHS.LT.0 ) THEN
148: INFO = -3
149: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
150: INFO = -7
151: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
152: INFO = -9
153: END IF
154: IF( INFO.NE.0 ) THEN
155: CALL XERBLA( 'ZPPRFS', -INFO )
156: RETURN
157: END IF
158: *
159: * Quick return if possible
160: *
161: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
162: DO 10 J = 1, NRHS
163: FERR( J ) = ZERO
164: BERR( J ) = ZERO
165: 10 CONTINUE
166: RETURN
167: END IF
168: *
169: * NZ = maximum number of nonzero elements in each row of A, plus 1
170: *
171: NZ = N + 1
172: EPS = DLAMCH( 'Epsilon' )
173: SAFMIN = DLAMCH( 'Safe minimum' )
174: SAFE1 = NZ*SAFMIN
175: SAFE2 = SAFE1 / EPS
176: *
177: * Do for each right hand side
178: *
179: DO 140 J = 1, NRHS
180: *
181: COUNT = 1
182: LSTRES = THREE
183: 20 CONTINUE
184: *
185: * Loop until stopping criterion is satisfied.
186: *
187: * Compute residual R = B - A * X
188: *
189: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
190: CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 )
191: *
192: * Compute componentwise relative backward error from formula
193: *
194: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
195: *
196: * where abs(Z) is the componentwise absolute value of the matrix
197: * or vector Z. If the i-th component of the denominator is less
198: * than SAFE2, then SAFE1 is added to the i-th components of the
199: * numerator and denominator before dividing.
200: *
201: DO 30 I = 1, N
202: RWORK( I ) = CABS1( B( I, J ) )
203: 30 CONTINUE
204: *
205: * Compute abs(A)*abs(X) + abs(B).
206: *
207: KK = 1
208: IF( UPPER ) THEN
209: DO 50 K = 1, N
210: S = ZERO
211: XK = CABS1( X( K, J ) )
212: IK = KK
213: DO 40 I = 1, K - 1
214: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
215: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
216: IK = IK + 1
217: 40 CONTINUE
218: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
219: $ XK + S
220: KK = KK + K
221: 50 CONTINUE
222: ELSE
223: DO 70 K = 1, N
224: S = ZERO
225: XK = CABS1( X( K, J ) )
226: RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
227: IK = KK + 1
228: DO 60 I = K + 1, N
229: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
230: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
231: IK = IK + 1
232: 60 CONTINUE
233: RWORK( K ) = RWORK( K ) + S
234: KK = KK + ( N-K+1 )
235: 70 CONTINUE
236: END IF
237: S = ZERO
238: DO 80 I = 1, N
239: IF( RWORK( I ).GT.SAFE2 ) THEN
240: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
241: ELSE
242: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
243: $ ( RWORK( I )+SAFE1 ) )
244: END IF
245: 80 CONTINUE
246: BERR( J ) = S
247: *
248: * Test stopping criterion. Continue iterating if
249: * 1) The residual BERR(J) is larger than machine epsilon, and
250: * 2) BERR(J) decreased by at least a factor of 2 during the
251: * last iteration, and
252: * 3) At most ITMAX iterations tried.
253: *
254: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
255: $ COUNT.LE.ITMAX ) THEN
256: *
257: * Update solution and try again.
258: *
259: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
260: CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
261: LSTRES = BERR( J )
262: COUNT = COUNT + 1
263: GO TO 20
264: END IF
265: *
266: * Bound error from formula
267: *
268: * norm(X - XTRUE) / norm(X) .le. FERR =
269: * norm( abs(inv(A))*
270: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
271: *
272: * where
273: * norm(Z) is the magnitude of the largest component of Z
274: * inv(A) is the inverse of A
275: * abs(Z) is the componentwise absolute value of the matrix or
276: * vector Z
277: * NZ is the maximum number of nonzeros in any row of A, plus 1
278: * EPS is machine epsilon
279: *
280: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
281: * is incremented by SAFE1 if the i-th component of
282: * abs(A)*abs(X) + abs(B) is less than SAFE2.
283: *
284: * Use ZLACN2 to estimate the infinity-norm of the matrix
285: * inv(A) * diag(W),
286: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
287: *
288: DO 90 I = 1, N
289: IF( RWORK( I ).GT.SAFE2 ) THEN
290: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
291: ELSE
292: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
293: $ SAFE1
294: END IF
295: 90 CONTINUE
296: *
297: KASE = 0
298: 100 CONTINUE
299: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
300: IF( KASE.NE.0 ) THEN
301: IF( KASE.EQ.1 ) THEN
302: *
303: * Multiply by diag(W)*inv(A').
304: *
305: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
306: DO 110 I = 1, N
307: WORK( I ) = RWORK( I )*WORK( I )
308: 110 CONTINUE
309: ELSE IF( KASE.EQ.2 ) THEN
310: *
311: * Multiply by inv(A)*diag(W).
312: *
313: DO 120 I = 1, N
314: WORK( I ) = RWORK( I )*WORK( I )
315: 120 CONTINUE
316: CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
317: END IF
318: GO TO 100
319: END IF
320: *
321: * Normalize error.
322: *
323: LSTRES = ZERO
324: DO 130 I = 1, N
325: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
326: 130 CONTINUE
327: IF( LSTRES.NE.ZERO )
328: $ FERR( J ) = FERR( J ) / LSTRES
329: *
330: 140 CONTINUE
331: *
332: RETURN
333: *
334: * End of ZPPRFS
335: *
336: END
CVSweb interface <joel.bertrand@systella.fr>
![[BACK]](/cvsweb/icons/back.gif){kind=icon}
Return to zpprfs.f CVS log ![[TXT]](/cvsweb/icons/text.gif){kind=icon}
![[DIR]](/cvsweb/icons/dir.gif){kind=icon}
Up to [local] / rpl / lapack / lapack