1SPORFSX(1) LAPACK routine (version 3.2) SPORFSX(1)
2
6 SPORFSX - SPORFSX improve the computed solution to a system of linear
7 equations when the coefficient matrix is symmetric positive definite,
8 and provides error bounds and backward error estimates for the solu‐
9 tion
10
12 SUBROUTINE SPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB,
13 X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
14 ERR_BNDS_COMP, NPARAMS, PARAMS, 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 REAL RCOND
24 INTEGER IWORK( * )
26 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), X( LDX, *
28 ), WORK( * )
29 REAL S( * ), PARAMS( * ), BERR( * ), ERR_BNDS_NORM(
31 NRHS, * ), ERR_BNDS_COMP( NRHS, * )
32
34 SPORFSX improves the computed solution to a system of linear
35 equations when the coefficient matrix is symmetric positive
36 definite, 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 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) REAL 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) REAL array, dimension (LDAF,N)
83 The triangular factor U or L from the Cholesky factorization A
84 = U**T*U or A = L*L**T, as computed by SPOTRF.
85 LDAF (input) INTEGER
87 The leading dimension of the array AF. LDAF >= max(1,N).
88 S (input or output) REAL array, dimension (N)
90 The row scale factors for A. If EQUED = 'Y', A is multiplied
91 on the left and right by diag(S). S is an input argument if
92 FACT = 'F'; otherwise, S is an output argument. If FACT = 'F'
93 and EQUED = 'Y', each element of S must be positive. If S is
94 output, each element of S is a power of the radix. If S is
95 input, each element of S should be a power of the radix to
96 ensure a reliable solution and error estimates. Scaling by pow‐
97 ers of the radix does not cause rounding errors unless the
98 result underflows or overflows. Rounding errors during scaling
99 lead to refining with a matrix that is not equivalent to the
100 input matrix, producing error estimates that may not be reli‐
101 able.
102 B (input) REAL array, dimension (LDB,NRHS)
104 The right hand side matrix B.
105 LDB (input) INTEGER
107 The leading dimension of the array B. LDB >= max(1,N).
108 X (input/output) REAL array, dimension (LDX,NRHS)
110 On entry, the solution matrix X, as computed by SGETRS. On
111 exit, the improved solution matrix X.
112 LDX (input) INTEGER
114 The leading dimension of the array X. LDX >= max(1,N).
115 RCOND (output) REAL
117 Reciprocal scaled condition number. This is an estimate of the
118 reciprocal Skeel condition number of the matrix A after equili‐
119 bration (if done). If this is less than the machine precision
120 (in particular, if it is zero), the matrix is singular to work‐
121 ing precision. Note that the error may still be small even if
122 this number is very small and the matrix appears ill- condi‐
123 tioned.
124 BERR (output) REAL array, dimension (NRHS)
126 Componentwise relative backward error. This is the component‐
127 wise relative backward error of each solution vector X(j)
128 (i.e., the smallest relative change in any element of A or B
129 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
130 Number of error bounds to return for each right hand side and
131 each type (normwise or componentwise). See ERR_BNDS_NORM and
132 ERR_BNDS_COMP below.
133 ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS)
135 For each right-hand side, this array contains informa‐
136 tion about various error bounds and condition numbers
137 corresponding to the normwise relative error, which is
138 defined as follows: Normwise relative error in the ith
139 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
140 ------------------------------ max_j abs(X(j,i)) The
141 array is indexed by the type of error information as
142 described below. There currently are up to three pieces
143 of information returned. The first index in
144 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
145 side. The second index in ERR_BNDS_NORM(:,err) contains
146 the following three fields: err = 1 "Trust/don't trust"
147 boolean. Trust the answer if the reciprocal condition
148 number is less than the threshold sqrt(n) *
149 slamch('Epsilon'). err = 2 "Guaranteed" error bound:
150 The estimated forward error, almost certainly within a
151 factor of 10 of the true error so long as the next entry
152 is greater than the threshold sqrt(n) *
153 slamch('Epsilon'). This error bound should only be
154 trusted if the previous boolean is true. err = 3
155 Reciprocal condition number: Estimated normwise recipro‐
156 cal condition number. Compared with the threshold
157 sqrt(n) * slamch('Epsilon') to determine if the error
158 estimate is "guaranteed". These reciprocal condition
159 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
160 some appropriately scaled matrix Z. Let Z = S*A, where
161 S scales each row by a power of the radix so all abso‐
162 lute row sums of Z are approximately 1. See Lapack
163 Working Note 165 for further details and extra cautions.
164 ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)
166 For each right-hand side, this array contains informa‐
167 tion about various error bounds and condition numbers
168 corresponding to the componentwise relative error, which
169 is defined as follows: Componentwise relative error in
170 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
171 ---------------------- abs(X(j,i)) The array is indexed
172 by the right-hand side i (on which the componentwise
173 relative error depends), and the type of error informa‐
174 tion as described below. There currently are up to three
175 pieces of information returned for each right-hand side.
176 If componentwise accuracy is not requested (PARAMS(3) =
177 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
178 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
179 are returned. The first index in ERR_BNDS_COMP(i,:)
180 corresponds to the ith right-hand side. The second
181 index in ERR_BNDS_COMP(:,err) contains the following
182 three fields: err = 1 "Trust/don't trust" boolean. Trust
183 the answer if the reciprocal condition number is less
184 than the threshold sqrt(n) * slamch('Epsilon'). err = 2
185 "Guaranteed" error bound: The estimated forward error,
186 almost certainly within a factor of 10 of the true error
187 so long as the next entry is greater than the threshold
188 sqrt(n) * slamch('Epsilon'). This error bound should
189 only be trusted if the previous boolean is true. err =
190 3 Reciprocal condition number: Estimated componentwise
191 reciprocal condition number. Compared with the thresh‐
192 old sqrt(n) * slamch('Epsilon') to determine if the
193 error estimate is "guaranteed". These reciprocal condi‐
194 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
195 for some appropriately scaled matrix Z. Let Z =
196 S*(A*diag(x)), where x is the solution for the current
197 right-hand side and S scales each row of A*diag(x) by a
198 power of the radix so all absolute row sums of Z are
199 approximately 1. See Lapack Working Note 165 for fur‐
200 ther details and extra cautions. NPARAMS (input) INTE‐
201 GER Specifies the number of parameters set in PARAMS.
202 If .LE. 0, the PARAMS array is never referenced and
203 default values are used.
204 PARAMS (input / output) REAL array, dimension NPARAMS
206 Specifies algorithm parameters. If an entry is .LT. 0.0, then
207 that entry will be filled with default value used for that
208 parameter. Only positions up to NPARAMS are accessed; defaults
209 are used for higher-numbered parameters.
210 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
211 refinement or not. Default: 1.0
212 = 0.0 : No refinement is performed, and no error bounds are
213 computed. = 1.0 : Use the double-precision refinement algo‐
214 rithm, possibly with doubled-single computations if the compi‐
215 lation environment does not support DOUBLE PRECISION. (other
216 values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I =
217 2) : Maximum number of residual computations allowed for
218 refinement. Default: 10
219 Aggressive: Set to 100 to permit convergence using approximate
220 factorizations or factorizations other than LU. If the factor‐
221 ization uses a technique other than Gaussian elimination, the
222 guarantees in err_bnds_norm and err_bnds_comp may no longer be
223 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
224 if the code will attempt to find a solution with small compo‐
225 nentwise relative error in the double-precision algorithm.
226 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
227 nentwise convergence)
228 WORK (workspace) REAL array, dimension (4*N)
230 IWORK (workspace) INTEGER array, dimension (N)
232 INFO (output) INTEGER
234 = 0: Successful exit. The solution to every right-hand side is
235 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
236 gal value
237 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
238 has been completed, but the factor U is exactly singular, so
239 the solution and error bounds could not be computed. RCOND = 0
240 is returned. = N+J: The solution corresponding to the Jth
241 right-hand side is not guaranteed. The solutions corresponding
242 to other right- hand sides K with K > J may not be guaranteed
243 as well, but only the first such right-hand side is reported.
244 If a small componentwise error is not requested (PARAMS(3) =
245 0.0) then the Jth right-hand side is the first with a normwise
246 error bound that is not guaranteed (the smallest J such that
247 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
248 right-hand side is the first with either a normwise or compo‐
249 nentwise error bound that is not guaranteed (the smallest J
250 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
251 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
252 ERR_BNDS_COMP(:,1). To get information about all of the right-
253 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
254 LAPACK routine (version 3.2) November 2008 SPORFSX(1)