
SGBSVX(l) ) SGBSVX(l)
NAME
SGBSVX  use the LU factorization to compute the solution to a real system of linear equa
tions A * X = B, A**T * X = B, or A**H * X = B,
SYNOPSIS
SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C,
B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
REAL RCOND
INTEGER IPIV( * ), IWORK( * )
REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ), C( * ), FERR( *
), R( * ), WORK( * ), X( LDX, * )
PURPOSE
SGBSVX uses the LU factorization to compute the solution to a real system of linear equa
tions A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with
KL subdiagonals and KU superdiagonals, and X and B are NbyNRHS matrices.
Error bounds on the solution and a condition estimate are also provided.
DESCRIPTION
The following steps are performed by this subroutine:
1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
or diag(C)*B (if TRANS = 'T' or 'C').
2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
matrix A (after equilibration if FACT = 'E') as
A = L * U,
where L is a product of permutation and unit lower triangular
matrices with KL subdiagonals, and U is upper triangular with
KL+KU superdiagonals.
3. If some U(i,i)=0, so that U is exactly singular, then the routine
returns with INFO = i. Otherwise, the factored form of A is used
to estimate the condition number of the matrix A. If the
reciprocal of the condition number is less than machine precision,
INFO = N+1 is returned as a warning, but the routine still goes on
to solve for X and compute error bounds as described below.
4. The system of equations is solved for X using the factored form
of A.
5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
6. If equilibration was used, the matrix X is premultiplied by
diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
that it solves the original system before equilibration.
ARGUMENTS
FACT (input) CHARACTER*1
Specifies whether or not the factored form of the matrix A is supplied on entry,
and if not, whether the matrix A should be equilibrated before it is factored. =
'F': On entry, AFB and IPIV contain the factored form of A. If EQUED is not 'N',
the matrix A has been equilibrated with scaling factors given by R and C. AB,
AFB, and IPIV are not modified. = 'N': The matrix A will be copied to AFB and
factored.
= 'E': The matrix A will be equilibrated if necessary, then copied to AFB and
factored.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations. = 'N': A * X = B (No trans
pose)
= 'T': A**T * X = B (Transpose)
= 'C': A**H * X = B (Transpose)
N (input) INTEGER
The number of linear equations, i.e., the order of the matrix A. N >= 0.
KL (input) INTEGER
The number of subdiagonals within the band of A. KL >= 0.
KU (input) INTEGER
The number of superdiagonals within the band of A. KU >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of the matrices B and
X. NRHS >= 0.
AB (input/output) REAL array, dimension (LDAB,N)
On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The jth column of
A is stored in the jth column of the array AB as follows: AB(KU+1+ij,j) = A(i,j)
for max(1,jKU)<=i<=min(N,j+kl)
If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the
scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if
FACT = 'E' and EQUED = 'N' on exit.
On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED = 'R': A := diag(R) * A
EQUED = 'C': A := A * diag(C)
EQUED = 'B': A := diag(R) * A * diag(C).
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KL+KU+1.
AFB (input or output) REAL array, dimension (LDAFB,N)
If FACT = 'F', then AFB is an input argument and on entry contains details of the
LU factorization of the band matrix A, as computed by SGBTRF. U is stored as an
upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
the multipliers used during the factorization are stored in rows KL+KU+2 to
2*KL+KU+1. If EQUED .ne. 'N', then AFB is the factored form of the equilibrated
matrix A.
If FACT = 'N', then AFB is an output argument and on exit returns details of the
LU factorization of A.
If FACT = 'E', then AFB is an output argument and on exit returns details of the
LU factorization of the equilibrated matrix A (see the description of AB for the
form of the equilibrated matrix).
LDAFB (input) INTEGER
The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry contains the pivot
indices from the factorization A = L*U as computed by SGBTRF; row i of the matrix
was interchanged with row IPIV(i).
If FACT = 'N', then IPIV is an output argument and on exit contains the pivot
indices from the factorization A = L*U of the original matrix A.
If FACT = 'E', then IPIV is an output argument and on exit contains the pivot
indices from the factorization A = L*U of the equilibrated matrix A.
EQUED (input or output) CHARACTER*1
Specifies the form of equilibration that was done. = 'N': No equilibration
(always true if FACT = 'N').
= 'R': Row equilibration, i.e., A has been premultiplied by diag(R). = 'C':
Column equilibration, i.e., A has been postmultiplied by diag(C). = 'B': Both
row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).
EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.
R (input or output) REAL array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left
by diag(R); if EQUED = 'N' or 'C', R is not accessed. R is an input argument if
FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or
'B', each element of R must be positive.
C (input or output) REAL array, dimension (N)
The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the
right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input argu
ment if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED =
'C' or 'B', each element of C must be positive.
B (input/output) REAL array, dimension (LDB,NRHS)
On entry, the right hand side matrix B. On exit, if EQUED = 'N', B is not modi
fied; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if
TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X (output) REAL array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the NbyNRHS solution matrix X to the original system
of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the
solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED =
'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND (output) REAL
The estimate of the reciprocal condition number of the matrix A after equilibra
tion (if done). If RCOND is less than the machine precision (in particular, if
RCOND = 0), the matrix is singular to working precision. This condition is indi
cated by a return code of INFO > 0.
FERR (output) REAL array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j) (the jth column
of the solution matrix X). If XTRUE is the true solution corresponding to X(j),
FERR(j) is an estimated upper bound for the magnitude of the largest element in
(X(j)  XTRUE) divided by the magnitude of the largest element in X(j). The esti
mate is as reliable as the estimate for RCOND, and is almost always a slight over
estimate of the true error.
BERR (output) REAL array, dimension (NRHS)
The componentwise relative backward error of each solution vector X(j) (i.e., the
smallest relative change in any element of A or B that makes X(j) an exact solu
tion).
WORK (workspace/output) REAL array, dimension (3*N)
On exit, WORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The
"max absolute element" norm is used. If WORK(1) is much less than 1, then the sta
bility of the LU factorization of the (equilibrated) matrix A could be poor. This
also means that the solution X, condition estimator RCOND, and forward error bound
FERR could be unreliable. If factorization fails with 0<INFO<=N, then WORK(1) con
tains the reciprocal pivot growth factor for the leading INFO columns of A.
IWORK (workspace) INTEGER array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: if INFO = i, and i is
<= N: U(i,i) is exactly zero. The factorization has been completed, but the fac
tor U is exactly singular, so the solution and error bounds could not be computed.
RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular to working precision. Neverthe
less, the solution and error bounds are computed because there are a number of
situations where the computed solution can be more accurate than the value of
RCOND would suggest.
LAPACK version 3.0 15 June 2000 SGBSVX(l) 
