1DSYRFSX(1) LAPACK routine (version 3.2) DSYRFSX(1)
2
6 DSYRFSX - DSYRFSX improve the computed solution to a system of linear
7 equations when the coefficient matrix is symmetric indefinite, and
8 provides error bounds and backward error estimates for the solution
9
11 SUBROUTINE DSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, S, B,
12 LDB, X, LDX, RCOND, BERR, N_ERR_BNDS,
13 ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
14 WORK, IWORK, INFO )
15 IMPLICIT NONE
17 CHARACTER UPLO, EQUED
19 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
21 N_ERR_BNDS
22 DOUBLE PRECISION RCOND
24 INTEGER IPIV( * ), IWORK( * )
26 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
28 X( LDX, * ), WORK( * )
29 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
31 ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * )
32
34 DSYRFSX improves the computed solution to a system of linear
35 equations when the coefficient matrix is symmetric indefinite, and
36 provides error bounds and backward error estimates for the
37 solution. In addition to normwise error bound, the code provides
38 maximum componentwise error bound if possible. See comments for
39 ERR_BNDS_N and ERR_BNDS_C for details of the error bounds.
40 The original system of linear equations may have been equilibrated
41 before calling this routine, as described by arguments EQUED and S
42 below. In this case, the solution and error bounds returned are
43 for the original unequilibrated system.
44
46 Some optional parameters are bundled in the PARAMS array. These set‐
47 tings determine how refinement is performed, but often the defaults are
48 acceptable. If the defaults are acceptable, users can pass NPARAMS = 0
49 which prevents the source code from accessing the PARAMS argument.
50 UPLO (input) CHARACTER*1
52 = 'U': Upper triangle of A is stored;
53 = 'L': Lower triangle of A is stored.
54 EQUED (input) CHARACTER*1
56 Specifies the form of equilibration that was done to A before
57 calling this routine. This is needed to compute the solution
58 and error bounds correctly. = 'N': No equilibration
59 = 'Y': Both row and column equilibration, i.e., A has been
60 replaced by diag(S) * A * diag(S). The right hand side B has
61 been changed accordingly.
62 N (input) INTEGER
64 The order of the matrix A. N >= 0.
65 NRHS (input) INTEGER
67 The number of right hand sides, i.e., the number of columns of
68 the matrices B and X. NRHS >= 0.
69 A (input) DOUBLE PRECISION array, dimension (LDA,N)
71 The symmetric matrix A. If UPLO = 'U', the leading N-by-N
72 upper triangular part of A contains the upper triangular part
73 of the matrix A, and the strictly lower triangular part of A is
74 not referenced. If UPLO = 'L', the leading N-by-N lower trian‐
75 gular part of A contains the lower triangular part of the
76 matrix A, and the strictly upper triangular part of A is not
77 referenced.
78 LDA (input) INTEGER
80 The leading dimension of the array A. LDA >= max(1,N).
81 AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
83 The factored form of the matrix A. AF contains the block diag‐
84 onal matrix D and the multipliers used to obtain the factor U
85 or L from the factorization A = U*D*U**T or A = L*D*L**T as
86 computed by DSYTRF.
87 LDAF (input) INTEGER
89 The leading dimension of the array AF. LDAF >= max(1,N).
90 IPIV (input) INTEGER array, dimension (N)
92 Details of the interchanges and the block structure of D as
93 determined by DSYTRF.
94 S (input or output) DOUBLE PRECISION array, dimension (N)
96 The scale factors for A. If EQUED = 'Y', A is multiplied on
97 the left and right by diag(S). S is an input argument if FACT
98 = 'F'; otherwise, S is an output argument. If FACT = 'F' and
99 EQUED = 'Y', each element of S must be positive. If S is out‐
100 put, each element of S is a power of the radix. If S is input,
101 each element of S should be a power of the radix to ensure a
102 reliable solution and error estimates. Scaling by powers of the
103 radix does not cause rounding errors unless the result under‐
104 flows or overflows. Rounding errors during scaling lead to
105 refining with a matrix that is not equivalent to the input
106 matrix, producing error estimates that may not be reliable.
107 B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
109 The right hand side matrix B.
110 LDB (input) INTEGER
112 The leading dimension of the array B. LDB >= max(1,N).
113 X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
115 On entry, the solution matrix X, as computed by DGETRS. On
116 exit, the improved solution matrix X.
117 LDX (input) INTEGER
119 The leading dimension of the array X. LDX >= max(1,N).
120 RCOND (output) DOUBLE PRECISION
122 Reciprocal scaled condition number. This is an estimate of the
123 reciprocal Skeel condition number of the matrix A after equili‐
124 bration (if done). If this is less than the machine precision
125 (in particular, if it is zero), the matrix is singular to work‐
126 ing precision. Note that the error may still be small even if
127 this number is very small and the matrix appears ill- condi‐
128 tioned.
129 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
131 Componentwise relative backward error. This is the component‐
132 wise relative backward error of each solution vector X(j)
133 (i.e., the smallest relative change in any element of A or B
134 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
135 Number of error bounds to return for each right hand side and
136 each type (normwise or componentwise). See ERR_BNDS_NORM and
137 ERR_BNDS_COMP below.
138 ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS,
140 N_ERR_BNDS)
141 For each right-hand side, this array contains informa‐
142 tion about various error bounds and condition numbers
143 corresponding to the normwise relative error, which is
144 defined as follows: Normwise relative error in the ith
145 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
146 ------------------------------ max_j abs(X(j,i)) The
147 array is indexed by the type of error information as
148 described below. There currently are up to three pieces
149 of information returned. The first index in
150 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
151 side. The second index in ERR_BNDS_NORM(:,err) contains
152 the following three fields: err = 1 "Trust/don't trust"
153 boolean. Trust the answer if the reciprocal condition
154 number is less than the threshold sqrt(n) *
155 dlamch('Epsilon'). err = 2 "Guaranteed" error bound:
156 The estimated forward error, almost certainly within a
157 factor of 10 of the true error so long as the next entry
158 is greater than the threshold sqrt(n) *
159 dlamch('Epsilon'). This error bound should only be
160 trusted if the previous boolean is true. err = 3
161 Reciprocal condition number: Estimated normwise recipro‐
162 cal condition number. Compared with the threshold
163 sqrt(n) * dlamch('Epsilon') to determine if the error
164 estimate is "guaranteed". These reciprocal condition
165 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
166 some appropriately scaled matrix Z. Let Z = S*A, where
167 S scales each row by a power of the radix so all abso‐
168 lute row sums of Z are approximately 1. See Lapack
169 Working Note 165 for further details and extra cautions.
170 ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS,
172 N_ERR_BNDS)
173 For each right-hand side, this array contains informa‐
174 tion about various error bounds and condition numbers
175 corresponding to the componentwise relative error, which
176 is defined as follows: Componentwise relative error in
177 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
178 ---------------------- abs(X(j,i)) The array is indexed
179 by the right-hand side i (on which the componentwise
180 relative error depends), and the type of error informa‐
181 tion as described below. There currently are up to three
182 pieces of information returned for each right-hand side.
183 If componentwise accuracy is not requested (PARAMS(3) =
184 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
185 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
186 are returned. The first index in ERR_BNDS_COMP(i,:)
187 corresponds to the ith right-hand side. The second
188 index in ERR_BNDS_COMP(:,err) contains the following
189 three fields: err = 1 "Trust/don't trust" boolean. Trust
190 the answer if the reciprocal condition number is less
191 than the threshold sqrt(n) * dlamch('Epsilon'). err = 2
192 "Guaranteed" error bound: The estimated forward error,
193 almost certainly within a factor of 10 of the true error
194 so long as the next entry is greater than the threshold
195 sqrt(n) * dlamch('Epsilon'). This error bound should
196 only be trusted if the previous boolean is true. err =
197 3 Reciprocal condition number: Estimated componentwise
198 reciprocal condition number. Compared with the thresh‐
199 old sqrt(n) * dlamch('Epsilon') to determine if the
200 error estimate is "guaranteed". These reciprocal condi‐
201 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
202 for some appropriately scaled matrix Z. Let Z =
203 S*(A*diag(x)), where x is the solution for the current
204 right-hand side and S scales each row of A*diag(x) by a
205 power of the radix so all absolute row sums of Z are
206 approximately 1. See Lapack Working Note 165 for fur‐
207 ther details and extra cautions. NPARAMS (input) INTE‐
208 GER Specifies the number of parameters set in PARAMS.
209 If .LE. 0, the PARAMS array is never referenced and
210 default values are used.
211 PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
213 Specifies algorithm parameters. If an entry is .LT. 0.0, then
214 that entry will be filled with default value used for that
215 parameter. Only positions up to NPARAMS are accessed; defaults
216 are used for higher-numbered parameters.
217 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
218 refinement or not. Default: 1.0D+0
219 = 0.0 : No refinement is performed, and no error bounds are
220 computed. = 1.0 : Use the double-precision refinement algo‐
221 rithm, possibly with doubled-single computations if the compi‐
222 lation environment does not support DOUBLE PRECISION. (other
223 values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I =
224 2) : Maximum number of residual computations allowed for
225 refinement. Default: 10
226 Aggressive: Set to 100 to permit convergence using approximate
227 factorizations or factorizations other than LU. If the factor‐
228 ization uses a technique other than Gaussian elimination, the
229 guarantees in err_bnds_norm and err_bnds_comp may no longer be
230 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
231 if the code will attempt to find a solution with small compo‐
232 nentwise relative error in the double-precision algorithm.
233 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
234 nentwise convergence)
235 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
237 IWORK (workspace) INTEGER array, dimension (N)
239 INFO (output) INTEGER
241 = 0: Successful exit. The solution to every right-hand side is
242 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
243 gal value
244 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
245 has been completed, but the factor U is exactly singular, so
246 the solution and error bounds could not be computed. RCOND = 0
247 is returned. = N+J: The solution corresponding to the Jth
248 right-hand side is not guaranteed. The solutions corresponding
249 to other right- hand sides K with K > J may not be guaranteed
250 as well, but only the first such right-hand side is reported.
251 If a small componentwise error is not requested (PARAMS(3) =
252 0.0) then the Jth right-hand side is the first with a normwise
253 error bound that is not guaranteed (the smallest J such that
254 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
255 right-hand side is the first with either a normwise or compo‐
256 nentwise error bound that is not guaranteed (the smallest J
257 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
258 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
259 ERR_BNDS_COMP(:,1). To get information about all of the right-
260 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
261 LAPACK routine (version 3.2) November 2008 DSYRFSX(1)