1: SUBROUTINE DSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
2: $ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
3: $ IWORK, INFO )
4: *
5: * -- LAPACK driver 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: * .. Scalar Arguments ..
11: CHARACTER FACT, UPLO
12: INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
13: DOUBLE PRECISION RCOND
14: * ..
15: * .. Array Arguments ..
16: INTEGER IPIV( * ), IWORK( * )
17: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * DSYSVX uses the diagonal pivoting factorization to compute the
25: * solution to a real system of linear equations A * X = B,
26: * where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
27: * matrices.
28: *
29: * Error bounds on the solution and a condition estimate are also
30: * provided.
31: *
32: * Description
33: * ===========
34: *
35: * The following steps are performed:
36: *
37: * 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
38: * The form of the factorization is
39: * A = U * D * U**T, if UPLO = 'U', or
40: * A = L * D * L**T, if UPLO = 'L',
41: * where U (or L) is a product of permutation and unit upper (lower)
42: * triangular matrices, and D is symmetric and block diagonal with
43: * 1-by-1 and 2-by-2 diagonal blocks.
44: *
45: * 2. If some D(i,i)=0, so that D is exactly singular, then the routine
46: * returns with INFO = i. Otherwise, the factored form of A is used
47: * to estimate the condition number of the matrix A. If the
48: * reciprocal of the condition number is less than machine precision,
49: * INFO = N+1 is returned as a warning, but the routine still goes on
50: * to solve for X and compute error bounds as described below.
51: *
52: * 3. The system of equations is solved for X using the factored form
53: * of A.
54: *
55: * 4. Iterative refinement is applied to improve the computed solution
56: * matrix and calculate error bounds and backward error estimates
57: * for it.
58: *
59: * Arguments
60: * =========
61: *
62: * FACT (input) CHARACTER*1
63: * Specifies whether or not the factored form of A has been
64: * supplied on entry.
65: * = 'F': On entry, AF and IPIV contain the factored form of
66: * A. AF and IPIV will not be modified.
67: * = 'N': The matrix A will be copied to AF and factored.
68: *
69: * UPLO (input) CHARACTER*1
70: * = 'U': Upper triangle of A is stored;
71: * = 'L': Lower triangle of A is stored.
72: *
73: * N (input) INTEGER
74: * The number of linear equations, i.e., the order of the
75: * matrix A. N >= 0.
76: *
77: * NRHS (input) INTEGER
78: * The number of right hand sides, i.e., the number of columns
79: * of the matrices B and X. NRHS >= 0.
80: *
81: * A (input) DOUBLE PRECISION array, dimension (LDA,N)
82: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
83: * upper triangular part of A contains the upper triangular part
84: * of the matrix A, and the strictly lower triangular part of A
85: * is not referenced. If UPLO = 'L', the leading N-by-N lower
86: * triangular part of A contains the lower triangular part of
87: * the matrix A, and the strictly upper triangular part of A is
88: * not referenced.
89: *
90: * LDA (input) INTEGER
91: * The leading dimension of the array A. LDA >= max(1,N).
92: *
93: * AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
94: * If FACT = 'F', then AF is an input argument and on entry
95: * contains the block diagonal matrix D and the multipliers used
96: * to obtain the factor U or L from the factorization
97: * A = U*D*U**T or A = L*D*L**T as computed by DSYTRF.
98: *
99: * If FACT = 'N', then AF is an output argument and on exit
100: * returns the block diagonal matrix D and the multipliers used
101: * to obtain the factor U or L from the factorization
102: * A = U*D*U**T or A = L*D*L**T.
103: *
104: * LDAF (input) INTEGER
105: * The leading dimension of the array AF. LDAF >= max(1,N).
106: *
107: * IPIV (input or output) INTEGER array, dimension (N)
108: * If FACT = 'F', then IPIV is an input argument and on entry
109: * contains details of the interchanges and the block structure
110: * of D, as determined by DSYTRF.
111: * If IPIV(k) > 0, then rows and columns k and IPIV(k) were
112: * interchanged and D(k,k) is a 1-by-1 diagonal block.
113: * If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
114: * columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
115: * is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
116: * IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
117: * interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
118: *
119: * If FACT = 'N', then IPIV is an output argument and on exit
120: * contains details of the interchanges and the block structure
121: * of D, as determined by DSYTRF.
122: *
123: * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
124: * The N-by-NRHS right hand side matrix B.
125: *
126: * LDB (input) INTEGER
127: * The leading dimension of the array B. LDB >= max(1,N).
128: *
129: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
130: * If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
131: *
132: * LDX (input) INTEGER
133: * The leading dimension of the array X. LDX >= max(1,N).
134: *
135: * RCOND (output) DOUBLE PRECISION
136: * The estimate of the reciprocal condition number of the matrix
137: * A. If RCOND is less than the machine precision (in
138: * particular, if RCOND = 0), the matrix is singular to working
139: * precision. This condition is indicated by a return code of
140: * INFO > 0.
141: *
142: * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
143: * The estimated forward error bound for each solution vector
144: * X(j) (the j-th column of the solution matrix X).
145: * If XTRUE is the true solution corresponding to X(j), FERR(j)
146: * is an estimated upper bound for the magnitude of the largest
147: * element in (X(j) - XTRUE) divided by the magnitude of the
148: * largest element in X(j). The estimate is as reliable as
149: * the estimate for RCOND, and is almost always a slight
150: * overestimate of the true error.
151: *
152: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
153: * The componentwise relative backward error of each solution
154: * vector X(j) (i.e., the smallest relative change in
155: * any element of A or B that makes X(j) an exact solution).
156: *
157: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
158: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
159: *
160: * LWORK (input) INTEGER
161: * The length of WORK. LWORK >= max(1,3*N), and for best
162: * performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where
163: * NB is the optimal blocksize for DSYTRF.
164: *
165: * If LWORK = -1, then a workspace query is assumed; the routine
166: * only calculates the optimal size of the WORK array, returns
167: * this value as the first entry of the WORK array, and no error
168: * message related to LWORK is issued by XERBLA.
169: *
170: * IWORK (workspace) INTEGER array, dimension (N)
171: *
172: * INFO (output) INTEGER
173: * = 0: successful exit
174: * < 0: if INFO = -i, the i-th argument had an illegal value
175: * > 0: if INFO = i, and i is
176: * <= N: D(i,i) is exactly zero. The factorization
177: * has been completed but the factor D is exactly
178: * singular, so the solution and error bounds could
179: * not be computed. RCOND = 0 is returned.
180: * = N+1: D is nonsingular, but RCOND is less than machine
181: * precision, meaning that the matrix is singular
182: * to working precision. Nevertheless, the
183: * solution and error bounds are computed because
184: * there are a number of situations where the
185: * computed solution can be more accurate than the
186: * value of RCOND would suggest.
187: *
188: * =====================================================================
189: *
190: * .. Parameters ..
191: DOUBLE PRECISION ZERO
192: PARAMETER ( ZERO = 0.0D+0 )
193: * ..
194: * .. Local Scalars ..
195: LOGICAL LQUERY, NOFACT
196: INTEGER LWKOPT, NB
197: DOUBLE PRECISION ANORM
198: * ..
199: * .. External Functions ..
200: LOGICAL LSAME
201: INTEGER ILAENV
202: DOUBLE PRECISION DLAMCH, DLANSY
203: EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
204: * ..
205: * .. External Subroutines ..
206: EXTERNAL DLACPY, DSYCON, DSYRFS, DSYTRF, DSYTRS, XERBLA
207: * ..
208: * .. Intrinsic Functions ..
209: INTRINSIC MAX
210: * ..
211: * .. Executable Statements ..
212: *
213: * Test the input parameters.
214: *
215: INFO = 0
216: NOFACT = LSAME( FACT, 'N' )
217: LQUERY = ( LWORK.EQ.-1 )
218: IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
219: INFO = -1
220: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
221: $ THEN
222: INFO = -2
223: ELSE IF( N.LT.0 ) THEN
224: INFO = -3
225: ELSE IF( NRHS.LT.0 ) THEN
226: INFO = -4
227: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
228: INFO = -6
229: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
230: INFO = -8
231: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
232: INFO = -11
233: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
234: INFO = -13
235: ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN
236: INFO = -18
237: END IF
238: *
239: IF( INFO.EQ.0 ) THEN
240: LWKOPT = MAX( 1, 3*N )
241: IF( NOFACT ) THEN
242: NB = ILAENV( 1, 'DSYTRF', UPLO, N, -1, -1, -1 )
243: LWKOPT = MAX( LWKOPT, N*NB )
244: END IF
245: WORK( 1 ) = LWKOPT
246: END IF
247: *
248: IF( INFO.NE.0 ) THEN
249: CALL XERBLA( 'DSYSVX', -INFO )
250: RETURN
251: ELSE IF( LQUERY ) THEN
252: RETURN
253: END IF
254: *
255: IF( NOFACT ) THEN
256: *
257: * Compute the factorization A = U*D*U' or A = L*D*L'.
258: *
259: CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
260: CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )
261: *
262: * Return if INFO is non-zero.
263: *
264: IF( INFO.GT.0 )THEN
265: RCOND = ZERO
266: RETURN
267: END IF
268: END IF
269: *
270: * Compute the norm of the matrix A.
271: *
272: ANORM = DLANSY( 'I', UPLO, N, A, LDA, WORK )
273: *
274: * Compute the reciprocal of the condition number of A.
275: *
276: CALL DSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, IWORK,
277: $ INFO )
278: *
279: * Compute the solution vectors X.
280: *
281: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
282: CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
283: *
284: * Use iterative refinement to improve the computed solutions and
285: * compute error bounds and backward error estimates for them.
286: *
287: CALL DSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
288: $ LDX, FERR, BERR, WORK, IWORK, INFO )
289: *
290: * Set INFO = N+1 if the matrix is singular to working precision.
291: *
292: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
293: $ INFO = N + 1
294: *
295: WORK( 1 ) = LWKOPT
296: *
297: RETURN
298: *
299: * End of DSYSVX
300: *
301: END
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