DGESVX(1) LAPACK driver routine (version 3.2) DGESVX(1)NAME
DGESVX - uses the LU factorization to compute the solution to a real
system of linear equations A * X = B,
SYNOPSIS
SUBROUTINE DGESVX( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, EQUED,
R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
IWORK, INFO )
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ), X(
LDX, * )
PURPOSE
DGESVX uses the LU factorization to compute the solution to a real sys‐
tem of linear equations
A * X = B, where A is an N-by-N matrix and X and B are N-by-NRHS
matrices. Error bounds on the solution and a condition estimate are
also provided.
DESCRIPTION
The following steps are performed:
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 = P * L * U,
where P is a permutation matrix, L is a unit lower triangular
matrix, and U is upper triangular.
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, AF 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. A, AF, and IPIV are not modified. = 'N': The matrix A
will be copied to AF and factored.
= 'E': The matrix A will be equilibrated if necessary, then
copied to AF and factored.
TRANS (input) CHARACTER*1
Specifies the form of the system of equations:
= 'N': A * X = B (No transpose)
= '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.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of
the matrices B and X. NRHS >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A. If FACT = 'F' and EQUED is not
'N', then A must have been equilibrated by the scaling factors
in R and/or C. A 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).
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
If FACT = 'F', then AF is an input argument and on entry con‐
tains the factors L and U from the factorization A = P*L*U as
computed by DGETRF. If EQUED .ne. 'N', then AF is the factored
form of the equilibrated matrix A. If FACT = 'N', then AF is
an output argument and on exit returns the factors L and U from
the factorization A = P*L*U of the original matrix A. If FACT
= 'E', then AF is an output argument and on exit returns the
factors L and U from the factorization A = P*L*U of the equili‐
brated matrix A (see the description of A for the form of the
equilibrated matrix).
LDAF (input) INTEGER
The leading dimension of the array AF. LDAF >= max(1,N).
IPIV (input or output) INTEGER array, dimension (N)
If FACT = 'F', then IPIV is an input argument and on entry con‐
tains the pivot indices from the factorization A = P*L*U as
computed by DGETRF; 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
= P*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 = P*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 post‐
multiplied by diag(C). = 'B': Both row and column equilibra‐
tion, 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) DOUBLE PRECISION array, dimension (N)
The row scale factors for A. If EQUED = 'R' or 'B', A is mul‐
tiplied 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) DOUBLE PRECISION 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 argument 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) DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B. On exit, if
EQUED = 'N', B is not modified; 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) DOUBLE PRECISION array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
the original system of equations. Note that A and B are modi‐
fied on exit if EQUED .ne. 'N', and the solution to the equili‐
brated 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) DOUBLE PRECISION
The estimate of the reciprocal condition number of the matrix A
after equilibration (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 indicated by
a return code of INFO > 0.
FERR (output) DOUBLE PRECISION array, dimension (NRHS)
The estimated forward error bound for each solution vector X(j)
(the j-th 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 estimate is as reliable as the estimate for RCOND, and is
almost always a slight overestimate of the true error.
BERR (output) DOUBLE PRECISION array, dimension (NRHS)
The componentwise relative backward error of each solution vec‐
tor X(j) (i.e., the smallest relative change in any element of
A or B that makes X(j) an exact solution).
WORK (workspace/output) DOUBLE PRECISION array, dimension (4*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 stability of the LU fac‐
torization 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) contains 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 i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: U(i,i) is exactly zero. The factorization has been com‐
pleted, but the factor 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 preci‐
sion, meaning that the matrix is singular to working precision.
Nevertheless, 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 driver routine (version 3November 2008 DGESVX(1)
