1ZPORFSX(1) LAPACK routine (version 3.2) ZPORFSX(1)
2
6 ZPORFSX - ZPORFSX 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 ZPORFSX( 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, RWORK, 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 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), X( LDX, *
26 ), WORK( * )
27 DOUBLE PRECISION RWORK( * ), S( * ), PARAMS(*), BERR( * ),
29 ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * )
30
32 ZPORFSX improves the computed solution to a system of linear
33 equations when the coefficient matrix is symmetric positive
34 definite, and provides error bounds and backward error estimates
35 for the solution. In addition to normwise error bound, the code
36 provides maximum componentwise error bound if possible. See
37 comments for ERR_BNDS for details of the error bounds.
38 The original system of linear equations may have been equilibrated
39 before calling this routine, as described by arguments EQUED and S
40 below. In this case, the solution and error bounds returned are
41 for the original unequilibrated system.
42
44 Some optional parameters are bundled in the PARAMS array. These set‐
45 tings determine how refinement is performed, but often the defaults are
46 acceptable. If the defaults are acceptable, users can pass NPARAMS = 0
47 which prevents the source code from accessing the PARAMS argument.
48 UPLO (input) CHARACTER*1
50 = 'U': Upper triangle of A is stored;
51 = 'L': Lower triangle of A is stored.
52 EQUED (input) CHARACTER*1
54 Specifies the form of equilibration that was done to A before
55 calling this routine. This is needed to compute the solution
56 and error bounds correctly. = 'N': No equilibration
57 = 'Y': Both row and column equilibration, i.e., A has been
58 replaced by diag(S) * A * diag(S). The right hand side B has
59 been changed accordingly.
60 N (input) INTEGER
62 The order of the matrix A. N >= 0.
63 NRHS (input) INTEGER
65 The number of right hand sides, i.e., the number of columns of
66 the matrices B and X. NRHS >= 0.
67 A (input) COMPLEX*16 array, dimension (LDA,N)
69 The symmetric matrix A. If UPLO = 'U', the leading N-by-N
70 upper triangular part of A contains the upper triangular part
71 of the matrix A, and the strictly lower triangular part of A is
72 not referenced. If UPLO = 'L', the leading N-by-N lower trian‐
73 gular part of A contains the lower triangular part of the
74 matrix A, and the strictly upper triangular part of A is not
75 referenced.
76 LDA (input) INTEGER
78 The leading dimension of the array A. LDA >= max(1,N).
79 AF (input) COMPLEX*16 array, dimension (LDAF,N)
81 The triangular factor U or L from the Cholesky factorization A
82 = U**T*U or A = L*L**T, as computed by DPOTRF.
83 LDAF (input) INTEGER
85 The leading dimension of the array AF. LDAF >= max(1,N).
86 S (input or output) DOUBLE PRECISION array, dimension (N)
88 The row scale factors for A. If EQUED = 'Y', A is multiplied
89 on the left and right by diag(S). S is an input argument if
90 FACT = 'F'; otherwise, S is an output argument. If FACT = 'F'
91 and EQUED = 'Y', each element of S must be positive. If S is
92 output, each element of S is a power of the radix. If S is
93 input, each element of S should be a power of the radix to
94 ensure a reliable solution and error estimates. Scaling by pow‐
95 ers of the radix does not cause rounding errors unless the
96 result underflows or overflows. Rounding errors during scaling
97 lead to refining with a matrix that is not equivalent to the
98 input matrix, producing error estimates that may not be reli‐
99 able.
100 B (input) COMPLEX*16 array, dimension (LDB,NRHS)
102 The right hand side matrix B.
103 LDB (input) INTEGER
105 The leading dimension of the array B. LDB >= max(1,N).
106 X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
108 On entry, the solution matrix X, as computed by DGETRS. On
109 exit, the improved solution matrix X.
110 LDX (input) INTEGER
112 The leading dimension of the array X. LDX >= max(1,N).
113 RCOND (output) DOUBLE PRECISION
115 Reciprocal scaled condition number. This is an estimate of the
116 reciprocal Skeel condition number of the matrix A after equili‐
117 bration (if done). If this is less than the machine precision
118 (in particular, if it is zero), the matrix is singular to work‐
119 ing precision. Note that the error may still be small even if
120 this number is very small and the matrix appears ill- condi‐
121 tioned.
122 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
124 Componentwise relative backward error. This is the component‐
125 wise relative backward error of each solution vector X(j)
126 (i.e., the smallest relative change in any element of A or B
127 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
128 Number of error bounds to return for each right hand side and
129 each type (normwise or componentwise). See ERR_BNDS_NORM and
130 ERR_BNDS_COMP below.
131 ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS,
133 N_ERR_BNDS)
134 For each right-hand side, this array contains informa‐
135 tion about various error bounds and condition numbers
136 corresponding to the normwise relative error, which is
137 defined as follows: Normwise relative error in the ith
138 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
139 ------------------------------ max_j abs(X(j,i)) The
140 array is indexed by the type of error information as
141 described below. There currently are up to three pieces
142 of information returned. The first index in
143 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
144 side. The second index in ERR_BNDS_NORM(:,err) contains
145 the following three fields: err = 1 "Trust/don't trust"
146 boolean. Trust the answer if the reciprocal condition
147 number is less than the threshold sqrt(n) *
148 dlamch('Epsilon'). err = 2 "Guaranteed" error bound:
149 The estimated forward error, almost certainly within a
150 factor of 10 of the true error so long as the next entry
151 is greater than the threshold sqrt(n) *
152 dlamch('Epsilon'). This error bound should only be
153 trusted if the previous boolean is true. err = 3
154 Reciprocal condition number: Estimated normwise recipro‐
155 cal condition number. Compared with the threshold
156 sqrt(n) * dlamch('Epsilon') to determine if the error
157 estimate is "guaranteed". These reciprocal condition
158 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
159 some appropriately scaled matrix Z. Let Z = S*A, where
160 S scales each row by a power of the radix so all abso‐
161 lute row sums of Z are approximately 1. See Lapack
162 Working Note 165 for further details and extra cautions.
163 ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS,
165 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) * dlamch('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) * dlamch('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) * dlamch('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) DOUBLE PRECISION 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: 1.0D+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) DOUBLE PRECISION 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 ZPORFSX(1)