1CGBSVXX(1) LAPACK driver routine (version 3.2) CGBSVXX(1)
2
6 CGBSVXX - CGBSVXX use the LU factorization to compute the solution to a
7 complex system of linear equations A * X = B, where A is an N-by-N
8 matrix and X and B are N-by-NRHS matrices
9
11 SUBROUTINE CGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
12 IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, RPVGRW,
13 BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
14 NPARAMS, PARAMS, WORK, RWORK, INFO )
15 IMPLICIT NONE
17 CHARACTER EQUED, FACT, TRANS
19 INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS,
21 N_ERR_BNDS
22 REAL RCOND, RPVGRW
24 INTEGER IPIV( * )
26 COMPLEX AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), X( LDX
28 , * ),WORK( * )
29 REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
31 ERR_BNDS_NORM( NRHS, * ), ERR_BNDS_COMP( NRHS, * ),
32 RWORK( * )
33
35 CGBSVXX uses the LU factorization to compute the solution to a
36 complex system of linear equations A * X = B, where A is an
37 N-by-N matrix and X and B are N-by-NRHS matrices.
38 If requested, both normwise and maximum componentwise error bounds
39 are returned. CGBSVXX will return a solution with a tiny
40 guaranteed error (O(eps) where eps is the working machine
41 precision) unless the matrix is very ill-conditioned, in which
42 case a warning is returned. Relevant condition numbers also are
43 calculated and returned.
44 CGBSVXX accepts user-provided factorizations and equilibration
45 factors; see the definitions of the FACT and EQUED options.
46 Solving with refinement and using a factorization from a previous
47 CGBSVXX call will also produce a solution with either O(eps)
48 errors or warnings, but we cannot make that claim for general
49 user-provided factorizations and equilibration factors if they
50 differ from what CGBSVXX would itself produce.
51
53 The following steps are performed:
54 1. If FACT = 'E', real scaling factors are computed to equilibrate
55 the system:
56 TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
57 TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
58 TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
59 Whether or not the system will be equilibrated depends on the
60 scaling of the matrix A, but if equilibration is used, A is
61 overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
62 or diag(C)*B (if TRANS = 'T' or 'C').
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 = P * L * U,
66 where P is a permutation matrix, L is a unit lower triangular
67 matrix, and U is upper triangular.
68 3. If some U(i,i)=0, so that U is exactly singular, then the
69 routine returns with INFO = i. Otherwise, the factored form of A
70 is used to estimate the condition number of the matrix A (see
71 argument RCOND). If the reciprocal of the condition number is less
72 than machine precision, the routine still goes on to solve for X
73 and compute error bounds as described below.
74 4. The system of equations is solved for X using the factored form
75 of A.
76 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
77 the routine will use iterative refinement to try to get a small
78 error and error bounds. Refinement calculates the residual to at
79 least twice the working precision.
80 6. If equilibration was used, the matrix X is premultiplied by
81 diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
82 that it solves the original system before equilibration.
83
85 Some optional parameters are bundled in the PARAMS array. These set‐
86 tings determine how refinement is performed, but often the defaults are
87 acceptable. If the defaults are acceptable, users can pass NPARAMS = 0
88 which prevents the source code from accessing the PARAMS argument.
89 FACT (input) CHARACTER*1
91 Specifies whether or not the factored form of the matrix A is
92 supplied on entry, and if not, whether the matrix A should be
93 equilibrated before it is factored. = 'F': On entry, AF and
94 IPIV contain the factored form of A. If EQUED is not 'N', the
95 matrix A has been equilibrated with scaling factors given by R
96 and C. A, AF, and IPIV are not modified. = 'N': The matrix A
97 will be copied to AF and factored.
98 = 'E': The matrix A will be equilibrated if necessary, then
99 copied to AF and factored.
100 TRANS (input) CHARACTER*1
102 Specifies the form of the system of equations:
103 = 'N': A * X = B (No transpose)
104 = 'T': A**T * X = B (Transpose)
105 = 'C': A**H * X = B (Conjugate Transpose = Transpose)
106 N (input) INTEGER
108 The number of linear equations, i.e., the order of the matrix
109 A. N >= 0.
110 KL (input) INTEGER
112 The number of subdiagonals within the band of A. KL >= 0.
113 KU (input) INTEGER
115 The number of superdiagonals within the band of A. KU >= 0.
116 NRHS (input) INTEGER
118 The number of right hand sides, i.e., the number of columns of
119 the matrices B and X. NRHS >= 0.
120 AB (input/output) REAL array, dimension (LDAB,N)
122 On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
123 The j-th column of A is stored in the j-th column of the array
124 AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-
125 KU)<=i<=min(N,j+kl) If FACT = 'F' and EQUED is not 'N', then AB
126 must have been equilibrated by the scaling factors in R and/or
127 C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E'
128 and EQUED = 'N' on exit. On exit, if EQUED .ne. 'N', A is
129 scaled as follows: EQUED = 'R': A := diag(R) * A
130 EQUED = 'C': A := A * diag(C)
131 EQUED = 'B': A := diag(R) * A * diag(C).
132 LDAB (input) INTEGER
134 The leading dimension of the array AB. LDAB >= KL+KU+1.
135 AFB (input or output) REAL array, dimension (LDAFB,N)
137 If FACT = 'F', then AFB is an input argument and on entry con‐
138 tains details of the LU factorization of the band matrix A, as
139 computed by CGBTRF. U is stored as an upper triangular band
140 matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
141 multipliers used during the factorization are stored in rows
142 KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is the fac‐
143 tored form of the equilibrated matrix A. If FACT = 'N', then
144 AF is an output argument and on exit returns the factors L and
145 U from the factorization A = P*L*U of the original matrix A.
146 If FACT = 'E', then AF is an output argument and on exit
147 returns the factors L and U from the factorization A = P*L*U of
148 the equilibrated matrix A (see the description of A for the
149 form of the equilibrated matrix).
150 LDAFB (input) INTEGER
152 The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
153 IPIV (input or output) INTEGER array, dimension (N)
155 If FACT = 'F', then IPIV is an input argument and on entry con‐
156 tains the pivot indices from the factorization A = P*L*U as
157 computed by SGETRF; row i of the matrix was interchanged with
158 row IPIV(i). If FACT = 'N', then IPIV is an output argument
159 and on exit contains the pivot indices from the factorization A
160 = P*L*U of the original matrix A. If FACT = 'E', then IPIV is
161 an output argument and on exit contains the pivot indices from
162 the factorization A = P*L*U of the equilibrated matrix A.
163 EQUED (input or output) CHARACTER*1
165 Specifies the form of equilibration that was done. = 'N': No
166 equilibration (always true if FACT = 'N').
167 = 'R': Row equilibration, i.e., A has been premultiplied by
168 diag(R). = 'C': Column equilibration, i.e., A has been post‐
169 multiplied by diag(C). = 'B': Both row and column equilibra‐
170 tion, i.e., A has been replaced by diag(R) * A * diag(C).
171 EQUED is an input argument if FACT = 'F'; otherwise, it is an
172 output argument.
173 R (input or output) REAL array, dimension (N)
175 The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
176 tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
177 accessed. R is an input argument if FACT = 'F'; otherwise, R
178 is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
179 each element of R must be positive. If R is output, each ele‐
180 ment of R is a power of the radix. If R is input, each element
181 of R should be a power of the radix to ensure a reliable solu‐
182 tion and error estimates. Scaling by powers of the radix does
183 not cause rounding errors unless the result underflows or over‐
184 flows. Rounding errors during scaling lead to refining with a
185 matrix that is not equivalent to the input matrix, producing
186 error estimates that may not be reliable.
187 C (input or output) REAL array, dimension (N)
189 The column scale factors for A. If EQUED = 'C' or 'B', A is
190 multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
191 not accessed. C is an input argument if FACT = 'F'; otherwise,
192 C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
193 each element of C must be positive. If C is output, each ele‐
194 ment of C is a power of the radix. If C is input, each element
195 of C should be a power of the radix to ensure a reliable solu‐
196 tion and error estimates. Scaling by powers of the radix does
197 not cause rounding errors unless the result underflows or over‐
198 flows. Rounding errors during scaling lead to refining with a
199 matrix that is not equivalent to the input matrix, producing
200 error estimates that may not be reliable.
201 B (input/output) REAL array, dimension (LDB,NRHS)
203 On entry, the N-by-NRHS right hand side matrix B. On exit, if
204 EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R'
205 or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C'
206 and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.
207 LDB (input) INTEGER
209 The leading dimension of the array B. LDB >= max(1,N).
210 X (output) REAL array, dimension (LDX,NRHS)
212 If INFO = 0, the N-by-NRHS solution matrix X to the original
213 system of equations. Note that A and B are modified on exit if
214 EQUED .ne. 'N', and the solution to the equilibrated system is
215 inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or
216 inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
217 LDX (input) INTEGER
219 The leading dimension of the array X. LDX >= max(1,N).
220 RCOND (output) REAL
222 Reciprocal scaled condition number. This is an estimate of the
223 reciprocal Skeel condition number of the matrix A after equili‐
224 bration (if done). If this is less than the machine precision
225 (in particular, if it is zero), the matrix is singular to work‐
226 ing precision. Note that the error may still be small even if
227 this number is very small and the matrix appears ill- condi‐
228 tioned.
229 RPVGRW (output) REAL
231 Reciprocal pivot growth. On exit, this contains the reciprocal
232 pivot growth factor norm(A)/norm(U). The "max absolute element"
233 norm is used. If this is much less than 1, then the stability
234 of the LU factorization of the (equilibrated) matrix A could be
235 poor. This also means that the solution X, estimated condition
236 numbers, and error bounds could be unreliable. If factorization
237 fails with 0<INFO<=N, then this contains the reciprocal pivot
238 growth factor for the leading INFO columns of A. In SGESVX,
239 this quantity is returned in WORK(1).
240 BERR (output) REAL array, dimension (NRHS)
242 Componentwise relative backward error. This is the component‐
243 wise relative backward error of each solution vector X(j)
244 (i.e., the smallest relative change in any element of A or B
245 that makes X(j) an exact solution). N_ERR_BNDS (input) INTEGER
246 Number of error bounds to return for each right hand side and
247 each type (normwise or componentwise). See ERR_BNDS_NORM and
248 ERR_BNDS_COMP below.
249 ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS)
251 For each right-hand side, this array contains informa‐
252 tion about various error bounds and condition numbers
253 corresponding to the normwise relative error, which is
254 defined as follows: Normwise relative error in the ith
255 solution vector: max_j (abs(XTRUE(j,i) - X(j,i)))
256 ------------------------------ max_j abs(X(j,i)) The
257 array is indexed by the type of error information as
258 described below. There currently are up to three pieces
259 of information returned. The first index in
260 ERR_BNDS_NORM(i,:) corresponds to the ith right-hand
261 side. The second index in ERR_BNDS_NORM(:,err) contains
262 the following three fields: err = 1 "Trust/don't trust"
263 boolean. Trust the answer if the reciprocal condition
264 number is less than the threshold sqrt(n) *
265 slamch('Epsilon'). err = 2 "Guaranteed" error bound:
266 The estimated forward error, almost certainly within a
267 factor of 10 of the true error so long as the next entry
268 is greater than the threshold sqrt(n) *
269 slamch('Epsilon'). This error bound should only be
270 trusted if the previous boolean is true. err = 3
271 Reciprocal condition number: Estimated normwise recipro‐
272 cal condition number. Compared with the threshold
273 sqrt(n) * slamch('Epsilon') to determine if the error
274 estimate is "guaranteed". These reciprocal condition
275 numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for
276 some appropriately scaled matrix Z. Let Z = S*A, where
277 S scales each row by a power of the radix so all abso‐
278 lute row sums of Z are approximately 1. See Lapack
279 Working Note 165 for further details and extra cautions.
280 ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)
282 For each right-hand side, this array contains informa‐
283 tion about various error bounds and condition numbers
284 corresponding to the componentwise relative error, which
285 is defined as follows: Componentwise relative error in
286 the ith solution vector: abs(XTRUE(j,i) - X(j,i)) max_j
287 ---------------------- abs(X(j,i)) The array is indexed
288 by the right-hand side i (on which the componentwise
289 relative error depends), and the type of error informa‐
290 tion as described below. There currently are up to three
291 pieces of information returned for each right-hand side.
292 If componentwise accuracy is not requested (PARAMS(3) =
293 0.0), then ERR_BNDS_COMP is not accessed. If N_ERR_BNDS
294 .LT. 3, then at most the first (:,N_ERR_BNDS) entries
295 are returned. The first index in ERR_BNDS_COMP(i,:)
296 corresponds to the ith right-hand side. The second
297 index in ERR_BNDS_COMP(:,err) contains the following
298 three fields: err = 1 "Trust/don't trust" boolean. Trust
299 the answer if the reciprocal condition number is less
300 than the threshold sqrt(n) * slamch('Epsilon'). err = 2
301 "Guaranteed" error bound: The estimated forward error,
302 almost certainly within a factor of 10 of the true error
303 so long as the next entry is greater than the threshold
304 sqrt(n) * slamch('Epsilon'). This error bound should
305 only be trusted if the previous boolean is true. err =
306 3 Reciprocal condition number: Estimated componentwise
307 reciprocal condition number. Compared with the thresh‐
308 old sqrt(n) * slamch('Epsilon') to determine if the
309 error estimate is "guaranteed". These reciprocal condi‐
310 tion numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf))
311 for some appropriately scaled matrix Z. Let Z =
312 S*(A*diag(x)), where x is the solution for the current
313 right-hand side and S scales each row of A*diag(x) by a
314 power of the radix so all absolute row sums of Z are
315 approximately 1. See Lapack Working Note 165 for fur‐
316 ther details and extra cautions. NPARAMS (input) INTE‐
317 GER Specifies the number of parameters set in PARAMS.
318 If .LE. 0, the PARAMS array is never referenced and
319 default values are used.
320 PARAMS (input / output) REAL array, dimension NPARAMS
322 Specifies algorithm parameters. If an entry is .LT. 0.0, then
323 that entry will be filled with default value used for that
324 parameter. Only positions up to NPARAMS are accessed; defaults
325 are used for higher-numbered parameters.
326 PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
327 refinement or not. Default: 1.0
328 = 0.0 : No refinement is performed, and no error bounds are
329 computed. = 1.0 : Use the double-precision refinement algo‐
330 rithm, possibly with doubled-single computations if the compi‐
331 lation environment does not support DOUBLE PRECISION. (other
332 values are reserved for future use) PARAMS(LA_LINRX_ITHRESH_I =
333 2) : Maximum number of residual computations allowed for
334 refinement. Default: 10
335 Aggressive: Set to 100 to permit convergence using approximate
336 factorizations or factorizations other than LU. If the factor‐
337 ization uses a technique other than Gaussian elimination, the
338 guarantees in err_bnds_norm and err_bnds_comp may no longer be
339 trustworthy. PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining
340 if the code will attempt to find a solution with small compo‐
341 nentwise relative error in the double-precision algorithm.
342 Positive is true, 0.0 is false. Default: 1.0 (attempt compo‐
343 nentwise convergence)
344 WORK (workspace) REAL array, dimension (4*N)
346 IWORK (workspace) INTEGER array, dimension (N)
348 INFO (output) INTEGER
350 = 0: Successful exit. The solution to every right-hand side is
351 guaranteed. < 0: If INFO = -i, the i-th argument had an ille‐
352 gal value
353 > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
354 has been completed, but the factor U is exactly singular, so
355 the solution and error bounds could not be computed. RCOND = 0
356 is returned. = N+J: The solution corresponding to the Jth
357 right-hand side is not guaranteed. The solutions corresponding
358 to other right- hand sides K with K > J may not be guaranteed
359 as well, but only the first such right-hand side is reported.
360 If a small componentwise error is not requested (PARAMS(3) =
361 0.0) then the Jth right-hand side is the first with a normwise
362 error bound that is not guaranteed (the smallest J such that
363 ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) the Jth
364 right-hand side is the first with either a normwise or compo‐
365 nentwise error bound that is not guaranteed (the smallest J
366 such that either ERR_BNDS_NORM(J,1) = 0.0 or ERR_BNDS_COMP(J,1)
367 = 0.0). See the definition of ERR_BNDS_NORM(:,1) and
368 ERR_BNDS_COMP(:,1). To get information about all of the right-
369 hand sides check ERR_BNDS_NORM or ERR_BNDS_COMP.
370 LAPACK driver routine (versionNo3v.e2m)ber 2008 CGBSVXX(1)