1DPOSVXX(1) LAPACK driver routine (version 3.2) DPOSVXX(1)
2
6 DPOSVXX - DPOSVXX use the Cholesky factorization A = U**T*U or A =
7 L*L**T to compute the solution to a double precision system of linear
8 equations A * X = B, where A is an N-by-N symmetric positive definite
9 matrix and X and B are N-by-NRHS matrices
10
12 SUBROUTINE DPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B,
13 LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
14 ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
15 WORK, IWORK, INFO )
16 IMPLICIT NONE
18 CHARACTER EQUED, FACT, UPLO
20 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
22 N_ERR_BNDS
23 DOUBLE PRECISION RCOND, RPVGRW
25 INTEGER IWORK( * )
27 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
29 X( LDX, * ), WORK( * )
30 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
32 ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * )
33
35 DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
36 to compute the solution to a double precision system of linear equa‐
37 tions
38 A * X = B, where A is an N-by-N symmetric positive definite matrix
39 and X and B are N-by-NRHS matrices.
40 If requested, both normwise and maximum componentwise error bounds
41 are returned. DPOSVXX will return a solution with a tiny
42 guaranteed error (O(eps) where eps is the working machine
43 precision) unless the matrix is very ill-conditioned, in which
44 case a warning is returned. Relevant condition numbers also are
45 calculated and returned.
46 DPOSVXX accepts user-provided factorizations and equilibration
47 factors; see the definitions of the FACT and EQUED options.
48 Solving with refinement and using a factorization from a previous
49 DPOSVXX call will also produce a solution with either O(eps)
50 errors or warnings, but we cannot make that claim for general
51 user-provided factorizations and equilibration factors if they
52 differ from what DPOSVXX would itself produce.
53
55 The following steps are performed:
56 1. If FACT = 'E', double precision scaling factors are computed to
57 equilibrate
58 the system:
59 diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
60 Whether or not the system will be equilibrated depends on the
61 scaling of the matrix A, but if equilibration is used, A is
62 overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
63 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
64 factor the matrix A (after equilibration if FACT = 'E') as
65 A = U**T* U, if UPLO = 'U', or
66 A = L * L**T, if UPLO = 'L',
67 where U is an upper triangular matrix and L is a lower triangular
68 matrix.
69 3. If the leading i-by-i principal minor is not positive definite,
70 then the routine returns with INFO = i. Otherwise, the factored
71 form of A is used to estimate the condition number of the matrix
72 A (see argument RCOND). If the reciprocal of the condition number
73 is less than machine precision, the routine still goes on to solve
74 for X and compute error bounds as described below.
75 4. The system of equations is solved for X using the factored form
76 of A.
77 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
78 the routine will use iterative refinement to try to get a small
79 error and error bounds. Refinement calculates the residual to at
80 least twice the working precision.
81 6. If equilibration was used, the matrix X is premultiplied by
82 diag(S) so that it solves the original system before
83 equilibration.
84
86 Some optional parameters are bundled in the PARAMS array. These set‐
87 tings determine how refinement is performed, but often the defaults are
88 acceptable. If the defaults are acceptable, users can pass NPARAMS = 0
89 which prevents the source code from accessing the PARAMS argument.
90 FACT (input) CHARACTER*1
92 Specifies whether or not the factored form of the matrix A is
93 supplied on entry, and if not, whether the matrix A should be
94 equilibrated before it is factored. = 'F': On entry, AF con‐
95 tains the factored form of A. If EQUED is not 'N', the matrix
96 A has been equilibrated with scaling factors given by S. A and
97 AF are not modified. = 'N': The matrix A will be copied to AF
98 and factored.
99 = 'E': The matrix A will be equilibrated if necessary, then
100 copied to AF and factored.
101 UPLO (input) CHARACTER*1
103 = 'U': Upper triangle of A is stored;
104 = 'L': Lower triangle of A is stored.
105 N (input) INTEGER
107 The number of linear equations, i.e., the order of the matrix
108 A. N >= 0.
109 NRHS (input) INTEGER
111 The number of right hand sides, i.e., the number of columns of
112 the matrices B and X. NRHS >= 0.
113 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
115 On entry, the symmetric matrix A, except if FACT = 'F' and
116 EQUED = 'Y', then A must contain the equilibrated matrix
117 diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
118 triangular part of A contains the upper triangular part of the
119 matrix A, and the strictly lower triangular part of A is not
120 referenced. If UPLO = 'L', the leading N-by-N lower triangular
121 part of A contains the lower triangular part of the matrix A,
122 and the strictly upper triangular part of A is not referenced.
123 A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and
124 EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A
125 is overwritten by diag(S)*A*diag(S).
126 LDA (input) INTEGER
128 The leading dimension of the array A. LDA >= max(1,N).
129 AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
131 If FACT = 'F', then AF is an input argument and on entry con‐
132 tains the triangular factor U or L from the Cholesky factoriza‐
133 tion A = U**T*U or A = L*L**T, in the same storage format as A.
134 If EQUED .ne. 'N', then AF is the factored form of the equili‐
135 brated matrix diag(S)*A*diag(S). If FACT = 'N', then AF is an
136 output argument and on exit returns the triangular factor U or
137 L from the Cholesky factorization A = U**T*U or A = L*L**T of
138 the original matrix A. If FACT = 'E', then AF is an output
139 argument and on exit returns the triangular factor U or L from
140 the Cholesky factorization A = U**T*U or A = L*L**T of the
141 equilibrated matrix A (see the description of A for the form of
142 the equilibrated matrix).
143 LDAF (input) INTEGER
145 The leading dimension of the array AF. LDAF >= max(1,N).
146 EQUED (input or output) CHARACTER*1
148 Specifies the form of equilibration that was done. = 'N': No
149 equilibration (always true if FACT = 'N').
150 = 'Y': Both row and column equilibration, i.e., A has been
151 replaced by diag(S) * A * diag(S). EQUED is an input argument
152 if FACT = 'F'; otherwise, it is an output argument.
153 S (input or output) DOUBLE PRECISION array, dimension (N)
155 The row scale factors for A. If EQUED = 'Y', A is multiplied
156 on the left and right by diag(S). S is an input argument if
157 FACT = 'F'; otherwise, S is an output argument. If FACT = 'F'
158 and EQUED = 'Y', each element of S must be positive. If S is
159 output, each element of S is a power of the radix. If S is
160 input, each element of S should be a power of the radix to
161 ensure a reliable solution and error estimates. Scaling by pow‐
162 ers of the radix does not cause rounding errors unless the
163 result underflows or overflows. Rounding errors during scaling
164 lead to refining with a matrix that is not equivalent to the
165 input matrix, producing error estimates that may not be reli‐
166 able.
167 B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
169 On entry, the N-by-NRHS right hand side matrix B. On exit, if
170 EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwrit‐
171 ten by diag(S)*B;
172 LDB (input) INTEGER
174 The leading dimension of the array B. LDB >= max(1,N).
175 X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
177 If INFO = 0, the N-by-NRHS solution matrix X to the original
178 system of equations. Note that A and B are modified on exit if
179 EQUED .ne. 'N', and the solution to the equilibrated system is
180 inv(diag(S))*X.
181 LDX (input) INTEGER
183 The leading dimension of the array X. LDX >= max(1,N).
184 RCOND (output) DOUBLE PRECISION
186 Reciprocal scaled condition number. This is an estimate of the
187 reciprocal Skeel condition number of the matrix A after equili‐
188 bration (if done). If this is less than the machine precision
189 (in particular, if it is zero), the matrix is singular to work‐
190 ing precision. Note that the error may still be small even if
191 this number is very small and the matrix appears ill- condi‐
192 tioned.
193 RPVGRW (output) DOUBLE PRECISION
195 Reciprocal pivot growth. On exit, this contains the reciprocal
196 pivot growth factor norm(A)/norm(U). The "max absolute element"
197 norm is used. If this is much less than 1, then the stability
198 of the LU factorization of the (equilibrated) matrix A could be
199 poor. This also means that the solution X, estimated condition
200 numbers, and error bounds could be unreliable. If factorization
201 fails with 0<INFO<=N, then this contains the reciprocal pivot
202 growth factor for the leading INFO columns of A.
203 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
205 Componentwise relative backward error. This is the component‐
206 wise relative backward error of each solution vector X(j)
207 (i.e., the smallest relative change in any element of A or B
208 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
209 Number of error bounds to return for each right hand side and
210 each type (normwise or componentwise). See ERR_BNDS_NORM and
211 ERR_BNDS_COMP below.
212 ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS,
214 N_ERR_BNDS)
215 For each right-hand side, this array contains informa‐
216 tion about various error bounds and condition numbers
217 corresponding to the normwise relative error, which is
218 defined as follows: Normwise relative error in the ith
219 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
220 ------------------------------ max_j abs(X(j,i)) The
221 array is indexed by the type of error information as
222 described below. There currently are up to three pieces
223 of information returned. The first index in
224 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
225 side. The second index in ERR_BNDS_NORM(:,err) contains
226 the following three fields: err = 1 "Trust/don't trust"
227 boolean. Trust the answer if the reciprocal condition
228 number is less than the threshold sqrt(n) *
229 dlamch('Epsilon'). err = 2 "Guaranteed" error bound:
230 The estimated forward error, almost certainly within a
231 factor of 10 of the true error so long as the next entry
232 is greater than the threshold sqrt(n) *
233 dlamch('Epsilon'). This error bound should only be
234 trusted if the previous boolean is true. err = 3
235 Reciprocal condition number: Estimated normwise recipro‐
236 cal condition number. Compared with the threshold
237 sqrt(n) * dlamch('Epsilon') to determine if the error
238 estimate is "guaranteed". These reciprocal condition
239 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
240 some appropriately scaled matrix Z. Let Z = S*A, where
241 S scales each row by a power of the radix so all abso‐
242 lute row sums of Z are approximately 1. See Lapack
243 Working Note 165 for further details and extra cautions.
244 ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS,
246 N_ERR_BNDS)
247 For each right-hand side, this array contains informa‐
248 tion about various error bounds and condition numbers
249 corresponding to the componentwise relative error, which
250 is defined as follows: Componentwise relative error in
251 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
252 ---------------------- abs(X(j,i)) The array is indexed
253 by the right-hand side i (on which the componentwise
254 relative error depends), and the type of error informa‐
255 tion as described below. There currently are up to three
256 pieces of information returned for each right-hand side.
257 If componentwise accuracy is not requested (PARAMS(3) =
258 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
259 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
260 are returned. The first index in ERR_BNDS_COMP(i,:)
261 corresponds to the ith right-hand side. The second
262 index in ERR_BNDS_COMP(:,err) contains the following
263 three fields: err = 1 "Trust/don't trust" boolean. Trust
264 the answer if the reciprocal condition number is less
265 than the threshold sqrt(n) * dlamch('Epsilon'). err = 2
266 "Guaranteed" error bound: The estimated forward error,
267 almost certainly within a factor of 10 of the true error
268 so long as the next entry is greater than the threshold
269 sqrt(n) * dlamch('Epsilon'). This error bound should
270 only be trusted if the previous boolean is true. err =
271 3 Reciprocal condition number: Estimated componentwise
272 reciprocal condition number. Compared with the thresh‐
273 old sqrt(n) * dlamch('Epsilon') to determine if the
274 error estimate is "guaranteed". These reciprocal condi‐
275 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
276 for some appropriately scaled matrix Z. Let Z =
277 S*(A*diag(x)), where x is the solution for the current
278 right-hand side and S scales each row of A*diag(x) by a
279 power of the radix so all absolute row sums of Z are
280 approximately 1. See Lapack Working Note 165 for fur‐
281 ther details and extra cautions. NPARAMS (input) INTE‐
282 GER Specifies the number of parameters set in PARAMS.
283 If .LE. 0, the PARAMS array is never referenced and
284 default values are used.
285 PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
287 Specifies algorithm parameters. If an entry is .LT. 0.0, then
288 that entry will be filled with default value used for that
289 parameter. Only positions up to NPARAMS are accessed; defaults
290 are used for higher-numbered parameters.
291 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
292 refinement or not. Default: 1.0D+0
293 = 0.0 : No refinement is performed, and no error bounds are
294 computed. = 1.0 : Use the extra-precise refinement algorithm.
295 (other values are reserved for future use)
296 PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
297 computations allowed for refinement. Default: 10
298 Aggressive: Set to 100 to permit convergence using approximate
299 factorizations or factorizations other than LU. If the factor‐
300 ization uses a technique other than Gaussian elimination, the
301 guarantees in err_bnds_norm and err_bnds_comp may no longer be
302 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
303 if the code will attempt to find a solution with small compo‐
304 nentwise relative error in the double-precision algorithm.
305 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
306 nentwise convergence)
307 WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
309 IWORK (workspace) INTEGER array, dimension (N)
311 INFO (output) INTEGER
313 = 0: Successful exit. The solution to every right-hand side is
314 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
315 gal value
316 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
317 has been completed, but the factor U is exactly singular, so
318 the solution and error bounds could not be computed. RCOND = 0
319 is returned. = N+J: The solution corresponding to the Jth
320 right-hand side is not guaranteed. The solutions corresponding
321 to other right- hand sides K with K > J may not be guaranteed
322 as well, but only the first such right-hand side is reported.
323 If a small componentwise error is not requested (PARAMS(3) =
324 0.0) then the Jth right-hand side is the first with a normwise
325 error bound that is not guaranteed (the smallest J such that
326 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
327 right-hand side is the first with either a normwise or compo‐
328 nentwise error bound that is not guaranteed (the smallest J
329 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
330 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
331 ERR_BNDS_COMP(:,1). To get information about all of the right-
332 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
333 LAPACK driver routine (versionNo3v.e2m)ber 2008 DPOSVXX(1)