NAG Library Function Document
nag_ode_bvp_coll_nlin_interp (d02tyc)
1
Purpose
nag_ode_bvp_coll_nlin_interp (d02tyc) interpolates on the solution of a general two-point boundary value problem computed by
nag_ode_bvp_coll_nlin_solve (d02tlc).
2
Specification
#include <nag.h> |
#include <nagd02.h> |
void |
nag_ode_bvp_coll_nlin_interp (double x,
double y[],
Integer neq,
Integer mmax,
double rcomm[],
const Integer icomm[],
NagError *fail) |
|
3
Description
nag_ode_bvp_coll_nlin_interp (d02tyc) and its associated functions (
nag_ode_bvp_coll_nlin_solve (d02tlc),
nag_ode_bvp_coll_nlin_setup (d02tvc),
nag_ode_bvp_coll_nlin_contin (d02txc) and
nag_ode_bvp_coll_nlin_diag (d02tzc)) solve the two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations
over an interval
subject to
(
) nonlinear boundary conditions at
and
(
) nonlinear boundary conditions at
, where
. Note that
is the
th derivative of the
th solution component. Hence
. The left boundary conditions at
are defined as
and the right boundary conditions at
as
where
and
First,
nag_ode_bvp_coll_nlin_setup (d02tvc) must be called to specify the initial mesh, error requirements and other details. Then,
nag_ode_bvp_coll_nlin_solve (d02tlc) can be used to solve the boundary value problem. After successful computation,
nag_ode_bvp_coll_nlin_diag (d02tzc) can be used to ascertain details about the final mesh and other details of the solution procedure, and
nag_ode_bvp_coll_nlin_interp (d02tyc) can be used to compute the approximate solution anywhere on the interval
using interpolation.
The functions are based on modified versions of the codes COLSYS and COLNEW (see
Ascher et al. (1979) and
Ascher and Bader (1987)). A comprehensive treatment of the numerical solution of boundary value problems can be found in
Ascher et al. (1988) and
Keller (1992).
4
References
Ascher U M and Bader G (1987) A new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comput. 8 483–500
Ascher U M, Christiansen J and Russell R D (1979) A collocation solver for mixed order systems of boundary value problems Math. Comput. 33 659–679
Ascher U M, Mattheij R M M and Russell R D (1988) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Prentice–Hall
Grossman C (1992) Enclosures of the solution of the Thomas–Fermi equation by monotone discretization J. Comput. Phys. 98 26–32
Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York
5
Arguments
- 1:
– doubleInput
-
On entry: , the independent variable.
Constraint:
, i.e., not outside the range of the original mesh specified in the initialization call to
nag_ode_bvp_coll_nlin_setup (d02tvc).
- 2:
– doubleOutput
-
Note: where appears in this document, it refers to the array element
.
On exit:
contains an approximation to
, for
and
. The remaining elements of
y (where
) are initialized to
.
- 3:
– IntegerInput
-
On entry: the number of differential equations.
Constraint:
must be the same value as supplied to
nag_ode_bvp_coll_nlin_setup (d02tvc).
- 4:
– IntegerInput
-
On entry: the maximal order of the differential equations,
, for .
Constraint:
must contain the maximum value of the components of the argument
m as supplied to
nag_ode_bvp_coll_nlin_setup (d02tvc).
- 5:
– doubleCommunication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
rcomm in the previous call to
nag_ode_bvp_coll_nlin_solve (d02tlc).
On entry: this must be the same array as supplied to
nag_ode_bvp_coll_nlin_solve (d02tlc) and
must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated functions.
- 6:
– const IntegerCommunication Array
-
Note: the dimension,
, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument
icomm in the previous call to
nag_ode_bvp_coll_nlin_solve (d02tlc).
On entry: this must be the same array as supplied to
nag_ode_bvp_coll_nlin_solve (d02tlc) and
must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated functions.
- 7:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE_SOL
-
The solver function did not produce any results suitable for interpolation.
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INT_CHANGED
-
- 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_MISSING_CALL
-
The solver function does not appear to have been called.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NW_NOT_CONVERGED
-
The solver function did not converge to a suitable solution.
A converged intermediate solution has been used.
Interpolated values should be treated with caution.
- NW_TOO_MUCH_ACC_REQUESTED
-
The solver function did not satisfy the error requirements.
Interpolated values should be treated with caution.
7
Accuracy
If
nag_ode_bvp_coll_nlin_interp (d02tyc) returns the value
NE_NOERROR, the computed values of the solution components
should be of similar accuracy to that specified by the argument
tols of
nag_ode_bvp_coll_nlin_setup (d02tvc). Note that during the solution process the error in the derivatives
, for
, has not been controlled and that the derivative values returned by
nag_ode_bvp_coll_nlin_interp (d02tyc) are computed via differentiation of the piecewise polynomial approximation to
. See also
Section 9.
8
Parallelism and Performance
nag_ode_bvp_coll_nlin_interp (d02tyc) 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
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
If
nag_ode_bvp_coll_nlin_interp (d02tyc) returns the value
fail.code ,
then the accuracy of the interpolated values may be proportional to the quantity
ermx as returned by
nag_ode_bvp_coll_nlin_diag (d02tzc).
If
nag_ode_bvp_coll_nlin_solve (d02tlc) returned a value for
fail.code
other than
NE_NOERROR, then nothing can be said regarding either the quality or accuracy of the values computed by
nag_ode_bvp_coll_nlin_interp (d02tyc).
10
Example
The following example is used to illustrate that a system with singular coefficients can be treated without modification of the system definition. See also
nag_ode_bvp_coll_nlin_solve (d02tlc),
nag_ode_bvp_coll_nlin_setup (d02tvc),
nag_ode_bvp_coll_nlin_contin (d02txc) and
nag_ode_bvp_coll_nlin_diag (d02tzc), for the illustration of other facilities.
Consider the Thomas–Fermi equation used in the investigation of potentials and charge densities of ionized atoms. See
Grossman (1992), for example, and the references therein. The equation is
with boundary conditions
The coefficient
implies a singularity at the left-hand boundary
.
We use the initial approximation
, which satisfies the boundary conditions, on a uniform mesh of six points. For illustration we choose
, as in
Grossman (1992). Note that in
ffun and
fjac (see
nag_ode_bvp_coll_nlin_solve (d02tlc)) we have taken the precaution of setting the function value and Jacobian value to
in case a value of
becomes negative, although starting from our initial solution profile this proves unnecessary during the solution phase. Of course the true solution
is positive for all
.
10.1
Program Text
Program Text (d02tyce.c)
10.2
Program Data
Program Data (d02tyce.d)
10.3
Program Results
Program Results (d02tyce.r)