Purpose
To solve (if well-conditioned) the matrix equations
op( A )*X = B,
where X and B are N-by-NRHS matrices, A is an N-by-N matrix and
op( A ) is one of
op( A ) = A or op( A ) = A'.
Error bounds on the solution and a condition estimate are also
provided.
Specification
SUBROUTINE MB02PD( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV,
$ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR,
$ IWORK, DWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
C .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), C( * ), DWORK( * ), FERR( * ),
$ R( * ), X( LDX, * )
Arguments
Mode Parameters
FACT 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 CHARACTER*1
Specifies the form of the system of equations as follows:
= 'N': A * X = B (No transpose);
= 'T': A**T * X = B (Transpose);
= 'C': A**H * X = B (Transpose).
Input/Output Parameters
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 leading N-by-N part of this array must
contain the 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', the leading N-by-N part of
this array contains the matrix A scaled as follows:
EQUED = 'R': A := diag(R) * A;
EQUED = 'C': A := A * diag(C);
EQUED = 'B': A := diag(R) * A * diag(C).
LDA 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
the leading N-by-N part of this array must contain 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
the leading N-by-N part of this array contains 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
the leading N-by-N part of this array contains the factors
L and U from the factorization A = P*L*U of the
equilibrated 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
it must contain 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
it 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
it 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 as
follows:
= '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) DOUBLE PRECISION 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) 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 leading N-by-NRHS part of this array must
contain the right-hand side matrix B.
On exit,
if EQUED = 'N', B is not modified;
if TRANS = 'N' and EQUED = 'R' or 'B', the leading
N-by-NRHS part of this array contains diag(R)*B;
if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', the leading
N-by-NRHS part of this array contains diag(C)*B.
LDB 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 leading N-by-NRHS part of
this array contains the 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) 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.
For efficiency reasons, RCOND is computed only when the
matrix A is factored, i.e., for FACT = 'N' or 'E'. For
FACT = 'F', RCOND is not used, but it is assumed that it
has been computed and checked before the routine call.
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
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
Workspace
IWORK INTEGER array, dimension (N)
DWORK DOUBLE PRECISION array, dimension (4*N)
On exit, DWORK(1) contains the reciprocal pivot growth
factor norm(A)/norm(U). The "max absolute element" norm is
used. If DWORK(1) is much less than 1, then the stability
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 DWORK(1) contains the reciprocal pivot
growth factor for the leading INFO columns of A.
Error Indicator
INFO 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 completed, 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 precision, 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.
The positive values for INFO are set only when the
matrix A is factored, i.e., for FACT = 'N' or 'E'.
Method
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.
References
[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J.,
Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A.,
Ostrouchov, S., Sorensen, D.
LAPACK Users' Guide: Second Edition, SIAM, Philadelphia, 1995.
Further Comments
This is a simplified version of the LAPACK Library routine DGESVX, useful when several sets of matrix equations with the same coefficient matrix A and/or A' should be solved.Numerical Aspects
3 The algorithm requires 0(N ) operations.Example
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