1ZHESVXX(1) LAPACK driver routine (version 3.2) ZHESVXX(1)
2
6 ZHESVXX - ZHESVXX use the diagonal pivoting factorization to compute
7 the solution to a complex*16 system of linear equations A * X = B,
8 where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
9 matrices
10
12 SUBROUTINE ZHESVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED,
13 S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS,
14 ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS,
15 WORK, RWORK, 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 IPIV( * )
27 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), WORK( * ),
29 X( LDX, * )
30 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( *
32 ), ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, *
33 )
34
36 ZHESVXX uses the diagonal pivoting factorization to compute the
37 solution to a complex*16 system of linear equations A * X = B, where
38 A is an N-by-N symmetric matrix and X and B are N-by-NRHS
39 matrices.
40 If requested, both normwise and maximum componentwise error bounds
41 are returned. ZHESVXX 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 ZHESVXX 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 ZHESVXX 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 ZHESVXX 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 LU decomposition is used to factor
64 the matrix A (after equilibration if FACT = 'E') as
65 A = U * D * U**T, if UPLO = 'U', or
66 A = L * D * L**T, if UPLO = 'L',
67 where U (or L) is a product of permutation and unit upper (lower)
68 triangular matrices, and D is symmetric and block diagonal with
69 1-by-1 and 2-by-2 diagonal blocks.
70 3. If some D(i,i)=0, so that D is exactly singular, then the
71 routine returns with INFO = i. Otherwise, the factored form of A
72 is used to estimate the condition number of the matrix A (see
73 argument RCOND). If the reciprocal of the condition number is
74 less than machine precision, the routine still goes on to solve
75 for X and compute error bounds as described below.
76 4. The system of equations is solved for X using the factored form
77 of A.
78 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
79 the routine will use iterative refinement to try to get a small
80 error and error bounds. Refinement calculates the residual to at
81 least twice the working precision.
82 6. If equilibration was used, the matrix X is premultiplied by
83 diag(R) so that it solves the original system before
84 equilibration.
85
87 Some optional parameters are bundled in the PARAMS array. These set‐
88 tings determine how refinement is performed, but often the defaults are
89 acceptable. If the defaults are acceptable, users can pass NPARAMS = 0
90 which prevents the source code from accessing the PARAMS argument.
91 FACT (input) CHARACTER*1
93 Specifies whether or not the factored form of the matrix A is
94 supplied on entry, and if not, whether the matrix A should be
95 equilibrated before it is factored. = 'F': On entry, AF and
96 IPIV contain the factored form of A. If EQUED is not 'N', the
97 matrix A has been equilibrated with scaling factors given by S.
98 A, AF, and IPIV are not modified. = 'N': The matrix A will be
99 copied to AF and factored.
100 = 'E': The matrix A will be equilibrated if necessary, then
101 copied to AF and factored.
102 N (input) INTEGER
104 The number of linear equations, i.e., the order of the matrix
105 A. N >= 0.
106 NRHS (input) INTEGER
108 The number of right hand sides, i.e., the number of columns of
109 the matrices B and X. NRHS >= 0.
110 A (input/output) COMPLEX*16 array, dimension (LDA,N)
112 The symmetric matrix A. If UPLO = 'U', the leading N-by-N
113 upper triangular part of A contains the upper triangular part
114 of the matrix A, and the strictly lower triangular part of A is
115 not referenced. If UPLO = 'L', the leading N-by-N lower trian‐
116 gular part of A contains the lower triangular part of the
117 matrix A, and the strictly upper triangular part of A is not
118 referenced. On exit, if FACT = 'E' and EQUED = 'Y', A is over‐
119 written by diag(S)*A*diag(S).
120 LDA (input) INTEGER
122 The leading dimension of the array A. LDA >= max(1,N).
123 AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
125 If FACT = 'F', then AF is an input argument and on entry con‐
126 tains the block diagonal matrix D and the multipliers used to
127 obtain the factor U or L from the factorization A = U*D*U**T or
128 A = L*D*L**T as computed by DSYTRF. If FACT = 'N', then AF is
129 an output argument and on exit returns the block diagonal
130 matrix D and the multipliers used to obtain the factor U or L
131 from the factorization A = U*D*U**T or A = L*D*L**T.
132 LDAF (input) INTEGER
134 The leading dimension of the array AF. LDAF >= max(1,N).
135 IPIV (input or output) INTEGER array, dimension (N)
137 If FACT = 'F', then IPIV is an input argument and on entry con‐
138 tains details of the interchanges and the block structure of D,
139 as determined by DSYTRF. If IPIV(k) > 0, then rows and columns
140 k and IPIV(k) were interchanged and D(k,k) is a 1-by-1 diagonal
141 block. If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows
142 and columns k-1 and -IPIV(k) were interchanged and
143 D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = 'L' and
144 IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k)
145 were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal
146 block. If FACT = 'N', then IPIV is an output argument and on
147 exit contains details of the interchanges and the block struc‐
148 ture of D, as determined by DSYTRF.
149 EQUED (input or output) CHARACTER*1
151 Specifies the form of equilibration that was done. = 'N': No
152 equilibration (always true if FACT = 'N').
153 = 'Y': Both row and column equilibration, i.e., A has been
154 replaced by diag(S) * A * diag(S). EQUED is an input argument
155 if FACT = 'F'; otherwise, it is an output argument.
156 S (input or output) DOUBLE PRECISION array, dimension (N)
158 The scale factors for A. If EQUED = 'Y', A is multiplied on
159 the left and right by diag(S). S is an input argument if FACT
160 = 'F'; otherwise, S is an output argument. If FACT = 'F' and
161 EQUED = 'Y', each element of S must be positive. If S is out‐
162 put, each element of S is a power of the radix. If S is input,
163 each element of S should be a power of the radix to ensure a
164 reliable solution and error estimates. Scaling by powers of the
165 radix does not cause rounding errors unless the result under‐
166 flows or overflows. Rounding errors during scaling lead to
167 refining with a matrix that is not equivalent to the input
168 matrix, producing error estimates that may not be reliable.
169 B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
171 On entry, the N-by-NRHS right hand side matrix B. On exit, if
172 EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwrit‐
173 ten by diag(S)*B;
174 LDB (input) INTEGER
176 The leading dimension of the array B. LDB >= max(1,N).
177 X (output) COMPLEX*16 array, dimension (LDX,NRHS)
179 If INFO = 0, the N-by-NRHS solution matrix X to the original
180 system of equations. Note that A and B are modified on exit if
181 EQUED .ne. 'N', and the solution to the equilibrated system is
182 inv(diag(S))*X.
183 LDX (input) INTEGER
185 The leading dimension of the array X. LDX >= max(1,N).
186 RCOND (output) DOUBLE PRECISION
188 Reciprocal scaled condition number. This is an estimate of the
189 reciprocal Skeel condition number of the matrix A after equili‐
190 bration (if done). If this is less than the machine precision
191 (in particular, if it is zero), the matrix is singular to work‐
192 ing precision. Note that the error may still be small even if
193 this number is very small and the matrix appears ill- condi‐
194 tioned.
195 RPVGRW (output) DOUBLE PRECISION
197 Reciprocal pivot growth. On exit, this contains the reciprocal
198 pivot growth factor norm(A)/norm(U). The "max absolute element"
199 norm is used. If this is much less than 1, then the stability
200 of the LU factorization of the (equilibrated) matrix A could be
201 poor. This also means that the solution X, estimated condition
202 numbers, and error bounds could be unreliable. If factorization
203 fails with 0<INFO<=N, then this contains the reciprocal pivot
204 growth factor for the leading INFO columns of A.
205 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
207 Componentwise relative backward error. This is the component‐
208 wise relative backward error of each solution vector X(j)
209 (i.e., the smallest relative change in any element of A or B
210 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
211 Number of error bounds to return for each right hand side and
212 each type (normwise or componentwise). See ERR_BNDS_NORM and
213 ERR_BNDS_COMP below.
214 ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS,
216 N_ERR_BNDS)
217 For each right-hand side, this array contains informa‐
218 tion about various error bounds and condition numbers
219 corresponding to the normwise relative error, which is
220 defined as follows: Normwise relative error in the ith
221 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
222 ------------------------------ max_j abs(X(j,i)) The
223 array is indexed by the type of error information as
224 described below. There currently are up to three pieces
225 of information returned. The first index in
226 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
227 side. The second index in ERR_BNDS_NORM(:,err) contains
228 the following three fields: err = 1 "Trust/don't trust"
229 boolean. Trust the answer if the reciprocal condition
230 number is less than the threshold sqrt(n) *
231 dlamch('Epsilon'). err = 2 "Guaranteed" error bound:
232 The estimated forward error, almost certainly within a
233 factor of 10 of the true error so long as the next entry
234 is greater than the threshold sqrt(n) *
235 dlamch('Epsilon'). This error bound should only be
236 trusted if the previous boolean is true. err = 3
237 Reciprocal condition number: Estimated normwise recipro‐
238 cal condition number. Compared with the threshold
239 sqrt(n) * dlamch('Epsilon') to determine if the error
240 estimate is "guaranteed". These reciprocal condition
241 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
242 some appropriately scaled matrix Z. Let Z = S*A, where
243 S scales each row by a power of the radix so all abso‐
244 lute row sums of Z are approximately 1. See Lapack
245 Working Note 165 for further details and extra cautions.
246 ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS,
248 N_ERR_BNDS)
249 For each right-hand side, this array contains informa‐
250 tion about various error bounds and condition numbers
251 corresponding to the componentwise relative error, which
252 is defined as follows: Componentwise relative error in
253 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
254 ---------------------- abs(X(j,i)) The array is indexed
255 by the right-hand side i (on which the componentwise
256 relative error depends), and the type of error informa‐
257 tion as described below. There currently are up to three
258 pieces of information returned for each right-hand side.
259 If componentwise accuracy is not requested (PARAMS(3) =
260 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
261 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
262 are returned. The first index in ERR_BNDS_COMP(i,:)
263 corresponds to the ith right-hand side. The second
264 index in ERR_BNDS_COMP(:,err) contains the following
265 three fields: err = 1 "Trust/don't trust" boolean. Trust
266 the answer if the reciprocal condition number is less
267 than the threshold sqrt(n) * dlamch('Epsilon'). err = 2
268 "Guaranteed" error bound: The estimated forward error,
269 almost certainly within a factor of 10 of the true error
270 so long as the next entry is greater than the threshold
271 sqrt(n) * dlamch('Epsilon'). This error bound should
272 only be trusted if the previous boolean is true. err =
273 3 Reciprocal condition number: Estimated componentwise
274 reciprocal condition number. Compared with the thresh‐
275 old sqrt(n) * dlamch('Epsilon') to determine if the
276 error estimate is "guaranteed". These reciprocal condi‐
277 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
278 for some appropriately scaled matrix Z. Let Z =
279 S*(A*diag(x)), where x is the solution for the current
280 right-hand side and S scales each row of A*diag(x) by a
281 power of the radix so all absolute row sums of Z are
282 approximately 1. See Lapack Working Note 165 for fur‐
283 ther details and extra cautions. NPARAMS (input) INTE‐
284 GER Specifies the number of parameters set in PARAMS.
285 If .LE. 0, the PARAMS array is never referenced and
286 default values are used.
287 PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
289 Specifies algorithm parameters. If an entry is .LT. 0.0, then
290 that entry will be filled with default value used for that
291 parameter. Only positions up to NPARAMS are accessed; defaults
292 are used for higher-numbered parameters.
293 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
294 refinement or not. Default: 1.0D+0
295 = 0.0 : No refinement is performed, and no error bounds are
296 computed. = 1.0 : Use the extra-precise refinement algorithm.
297 (other values are reserved for future use)
298 PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
299 computations allowed for refinement. Default: 10
300 Aggressive: Set to 100 to permit convergence using approximate
301 factorizations or factorizations other than LU. If the factor‐
302 ization uses a technique other than Gaussian elimination, the
303 guarantees in err_bnds_norm and err_bnds_comp may no longer be
304 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
305 if the code will attempt to find a solution with small compo‐
306 nentwise relative error in the double-precision algorithm.
307 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
308 nentwise convergence)
309 WORK (workspace) COMPLEX*16 array, dimension (2*N)
311 RWORK (workspace) DOUBLE PRECISION array, dimension (3*N)
313 INFO (output) INTEGER
315 = 0: Successful exit. The solution to every right-hand side is
316 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
317 gal value
318 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
319 has been completed, but the factor U is exactly singular, so
320 the solution and error bounds could not be computed. RCOND = 0
321 is returned. = N+J: The solution corresponding to the Jth
322 right-hand side is not guaranteed. The solutions corresponding
323 to other right- hand sides K with K > J may not be guaranteed
324 as well, but only the first such right-hand side is reported.
325 If a small componentwise error is not requested (PARAMS(3) =
326 0.0) then the Jth right-hand side is the first with a normwise
327 error bound that is not guaranteed (the smallest J such that
328 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
329 right-hand side is the first with either a normwise or compo‐
330 nentwise error bound that is not guaranteed (the smallest J
331 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
332 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
333 ERR_BNDS_COMP(:,1). To get information about all of the right-
334 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
335 LAPACK driver routine (versionNo3v.e2m)ber 2008 ZHESVXX(1)