1CGBRFSX(1) LAPACK routine (version 3.2) CGBRFSX(1)
2
6 CGBRFSX - CGBRFSX 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 CGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB,
12 LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, BERR,
13 N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS,
14 PARAMS, WORK, RWORK, INFO )
15 IMPLICIT NONE
17 CHARACTER TRANS, EQUED
19 INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS,
21 NPARAMS, N_ERR_BNDS
22 REAL RCOND
24 INTEGER IPIV( * )
26 COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), 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 CGBRFSX 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65 tion, i.e., A has been replaced by diag(R) * A * diag(C).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 KL (input) INTEGER
72 The number of subdiagonals within the band of A. KL >= 0.
73 KU (input) INTEGER
75 The number of superdiagonals within the band of A. KU >= 0.
76 NRHS (input) INTEGER
78 The number of right hand sides, i.e., the number of columns of
79 the matrices B and X. NRHS >= 0.
80 AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
82 The original band matrix A, stored in rows 1 to KL+KU+1. The
83 j-th column of A is stored in the j-th column of the array AB
84 as follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-
85 ku)<=i<=min(n,j+kl).
86 LDAB (input) INTEGER
88 The leading dimension of the array AB. LDAB >= KL+KU+1.
89 AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N)
91 Details of the LU factorization of the band matrix A, as com‐
92 puted by DGBTRF. U is stored as an upper triangular band
93 matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
94 multipliers used during the factorization are stored in rows
95 KL+KU+2 to 2*KL+KU+1.
96 LDAFB (input) INTEGER
98 The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
99 IPIV (input) INTEGER array, dimension (N)
101 The pivot indices from SGETRF; for 1<=i<=N, row i of the matrix
102 was interchanged with row IPIV(i).
103 R (input or output) REAL array, dimension (N)
105 The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
106 tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
107 accessed. R is an input argument if FACT = 'F'; otherwise, R
108 is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
109 each element of R must be positive. If R is output, each ele‐
110 ment of R is a power of the radix. If R is input, each element
111 of R should be a power of the radix to ensure a reliable solu‐
112 tion and error estimates. Scaling by powers of the radix does
113 not cause rounding errors unless the result underflows or over‐
114 flows. Rounding errors during scaling lead to refining with a
115 matrix that is not equivalent to the input matrix, producing
116 error estimates that may not be reliable.
117 C (input or output) REAL array, dimension (N)
119 The column scale factors for A. If EQUED = 'C' or 'B', A is
120 multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
121 not accessed. C is an input argument if FACT = 'F'; otherwise,
122 C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
123 each element of C must be positive. If C is output, each ele‐
124 ment of C is a power of the radix. If C is input, each element
125 of C should be a power of the radix to ensure a reliable solu‐
126 tion and error estimates. Scaling by powers of the radix does
127 not cause rounding errors unless the result underflows or over‐
128 flows. Rounding errors during scaling lead to refining with a
129 matrix that is not equivalent to the input matrix, producing
130 error estimates that may not be reliable.
131 B (input) REAL array, dimension (LDB,NRHS)
133 The right hand side matrix B.
134 LDB (input) INTEGER
136 The leading dimension of the array B. LDB >= max(1,N).
137 X (input/output) REAL array, dimension (LDX,NRHS)
139 On entry, the solution matrix X, as computed by SGETRS. On
140 exit, the improved solution matrix X.
141 LDX (input) INTEGER
143 The leading dimension of the array X. LDX >= max(1,N).
144 RCOND (output) REAL
146 Reciprocal scaled condition number. This is an estimate of the
147 reciprocal Skeel condition number of the matrix A after equili‐
148 bration (if done). If this is less than the machine precision
149 (in particular, if it is zero), the matrix is singular to work‐
150 ing precision. Note that the error may still be small even if
151 this number is very small and the matrix appears ill- condi‐
152 tioned.
153 BERR (output) REAL array, dimension (NRHS)
155 Componentwise relative backward error. This is the component‐
156 wise relative backward error of each solution vector X(j)
157 (i.e., the smallest relative change in any element of A or B
158 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
159 Number of error bounds to return for each right hand side and
160 each type (normwise or componentwise). See ERR_BNDS_NORM and
161 ERR_BNDS_COMP below.
162 ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS)
164 For each right-hand side, this array contains informa‐
165 tion about various error bounds and condition numbers
166 corresponding to the normwise relative error, which is
167 defined as follows: Normwise relative error in the ith
168 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
169 ------------------------------ max_j abs(X(j,i)) The
170 array is indexed by the type of error information as
171 described below. There currently are up to three pieces
172 of information returned. The first index in
173 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
174 side. The second index in ERR_BNDS_NORM(:,err) contains
175 the following three fields: err = 1 "Trust/don't trust"
176 boolean. Trust the answer if the reciprocal condition
177 number is less than the threshold sqrt(n) *
178 slamch('Epsilon'). err = 2 "Guaranteed" error bound:
179 The estimated forward error, almost certainly within a
180 factor of 10 of the true error so long as the next entry
181 is greater than the threshold sqrt(n) *
182 slamch('Epsilon'). This error bound should only be
183 trusted if the previous boolean is true. err = 3
184 Reciprocal condition number: Estimated normwise recipro‐
185 cal condition number. Compared with the threshold
186 sqrt(n) * slamch('Epsilon') to determine if the error
187 estimate is "guaranteed". These reciprocal condition
188 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
189 some appropriately scaled matrix Z. Let Z = S*A, where
190 S scales each row by a power of the radix so all abso‐
191 lute row sums of Z are approximately 1. See Lapack
192 Working Note 165 for further details and extra cautions.
193 ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)
195 For each right-hand side, this array contains informa‐
196 tion about various error bounds and condition numbers
197 corresponding to the componentwise relative error, which
198 is defined as follows: Componentwise relative error in
199 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
200 ---------------------- abs(X(j,i)) The array is indexed
201 by the right-hand side i (on which the componentwise
202 relative error depends), and the type of error informa‐
203 tion as described below. There currently are up to three
204 pieces of information returned for each right-hand side.
205 If componentwise accuracy is not requested (PARAMS(3) =
206 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
207 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
208 are returned. The first index in ERR_BNDS_COMP(i,:)
209 corresponds to the ith right-hand side. The second
210 index in ERR_BNDS_COMP(:,err) contains the following
211 three fields: err = 1 "Trust/don't trust" boolean. Trust
212 the answer if the reciprocal condition number is less
213 than the threshold sqrt(n) * slamch('Epsilon'). err = 2
214 "Guaranteed" error bound: The estimated forward error,
215 almost certainly within a factor of 10 of the true error
216 so long as the next entry is greater than the threshold
217 sqrt(n) * slamch('Epsilon'). This error bound should
218 only be trusted if the previous boolean is true. err =
219 3 Reciprocal condition number: Estimated componentwise
220 reciprocal condition number. Compared with the thresh‐
221 old sqrt(n) * slamch('Epsilon') to determine if the
222 error estimate is "guaranteed". These reciprocal condi‐
223 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
224 for some appropriately scaled matrix Z. Let Z =
225 S*(A*diag(x)), where x is the solution for the current
226 right-hand side and S scales each row of A*diag(x) by a
227 power of the radix so all absolute row sums of Z are
228 approximately 1. See Lapack Working Note 165 for fur‐
229 ther details and extra cautions. NPARAMS (input) INTE‐
230 GER Specifies the number of parameters set in PARAMS.
231 If .LE. 0, the PARAMS array is never referenced and
232 default values are used.
233 PARAMS (input / output) REAL array, dimension NPARAMS
235 Specifies algorithm parameters. If an entry is .LT. 0.0, then
236 that entry will be filled with default value used for that
237 parameter. Only positions up to NPARAMS are accessed; defaults
238 are used for higher-numbered parameters.
239 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
240 refinement or not. Default: 1.0
241 = 0.0 : No refinement is performed, and no error bounds are
242 computed. = 1.0 : Use the double-precision refinement algo‐
243 rithm, possibly with doubled-single computations if the compi‐
244 lation environment does not support DOUBLE PRECISION. (other
245 values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I =
246 2) : Maximum number of residual computations allowed for
247 refinement. Default: 10
248 Aggressive: Set to 100 to permit convergence using approximate
249 factorizations or factorizations other than LU. If the factor‐
250 ization uses a technique other than Gaussian elimination, the
251 guarantees in err_bnds_norm and err_bnds_comp may no longer be
252 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
253 if the code will attempt to find a solution with small compo‐
254 nentwise relative error in the double-precision algorithm.
255 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
256 nentwise convergence)
257 WORK (workspace) REAL array, dimension (4*N)
259 IWORK (workspace) INTEGER array, dimension (N)
261 INFO (output) INTEGER
263 = 0: Successful exit. The solution to every right-hand side is
264 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
265 gal value
266 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
267 has been completed, but the factor U is exactly singular, so
268 the solution and error bounds could not be computed. RCOND = 0
269 is returned. = N+J: The solution corresponding to the Jth
270 right-hand side is not guaranteed. The solutions corresponding
271 to other right- hand sides K with K > J may not be guaranteed
272 as well, but only the first such right-hand side is reported.
273 If a small componentwise error is not requested (PARAMS(3) =
274 0.0) then the Jth right-hand side is the first with a normwise
275 error bound that is not guaranteed (the smallest J such that
276 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
277 right-hand side is the first with either a normwise or compo‐
278 nentwise error bound that is not guaranteed (the smallest J
279 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
280 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
281 ERR_BNDS_COMP(:,1). To get information about all of the right-
282 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
283 LAPACK routine (version 3.2) November 2008 CGBRFSX(1)