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