1CGERFSX(1) LAPACK routine (version 3.2) CGERFSX(1)
2
6 CGERFSX - CGERFSX 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 CGERFSX( 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, RWORK, INFO )
15 IMPLICIT NONE
17 CHARACTER TRANS, EQUED
19 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
21 N_ERR_BNDS
22 REAL RCOND
24 INTEGER IPIV( * )
26 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ), X( LDX , *
28 ), WORK( * )
29 REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
31 ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * ),
32 RWORK( * )
33
35 CGERFSX improves the computed solution to a system of linear
36 equations and provides error bounds and backward error estimates
37 for the solution. In addition to normwise error bound, the code
38 provides maximum componentwise error bound if possible. See
39 comments for ERR_BNDS_N and ERR_BNDS_C for details of the error
40 bounds.
41 The original system of linear equations may have been equilibrated
42 before calling this routine, as described by arguments EQUED, R
43 and C below. In this case, the solution and error bounds returned
44 are for the original unequilibrated system.
45
47 Some optional parameters are bundled in the PARAMS array. These set‐
48 tings determine how refinement is performed, but often the defaults are
49 acceptable. If the defaults are acceptable, users can pass NPARAMS = 0
50 which prevents the source code from accessing the PARAMS argument.
51 TRANS (input) CHARACTER*1
53 Specifies the form of the system of equations:
54 = 'N': A * X = B (No transpose)
55 = 'T': A**T * X = B (Transpose)
56 = 'C': A**H * X = B (Conjugate transpose = Transpose)
57 EQUED (input) CHARACTER*1
59 Specifies the form of equilibration that was done to A before
60 calling this routine. This is needed to compute the solution
61 and error bounds correctly. = 'N': No equilibration
62 = 'R': Row equilibration, i.e., A has been premultiplied by
63 diag(R). = 'C': Column equilibration, i.e., A has been post‐
64 multiplied by diag(C). = 'B': Both row and column equilibra‐
65 tion, i.e., A has been replaced by diag(R) * A * diag(C). The
66 right hand side B has been changed accordingly.
67 N (input) INTEGER
69 The order of the matrix A. N >= 0.
70 NRHS (input) INTEGER
72 The number of right hand sides, i.e., the number of columns of
73 the matrices B and X. NRHS >= 0.
74 A (input) COMPLEX array, dimension (LDA,N)
76 The original N-by-N matrix A.
77 LDA (input) INTEGER
79 The leading dimension of the array A. LDA >= max(1,N).
80 AF (input) COMPLEX array, dimension (LDAF,N)
82 The factors L and U from the factorization A = P*L*U as com‐
83 puted by CGETRF.
84 LDAF (input) INTEGER
86 The leading dimension of the array AF. LDAF >= max(1,N).
87 IPIV (input) INTEGER array, dimension (N)
89 The pivot indices from CGETRF; for 1<=i<=N, row i of the matrix
90 was interchanged with row IPIV(i).
91 R (input or output) REAL array, dimension (N)
93 The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
94 tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
95 accessed. R is an input argument if FACT = 'F'; otherwise, R
96 is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
97 each element of R must be positive. If R is output, each ele‐
98 ment of R is a power of the radix. If R is input, each element
99 of R should be a power of the radix to ensure a reliable solu‐
100 tion and error estimates. Scaling by powers of the radix does
101 not cause rounding errors unless the result underflows or over‐
102 flows. Rounding errors during scaling lead to refining with a
103 matrix that is not equivalent to the input matrix, producing
104 error estimates that may not be reliable.
105 C (input or output) REAL array, dimension (N)
107 The column scale factors for A. If EQUED = 'C' or 'B', A is
108 multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
109 not accessed. C is an input argument if FACT = 'F'; otherwise,
110 C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
111 each element of C must be positive. If C is output, each ele‐
112 ment of C is a power of the radix. If C is input, each element
113 of C should be a power of the radix to ensure a reliable solu‐
114 tion and error estimates. Scaling by powers of the radix does
115 not cause rounding errors unless the result underflows or over‐
116 flows. Rounding errors during scaling lead to refining with a
117 matrix that is not equivalent to the input matrix, producing
118 error estimates that may not be reliable.
119 B (input) COMPLEX array, dimension (LDB,NRHS)
121 The right hand side matrix B.
122 LDB (input) INTEGER
124 The leading dimension of the array B. LDB >= max(1,N).
125 X (input/output) COMPLEX array, dimension (LDX,NRHS)
127 On entry, the solution matrix X, as computed by CGETRS. On
128 exit, the improved solution matrix X.
129 LDX (input) INTEGER
131 The leading dimension of the array X. LDX >= max(1,N).
132 RCOND (output) REAL
134 Reciprocal scaled condition number. This is an estimate of the
135 reciprocal Skeel condition number of the matrix A after equili‐
136 bration (if done). If this is less than the machine precision
137 (in particular, if it is zero), the matrix is singular to work‐
138 ing precision. Note that the error may still be small even if
139 this number is very small and the matrix appears ill- condi‐
140 tioned.
141 BERR (output) REAL array, dimension (NRHS)
143 Componentwise relative backward error. This is the component‐
144 wise relative backward error of each solution vector X(j)
145 (i.e., the smallest relative change in any element of A or B
146 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
147 Number of error bounds to return for each right hand side and
148 each type (normwise or componentwise). See ERR_BNDS_NORM and
149 ERR_BNDS_COMP below.
150 ERR_BNDS_NORM (output) REAL array, dimension (NRHS, 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 slamch('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 slamch('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) * slamch('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) REAL array, dimension (NRHS, N_ERR_BNDS)
183 For each right-hand side, this array contains informa‐
184 tion about various error bounds and condition numbers
185 corresponding to the componentwise relative error, which
186 is defined as follows: Componentwise relative error in
187 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
188 ---------------------- abs(X(j,i)) The array is indexed
189 by the right-hand side i (on which the componentwise
190 relative error depends), and the type of error informa‐
191 tion as described below. There currently are up to three
192 pieces of information returned for each right-hand side.
193 If componentwise accuracy is not requested (PARAMS(3) =
194 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
195 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
196 are returned. The first index in ERR_BNDS_COMP(i,:)
197 corresponds to the ith right-hand side. The second
198 index in ERR_BNDS_COMP(:,err) contains the following
199 three fields: err = 1 "Trust/don't trust" boolean. Trust
200 the answer if the reciprocal condition number is less
201 than the threshold sqrt(n) * slamch('Epsilon'). err = 2
202 "Guaranteed" error bound: The estimated forward error,
203 almost certainly within a factor of 10 of the true error
204 so long as the next entry is greater than the threshold
205 sqrt(n) * slamch('Epsilon'). This error bound should
206 only be trusted if the previous boolean is true. err =
207 3 Reciprocal condition number: Estimated componentwise
208 reciprocal condition number. Compared with the thresh‐
209 old sqrt(n) * slamch('Epsilon') to determine if the
210 error estimate is "guaranteed". These reciprocal condi‐
211 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
212 for some appropriately scaled matrix Z. Let Z =
213 S*(A*diag(x)), where x is the solution for the current
214 right-hand side and S scales each row of A*diag(x) by a
215 power of the radix so all absolute row sums of Z are
216 approximately 1. See Lapack Working Note 165 for fur‐
217 ther details and extra cautions. NPARAMS (input) INTE‐
218 GER Specifies the number of parameters set in PARAMS.
219 If .LE. 0, the PARAMS array is never referenced and
220 default values are used.
221 PARAMS (input / output) REAL array, dimension NPARAMS
223 Specifies algorithm parameters. If an entry is .LT. 0.0, then
224 that entry will be filled with default value used for that
225 parameter. Only positions up to NPARAMS are accessed; defaults
226 are used for higher-numbered parameters.
227 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
228 refinement or not. Default: 1.0
229 = 0.0 : No refinement is performed, and no error bounds are
230 computed. = 1.0 : Use the double-precision refinement algo‐
231 rithm, possibly with doubled-single computations if the compi‐
232 lation environment does not support DOUBLE PRECISION. (other
233 values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I =
234 2) : Maximum number of residual computations allowed for
235 refinement. Default: 10
236 Aggressive: Set to 100 to permit convergence using approximate
237 factorizations or factorizations other than LU. If the factor‐
238 ization uses a technique other than Gaussian elimination, the
239 guarantees in err_bnds_norm and err_bnds_comp may no longer be
240 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
241 if the code will attempt to find a solution with small compo‐
242 nentwise relative error in the double-precision algorithm.
243 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
244 nentwise convergence)
245 WORK (workspace) REAL array, dimension (4*N)
247 IWORK (workspace) INTEGER array, dimension (N)
249 INFO (output) INTEGER
251 = 0: Successful exit. The solution to every right-hand side is
252 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
253 gal value
254 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
255 has been completed, but the factor U is exactly singular, so
256 the solution and error bounds could not be computed. RCOND = 0
257 is returned. = N+J: The solution corresponding to the Jth
258 right-hand side is not guaranteed. The solutions corresponding
259 to other right- hand sides K with K > J may not be guaranteed
260 as well, but only the first such right-hand side is reported.
261 If a small componentwise error is not requested (PARAMS(3) =
262 0.0) then the Jth right-hand side is the first with a normwise
263 error bound that is not guaranteed (the smallest J such that
264 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
265 right-hand side is the first with either a normwise or compo‐
266 nentwise error bound that is not guaranteed (the smallest J
267 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
268 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
269 ERR_BNDS_COMP(:,1). To get information about all of the right-
270 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
271 LAPACK routine (version 3.2) November 2008 CGERFSX(1)