NAG Library Function Document
nag_real_gen_lin_solve (f04bac)
1 Purpose
nag_real_gen_lin_solve (f04bac) computes the solution to a real system of linear equations , where is an by matrix and and are by matrices. An estimate of the condition number of and an error bound for the computed solution are also returned.
2 Specification
#include <nag.h> |
#include <nagf04.h> |
void |
nag_real_gen_lin_solve (Nag_OrderType order,
Integer n,
Integer nrhs,
double a[],
Integer pda,
Integer ipiv[],
double b[],
Integer pdb,
double *rcond,
double *errbnd,
NagError *fail) |
|
3 Description
The decomposition with partial pivoting and row interchanges is used to factor as , where is a permutation matrix, is unit lower triangular, and is upper triangular. The factored form of is then used to solve the system of equations .
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5 Arguments
- 1:
order – Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
n – IntegerInput
On entry: the number of linear equations , i.e., the order of the matrix .
Constraint:
.
- 3:
nrhs – IntegerInput
On entry: the number of right-hand sides , i.e., the number of columns of the matrix .
Constraint:
.
- 4:
a[] – doubleInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by coefficient matrix .
On exit: if NE_NOERROR, the factors and from the factorization . The unit diagonal elements of are not stored.
- 5:
pda – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 6:
ipiv[n] – IntegerOutput
On exit: if NE_NOERROR, the pivot indices that define the permutation matrix ; at the th step row of the matrix was interchanged with row . indicates a row interchange was not required.
- 7:
b[] – doubleInput/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix of right-hand sides .
On exit: if
NE_NOERROR or
NE_RCOND, the
by
solution matrix
.
- 8:
pdb – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 9:
rcond – double *Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix , computed as .
- 10:
errbnd – double *Output
On exit: if
NE_NOERROR or
NE_RCOND, an estimate of the forward error bound for a computed solution
, such that
, where
is a column of the computed solution returned in the array
b and
is the corresponding column of the exact solution
. If
rcond is less than
machine precision, then
errbnd is returned as unity.
- 11:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
- NE_RCOND
-
A solution has been computed, but
rcond is less than
machine precision so that the matrix
is numerically singular.
- NE_SINGULAR
-
Diagonal element of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
7 Accuracy
The computed solution for a single right-hand side,
, satisfies an equation of the form
where
and
is the
machine precision. An approximate error bound for the computed solution is given by
where
, the condition number of
with respect to the solution of the linear equations. nag_real_gen_lin_solve (f04bac) uses the approximation
to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999) for further details.
8 Parallelism and Performance
nag_real_gen_lin_solve (f04bac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_real_gen_lin_solve (f04bac) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations required to solve the equations is proportional to . The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The complex analogue of nag_real_gen_lin_solve (f04bac) is
nag_complex_gen_lin_solve (f04cac).
10 Example
This example solves the equations
where
An estimate of the condition number of and an approximate error bound for the computed solutions are also printed.
10.1 Program Text
Program Text (f04bace.c)
10.2 Program Data
Program Data (f04bace.d)
10.3 Program Results
Program Results (f04bace.r)