1DGBSVX(1) LAPACK driver routine (version 3.1) DGBSVX(1)
2
6 DGBSVX - the LU factorization to compute the solution to a real system
7 of linear equations A * X = B, A**T * X = B, or A**H * X = B,
8
10 SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
11 IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR,
12 BERR, WORK, IWORK, INFO )
13 CHARACTER EQUED, FACT, TRANS
15 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
17 DOUBLE PRECISION RCOND
19 INTEGER IPIV( * ), IWORK( * )
21 DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, *
23 ), BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ),
24 X( LDX, * )
25
27 DGBSVX uses the LU factorization to compute the solution to a real sys‐
28 tem of linear equations A * X = B, A**T * X = B, or A**H * X = B, where
29 A is a band matrix of order N with KL subdiagonals and KU superdiago‐
30 nals, and X and B are N-by-NRHS matrices.
31 Error bounds on the solution and a condition estimate are also pro‐
33 vided.
34
37 The following steps are performed by this subroutine:
38 1. If FACT = 'E', real scaling factors are computed to equilibrate
40 the system:
41 TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
42 TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
43 TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
44 Whether or not the system will be equilibrated depends on the
45 scaling of the matrix A, but if equilibration is used, A is
46 overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
47 or diag(C)*B (if TRANS = 'T' or 'C').
48 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
50 matrix A (after equilibration if FACT = 'E') as
51 A = L * U,
52 where L is a product of permutation and unit lower triangular
53 matrices with KL subdiagonals, and U is upper triangular with
54 KL+KU superdiagonals.
55 3. If some U(i,i)=0, so that U is exactly singular, then the routine
57 returns with INFO = i. Otherwise, the factored form of A is used
58 to estimate the condition number of the matrix A. If the
59 reciprocal of the condition number is less than machine precision,
60 INFO = N+1 is returned as a warning, but the routine still goes on
61 to solve for X and compute error bounds as described below.
62 4. The system of equations is solved for X using the factored form
64 of A.
65 5. Iterative refinement is applied to improve the computed solution
67 matrix and calculate error bounds and backward error estimates
68 for it.
69 6. If equilibration was used, the matrix X is premultiplied by
71 diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
72 that it solves the original system before equilibration.
73
76 FACT (input) CHARACTER*1
77 Specifies whether or not the factored form of the matrix A is
78 supplied on entry, and if not, whether the matrix A should be
79 equilibrated before it is factored. = 'F': On entry, AFB and
80 IPIV contain the factored form of A. If EQUED is not 'N', the
81 matrix A has been equilibrated with scaling factors given by R
82 and C. AB, AFB, and IPIV are not modified. = 'N': The matrix
83 A will be copied to AFB and factored.
84 = 'E': The matrix A will be equilibrated if necessary, then
85 copied to AFB and factored.
86 TRANS (input) CHARACTER*1
88 Specifies the form of the system of equations. = 'N': A * X =
89 B (No transpose)
90 = 'T': A**T * X = B (Transpose)
91 = 'C': A**H * X = B (Transpose)
92 N (input) INTEGER
94 The number of linear equations, i.e., the order of the matrix
95 A. N >= 0.
96 KL (input) INTEGER
98 The number of subdiagonals within the band of A. KL >= 0.
99 KU (input) INTEGER
101 The number of superdiagonals within the band of A. KU >= 0.
102 NRHS (input) INTEGER
104 The number of right hand sides, i.e., the number of columns of
105 the matrices B and X. NRHS >= 0.
106 AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
108 On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
109 The j-th column of A is stored in the j-th column of the array
110 AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-
111 KU)<=i<=min(N,j+kl)
112 If FACT = 'F' and EQUED is not 'N', then A must have been equi‐
114 librated by the scaling factors in R and/or C. AB is not modi‐
115 fied if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on
116 exit.
117 On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED =
119 'R': A := diag(R) * A
120 EQUED = 'C': A := A * diag(C)
121 EQUED = 'B': A := diag(R) * A * diag(C).
122 LDAB (input) INTEGER
124 The leading dimension of the array AB. LDAB >= KL+KU+1.
125 AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
127 If FACT = 'F', then AFB is an input argument and on entry con‐
128 tains details of the LU factorization of the band matrix A, as
129 computed by DGBTRF. U is stored as an upper triangular band
130 matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
131 multipliers used during the factorization are stored in rows
132 KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is the fac‐
133 tored form of the equilibrated matrix A.
134 If FACT = 'N', then AFB is an output argument and on exit
136 returns details of the LU factorization of A.
137 If FACT = 'E', then AFB is an output argument and on exit
139 returns details of the LU factorization of the equilibrated
140 matrix A (see the description of AB for the form of the equili‐
141 brated matrix).
142 LDAFB (input) INTEGER
144 The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
145 IPIV (input or output) INTEGER array, dimension (N)
147 If FACT = 'F', then IPIV is an input argument and on entry con‐
148 tains the pivot indices from the factorization A = L*U as com‐
149 puted by DGBTRF; row i of the matrix was interchanged with row
150 IPIV(i).
151 If FACT = 'N', then IPIV is an output argument and on exit con‐
153 tains the pivot indices from the factorization A = L*U of the
154 original matrix A.
155 If FACT = 'E', then IPIV is an output argument and on exit con‐
157 tains the pivot indices from the factorization A = L*U of the
158 equilibrated matrix A.
159 EQUED (input or output) CHARACTER*1
161 Specifies the form of equilibration that was done. = 'N': No
162 equilibration (always true if FACT = 'N').
163 = 'R': Row equilibration, i.e., A has been premultiplied by
164 diag(R). = 'C': Column equilibration, i.e., A has been post‐
165 multiplied by diag(C). = 'B': Both row and column equilibra‐
166 tion, i.e., A has been replaced by diag(R) * A * diag(C).
167 EQUED is an input argument if FACT = 'F'; otherwise, it is an
168 output argument.
169 R (input or output) DOUBLE PRECISION array, dimension (N)
171 The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
172 tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
173 accessed. R is an input argument if FACT = 'F'; otherwise, R
174 is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
175 each element of R must be positive.
176 C (input or output) DOUBLE PRECISION array, dimension (N)
178 The column scale factors for A. If EQUED = 'C' or 'B', A is
179 multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
180 not accessed. C is an input argument if FACT = 'F'; otherwise,
181 C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
182 each element of C must be positive.
183 B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
185 On entry, the right hand side matrix B. On exit, if EQUED =
186 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B',
187 B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED
188 = 'C' or 'B', B is overwritten by diag(C)*B.
189 LDB (input) INTEGER
191 The leading dimension of the array B. LDB >= max(1,N).
192 X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
194 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
195 the original system of equations. Note that A and B are modi‐
196 fied on exit if EQUED .ne. 'N', and the solution to the equili‐
197 brated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C'
198 or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R'
199 or 'B'.
200 LDX (input) INTEGER
202 The leading dimension of the array X. LDX >= max(1,N).
203 RCOND (output) DOUBLE PRECISION
205 The estimate of the reciprocal condition number of the matrix A
206 after equilibration (if done). If RCOND is less than the
207 machine precision (in particular, if RCOND = 0), the matrix is
208 singular to working precision. This condition is indicated by
209 a return code of INFO > 0.
210 FERR (output) DOUBLE PRECISION array, dimension (NRHS)
212 The estimated forward error bound for each solution vector X(j)
213 (the j-th column of the solution matrix X). If XTRUE is the
214 true solution corresponding to X(j), FERR(j) is an estimated
215 upper bound for the magnitude of the largest element in (X(j) -
216 XTRUE) divided by the magnitude of the largest element in X(j).
217 The estimate is as reliable as the estimate for RCOND, and is
218 almost always a slight overestimate of the true error.
219 BERR (output) DOUBLE PRECISION array, dimension (NRHS)
221 The componentwise relative backward error of each solution vec‐
222 tor X(j) (i.e., the smallest relative change in any element of
223 A or B that makes X(j) an exact solution).
224 WORK (workspace/output) DOUBLE PRECISION array, dimension (3*N)
226 On exit, WORK(1) contains the reciprocal pivot growth factor
227 norm(A)/norm(U). The "max absolute element" norm is used. If
228 WORK(1) is much less than 1, then the stability of the LU fac‐
229 torization of the (equilibrated) matrix A could be poor. This
230 also means that the solution X, condition estimator RCOND, and
231 forward error bound FERR could be unreliable. If factorization
232 fails with 0<INFO<=N, then WORK(1) contains the reciprocal
233 pivot growth factor for the leading INFO columns of A.
234 IWORK (workspace) INTEGER array, dimension (N)
236 INFO (output) INTEGER
238 = 0: successful exit
239 < 0: if INFO = -i, the i-th argument had an illegal value
240 > 0: if INFO = i, and i is
241 <= N: U(i,i) is exactly zero. The factorization has been com‐
242 pleted, but the factor U is exactly singular, so the solution
243 and error bounds could not be computed. RCOND = 0 is returned.
244 = N+1: U is nonsingular, but RCOND is less than machine preci‐
245 sion, meaning that the matrix is singular to working precision.
246 Nevertheless, the solution and error bounds are computed
247 because there are a number of situations where the computed
248 solution can be more accurate than the value of RCOND would
249 suggest.
250 LAPACK driver routine (version 3.N1o)vember 2006 DGBSVX(1)