1DGERFSX(1) LAPACK routine (version 3.2) DGERFSX(1)
2
6 DGERFSX - DGERFSX improve the computed solution to a system of linear
7 equations and provides error bounds and backward error estimates for
8 the solution
9
11 SUBROUTINE DGERFSX( TRANS, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, R,
12 C, B, 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 TRANS, 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 R( * ), C( * ), PARAMS( * ), BERR( * ),
31 ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * )
32
34 DGERFSX improves the computed solution to a system of linear
35 equations and provides error bounds and backward error estimates
36 for the solution. In addition to normwise error bound, the code
37 provides maximum componentwise error bound if possible. See
38 comments for ERR_BNDS_N and ERR_BNDS_C for details of the error
39 bounds.
40 The original system of linear equations may have been equilibrated
41 before calling this routine, as described by arguments EQUED, R
42 and C below. In this case, the solution and error bounds returned
43 are 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 TRANS (input) CHARACTER*1
52 Specifies the form of the system of equations:
53 = 'N': A * X = B (No transpose)
54 = 'T': A**T * X = B (Transpose)
55 = 'C': A**H * X = B (Conjugate transpose = Transpose)
56 EQUED (input) CHARACTER*1
58 Specifies the form of equilibration that was done to A before
59 calling this routine. This is needed to compute the solution
60 and error bounds correctly. = 'N': No equilibration
61 = 'R': Row equilibration, i.e., A has been premultiplied by
62 diag(R). = 'C': Column equilibration, i.e., A has been post‐
63 multiplied by diag(C). = 'B': Both row and column equilibra‐
64 tion, i.e., A has been replaced by diag(R) * A * diag(C). The
65 right hand side B has been changed accordingly.
66 N (input) INTEGER
68 The order of the matrix A. N >= 0.
69 NRHS (input) INTEGER
71 The number of right hand sides, i.e., the number of columns of
72 the matrices B and X. NRHS >= 0.
73 A (input) DOUBLE PRECISION array, dimension (LDA,N)
75 The original N-by-N matrix A.
76 LDA (input) INTEGER
78 The leading dimension of the array A. LDA >= max(1,N).
79 AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
81 The factors L and U from the factorization A = P*L*U as com‐
82 puted by DGETRF.
83 LDAF (input) INTEGER
85 The leading dimension of the array AF. LDAF >= max(1,N).
86 IPIV (input) INTEGER array, dimension (N)
88 The pivot indices from DGETRF; for 1<=i<=N, row i of the matrix
89 was interchanged with row IPIV(i).
90 R (input or output) DOUBLE PRECISION array, dimension (N)
92 The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
93 tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
94 accessed. R is an input argument if FACT = 'F'; otherwise, R
95 is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
96 each element of R must be positive. If R is output, each ele‐
97 ment of R is a power of the radix. If R is input, each element
98 of R should be a power of the radix to ensure a reliable solu‐
99 tion and error estimates. Scaling by powers of the radix does
100 not cause rounding errors unless the result underflows or over‐
101 flows. Rounding errors during scaling lead to refining with a
102 matrix that is not equivalent to the input matrix, producing
103 error estimates that may not be reliable.
104 C (input or output) DOUBLE PRECISION array, dimension (N)
106 The column scale factors for A. If EQUED = 'C' or 'B', A is
107 multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
108 not accessed. C is an input argument if FACT = 'F'; otherwise,
109 C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
110 each element of C must be positive. If C is output, each ele‐
111 ment of C is a power of the radix. If C is input, each element
112 of C should be a power of the radix to ensure a reliable solu‐
113 tion and error estimates. Scaling by powers of the radix does
114 not cause rounding errors unless the result underflows or over‐
115 flows. Rounding errors during scaling lead to refining with a
116 matrix that is not equivalent to the input matrix, producing
117 error estimates that may not be reliable.
118 B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
120 The right hand side matrix B.
121 LDB (input) INTEGER
123 The leading dimension of the array B. LDB >= max(1,N).
124 X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
126 On entry, the solution matrix X, as computed by DGETRS. On
127 exit, the improved solution matrix X.
128 LDX (input) INTEGER
130 The leading dimension of the array X. LDX >= max(1,N).
131 RCOND (output) DOUBLE PRECISION
133 Reciprocal scaled condition number. This is an estimate of the
134 reciprocal Skeel condition number of the matrix A after equili‐
135 bration (if done). If this is less than the machine precision
136 (in particular, if it is zero), the matrix is singular to work‐
137 ing precision. Note that the error may still be small even if
138 this number is very small and the matrix appears ill- condi‐
139 tioned.
140 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
142 Componentwise relative backward error. This is the component‐
143 wise relative backward error of each solution vector X(j)
144 (i.e., the smallest relative change in any element of A or B
145 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
146 Number of error bounds to return for each right hand side and
147 each type (normwise or componentwise). See ERR_BNDS_NORM and
148 ERR_BNDS_COMP below.
149 ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS,
151 N_ERR_BNDS)
152 For each right-hand side, this array contains informa‐
153 tion about various error bounds and condition numbers
154 corresponding to the normwise relative error, which is
155 defined as follows: Normwise relative error in the ith
156 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
157 ------------------------------ max_j abs(X(j,i)) The
158 array is indexed by the type of error information as
159 described below. There currently are up to three pieces
160 of information returned. The first index in
161 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
162 side. The second index in ERR_BNDS_NORM(:,err) contains
163 the following three fields: err = 1 "Trust/don't trust"
164 boolean. Trust the answer if the reciprocal condition
165 number is less than the threshold sqrt(n) *
166 dlamch('Epsilon'). err = 2 "Guaranteed" error bound:
167 The estimated forward error, almost certainly within a
168 factor of 10 of the true error so long as the next entry
169 is greater than the threshold sqrt(n) *
170 dlamch('Epsilon'). This error bound should only be
171 trusted if the previous boolean is true. err = 3
172 Reciprocal condition number: Estimated normwise recipro‐
173 cal condition number. Compared with the threshold
174 sqrt(n) * dlamch('Epsilon') to determine if the error
175 estimate is "guaranteed". These reciprocal condition
176 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
177 some appropriately scaled matrix Z. Let Z = S*A, where
178 S scales each row by a power of the radix so all abso‐
179 lute row sums of Z are approximately 1. See Lapack
180 Working Note 165 for further details and extra cautions.
181 ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS,
183 N_ERR_BNDS)
184 For each right-hand side, this array contains informa‐
185 tion about various error bounds and condition numbers
186 corresponding to the componentwise relative error, which
187 is defined as follows: Componentwise relative error in
188 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
189 ---------------------- abs(X(j,i)) The array is indexed
190 by the right-hand side i (on which the componentwise
191 relative error depends), and the type of error informa‐
192 tion as described below. There currently are up to three
193 pieces of information returned for each right-hand side.
194 If componentwise accuracy is not requested (PARAMS(3) =
195 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
196 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
197 are returned. The first index in ERR_BNDS_COMP(i,:)
198 corresponds to the ith right-hand side. The second
199 index in ERR_BNDS_COMP(:,err) contains the following
200 three fields: err = 1 "Trust/don't trust" boolean. Trust
201 the answer if the reciprocal condition number is less
202 than the threshold sqrt(n) * dlamch('Epsilon'). err = 2
203 "Guaranteed" error bound: The estimated forward error,
204 almost certainly within a factor of 10 of the true error
205 so long as the next entry is greater than the threshold
206 sqrt(n) * dlamch('Epsilon'). This error bound should
207 only be trusted if the previous boolean is true. err =
208 3 Reciprocal condition number: Estimated componentwise
209 reciprocal condition number. Compared with the thresh‐
210 old sqrt(n) * dlamch('Epsilon') to determine if the
211 error estimate is "guaranteed". These reciprocal condi‐
212 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
213 for some appropriately scaled matrix Z. Let Z =
214 S*(A*diag(x)), where x is the solution for the current
215 right-hand side and S scales each row of A*diag(x) by a
216 power of the radix so all absolute row sums of Z are
217 approximately 1. See Lapack Working Note 165 for fur‐
218 ther details and extra cautions. NPARAMS (input) INTE‐
219 GER Specifies the number of parameters set in PARAMS.
220 If .LE. 0, the PARAMS array is never referenced and
221 default values are used.
222 PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
224 Specifies algorithm parameters. If an entry is .LT. 0.0, then
225 that entry will be filled with default value used for that
226 parameter. Only positions up to NPARAMS are accessed; defaults
227 are used for higher-numbered parameters.
228 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
229 refinement or not. Default: 1.0D+0
230 = 0.0 : No refinement is performed, and no error bounds are
231 computed. = 1.0 : Use the double-precision refinement algo‐
232 rithm, possibly with doubled-single computations if the compi‐
233 lation environment does not support DOUBLE PRECISION. (other
234 values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I =
235 2) : Maximum number of residual computations allowed for
236 refinement. Default: 10
237 Aggressive: Set to 100 to permit convergence using approximate
238 factorizations or factorizations other than LU. If the factor‐
239 ization uses a technique other than Gaussian elimination, the
240 guarantees in err_bnds_norm and err_bnds_comp may no longer be
241 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
242 if the code will attempt to find a solution with small compo‐
243 nentwise relative error in the double-precision algorithm.
244 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
245 nentwise convergence)
246 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
248 IWORK (workspace) INTEGER array, dimension (N)
250 INFO (output) INTEGER
252 = 0: Successful exit. The solution to every right-hand side is
253 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
254 gal value
255 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
256 has been completed, but the factor U is exactly singular, so
257 the solution and error bounds could not be computed. RCOND = 0
258 is returned. = N+J: The solution corresponding to the Jth
259 right-hand side is not guaranteed. The solutions corresponding
260 to other right- hand sides K with K > J may not be guaranteed
261 as well, but only the first such right-hand side is reported.
262 If a small componentwise error is not requested (PARAMS(3) =
263 0.0) then the Jth right-hand side is the first with a normwise
264 error bound that is not guaranteed (the smallest J such that
265 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
266 right-hand side is the first with either a normwise or compo‐
267 nentwise error bound that is not guaranteed (the smallest J
268 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
269 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
270 ERR_BNDS_COMP(:,1). To get information about all of the right-
271 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
272 LAPACK routine (version 3.2) November 2008 DGERFSX(1)