1SGBSVX(1) LAPACK driver routine (version 3.2) SGBSVX(1)
2
6 SGBSVX - uses the LU factorization to compute the solution to a real
7 system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
8
10 SUBROUTINE SGBSVX( 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 REAL RCOND
19 INTEGER IPIV( * ), IWORK( * )
21 REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( *
23 ), C( * ), FERR( * ), R( * ), WORK( * ), X( LDX, * )
24
26 SGBSVX uses the LU factorization to compute the solution to a real sys‐
27 tem of linear equations A * X = B, A**T * X = B, or A**H * X = B, where
28 A is a band matrix of order N with KL subdiagonals and KU superdiago‐
29 nals, and X and B are N-by-NRHS matrices.
30 Error bounds on the solution and a condition estimate are also pro‐
31 vided.
32
34 The following steps are performed by this subroutine:
35 1. If FACT = 'E', real scaling factors are computed to equilibrate
36 the system:
37 TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
38 TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
39 TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
40 Whether or not the system will be equilibrated depends on the
41 scaling of the matrix A, but if equilibration is used, A is
42 overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
43 or diag(C)*B (if TRANS = 'T' or 'C').
44 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
45 matrix A (after equilibration if FACT = 'E') as
46 A = L * U,
47 where L is a product of permutation and unit lower triangular
48 matrices with KL subdiagonals, and U is upper triangular with
49 KL+KU superdiagonals.
50 3. If some U(i,i)=0, so that U is exactly singular, then the routine
51 returns with INFO = i. Otherwise, the factored form of A is used
52 to estimate the condition number of the matrix A. If the
53 reciprocal of the condition number is less than machine precision,
54 INFO = N+1 is returned as a warning, but the routine still goes on
55 to solve for X and compute error bounds as described below. 4. The
56 system of equations is solved for X using the factored form
57 of A.
58 5. Iterative refinement is applied to improve the computed solution
59 matrix and calculate error bounds and backward error estimates
60 for it.
61 6. If equilibration was used, the matrix X is premultiplied by
62 diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
63 that it solves the original system before equilibration.
64
66 FACT (input) CHARACTER*1
67 Specifies whether or not the factored form of the matrix A is
68 supplied on entry, and if not, whether the matrix A should be
69 equilibrated before it is factored. = 'F': On entry, AFB and
70 IPIV contain the factored form of A. If EQUED is not 'N', the
71 matrix A has been equilibrated with scaling factors given by R
72 and C. AB, AFB, and IPIV are not modified. = 'N': The matrix
73 A will be copied to AFB and factored.
74 = 'E': The matrix A will be equilibrated if necessary, then
75 copied to AFB and factored.
76 TRANS (input) CHARACTER*1
78 Specifies the form of the system of equations. = 'N': A * X =
79 B (No transpose)
80 = 'T': A**T * X = B (Transpose)
81 = 'C': A**H * X = B (Transpose)
82 N (input) INTEGER
84 The number of linear equations, i.e., the order of the matrix
85 A. N >= 0.
86 KL (input) INTEGER
88 The number of subdiagonals within the band of A. KL >= 0.
89 KU (input) INTEGER
91 The number of superdiagonals within the band of A. KU >= 0.
92 NRHS (input) INTEGER
94 The number of right hand sides, i.e., the number of columns of
95 the matrices B and X. NRHS >= 0.
96 AB (input/output) REAL array, dimension (LDAB,N)
98 On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
99 The j-th column of A is stored in the j-th column of the array
100 AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-
101 KU)<=i<=min(N,j+kl) If FACT = 'F' and EQUED is not 'N', then A
102 must have been equilibrated by the scaling factors in R and/or
103 C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E'
104 and EQUED = 'N' on exit. On exit, if EQUED .ne. 'N', A is
105 scaled as follows: EQUED = 'R': A := diag(R) * A
106 EQUED = 'C': A := A * diag(C)
107 EQUED = 'B': A := diag(R) * A * diag(C).
108 LDAB (input) INTEGER
110 The leading dimension of the array AB. LDAB >= KL+KU+1.
111 AFB (input or output) REAL array, dimension (LDAFB,N)
113 If FACT = 'F', then AFB is an input argument and on entry con‐
114 tains details of the LU factorization of the band matrix A, as
115 computed by SGBTRF. U is stored as an upper triangular band
116 matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the
117 multipliers used during the factorization are stored in rows
118 KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is the fac‐
119 tored form of the equilibrated matrix A. If FACT = 'N', then
120 AFB is an output argument and on exit returns details of the LU
121 factorization of A. If FACT = 'E', then AFB is an output argu‐
122 ment and on exit returns details of the LU factorization of the
123 equilibrated matrix A (see the description of AB for the form
124 of the equilibrated matrix).
125 LDAFB (input) INTEGER
127 The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
128 IPIV (input or output) INTEGER array, dimension (N)
130 If FACT = 'F', then IPIV is an input argument and on entry con‐
131 tains the pivot indices from the factorization A = L*U as com‐
132 puted by SGBTRF; row i of the matrix was interchanged with row
133 IPIV(i). If FACT = 'N', then IPIV is an output argument and on
134 exit contains the pivot indices from the factorization A = L*U
135 of the original matrix A. If FACT = 'E', then IPIV is an out‐
136 put argument and on exit contains the pivot indices from the
137 factorization A = L*U of the equilibrated matrix A.
138 EQUED (input or output) CHARACTER*1
140 Specifies the form of equilibration that was done. = 'N': No
141 equilibration (always true if FACT = 'N').
142 = 'R': Row equilibration, i.e., A has been premultiplied by
143 diag(R). = 'C': Column equilibration, i.e., A has been post‐
144 multiplied by diag(C). = 'B': Both row and column equilibra‐
145 tion, i.e., A has been replaced by diag(R) * A * diag(C).
146 EQUED is an input argument if FACT = 'F'; otherwise, it is an
147 output argument.
148 R (input or output) REAL array, dimension (N)
150 The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
151 tiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not
152 accessed. R is an input argument if FACT = 'F'; otherwise, R
153 is an output argument. If FACT = 'F' and EQUED = 'R' or 'B',
154 each element of R must be positive.
155 C (input or output) REAL array, dimension (N)
157 The column scale factors for A. If EQUED = 'C' or 'B', A is
158 multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is
159 not accessed. C is an input argument if FACT = 'F'; otherwise,
160 C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B',
161 each element of C must be positive.
162 B (input/output) REAL array, dimension (LDB,NRHS)
164 On entry, the right hand side matrix B. On exit, if EQUED =
165 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B',
166 B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED
167 = 'C' or 'B', B is overwritten by diag(C)*B.
168 LDB (input) INTEGER
170 The leading dimension of the array B. LDB >= max(1,N).
171 X (output) REAL array, dimension (LDX,NRHS)
173 If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
174 the original system of equations. Note that A and B are modi‐
175 fied on exit if EQUED .ne. 'N', and the solution to the equili‐
176 brated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C'
177 or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R'
178 or 'B'.
179 LDX (input) INTEGER
181 The leading dimension of the array X. LDX >= max(1,N).
182 RCOND (output) REAL
184 The estimate of the reciprocal condition number of the matrix A
185 after equilibration (if done). If RCOND is less than the
186 machine precision (in particular, if RCOND = 0), the matrix is
187 singular to working precision. This condition is indicated by
188 a return code of INFO > 0.
189 FERR (output) REAL array, dimension (NRHS)
191 The estimated forward error bound for each solution vector X(j)
192 (the j-th column of the solution matrix X). If XTRUE is the
193 true solution corresponding to X(j), FERR(j) is an estimated
194 upper bound for the magnitude of the largest element in (X(j) -
195 XTRUE) divided by the magnitude of the largest element in X(j).
196 The estimate is as reliable as the estimate for RCOND, and is
197 almost always a slight overestimate of the true error.
198 BERR (output) REAL array, dimension (NRHS)
200 The componentwise relative backward error of each solution vec‐
201 tor X(j) (i.e., the smallest relative change in any element of
202 A or B that makes X(j) an exact solution).
203 WORK (workspace/output) REAL array, dimension (3*N)
205 On exit, WORK(1) contains the reciprocal pivot growth factor
206 norm(A)/norm(U). The "max absolute element" norm is used. If
207 WORK(1) is much less than 1, then the stability of the LU fac‐
208 torization of the (equilibrated) matrix A could be poor. This
209 also means that the solution X, condition estimator RCOND, and
210 forward error bound FERR could be unreliable. If factorization
211 fails with 0<INFO<=N, then WORK(1) contains the reciprocal
212 pivot growth factor for the leading INFO columns of A.
213 IWORK (workspace) INTEGER array, dimension (N)
215 INFO (output) INTEGER
217 = 0: successful exit
218 < 0: if INFO = -i, the i-th argument had an illegal value
219 > 0: if INFO = i, and i is
220 <= N: U(i,i) is exactly zero. The factorization has been com‐
221 pleted, but the factor U is exactly singular, so the solution
222 and error bounds could not be computed. RCOND = 0 is returned.
223 = N+1: U is nonsingular, but RCOND is less than machine preci‐
224 sion, meaning that the matrix is singular to working precision.
225 Nevertheless, the solution and error bounds are computed
226 because there are a number of situations where the computed
227 solution can be more accurate than the value of RCOND would
228 suggest.
229 LAPACK driver routine (version 3.N2o)vember 2008 SGBSVX(1)