naginterfaces.library.ode.bvp_​coll_​nlin_​diag

naginterfaces.library.ode.bvp_coll_nlin_diag(mxmesh, comm)

bvp_coll_nlin_diag returns information about the solution of a general two-point boundary value problem computed by bvp_coll_nlin_solve().

For full information please refer to the NAG Library document for d02tz

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d02/d02tzf.html

Parameters

mxmeshint

The maximum number of points allowed in the mesh.

commdict, communication object

Communication structure.

This argument must have been initialized by a prior call to bvp_coll_nlin_setup().

Returns

nmeshint

The number of points in the mesh last used by bvp_coll_nlin_solve().

meshfloat, ndarray, shape $\left(mxmesh\right)$

$mesh[\text{i}-1]$ contains the $\text{i}$th point of the mesh last used by bvp_coll_nlin_solve(), for $\text{i}=1,2,\dots ,nmesh$. $mesh\left[0\right]$ will contain $a$ and $mesh[nmesh-1]$ will contain $b$. The remaining elements of $mesh$ are not initialized.

ipmeshint, ndarray, shape $\left(mxmesh\right)$

$ipmesh[\text{i}-1]$ specifies the nature of the point $mesh[\text{i}-1]$, for $\text{i}=1,2,\dots ,nmesh$, in the final mesh computed by bvp_coll_nlin_solve().

$ipmesh[i-1]=1$

Indicates that the $i$th point is a fixed point and was used by the solver before an extrapolation-like error test.

$ipmesh[i-1]=2$

Indicates that the $i$th point was used by the solver before an extrapolation-like error test.

$ipmesh[i-1]=3$

Indicates that the $i$th point was used by the solver only as part of an extrapolation-like error test.

The remaining elements of $ipmesh$ are initialized to $-1$.

See Further Comments for advice on how these values may be used in conjunction with a continuation process.

ermxfloat

An estimate of the maximum error in the solution computed by bvp_coll_nlin_solve(), that is

$$ermx=max\left(\frac{\parallel y_i-v_i\parallel }{(1.0+\parallel v_i\parallel )}\right)$$

where $v_i$ is the approximate solution for the $i$th solution component. If bvp_coll_nlin_solve() returned successfully with no exception or warning is raised, $ermx$ will be less than $\text{tols}[ijermx-1]$ in bvp_coll_nlin_setup() where $\text{tols}$ contains the error requirements as specified in Notes for bvp_coll_nlin_setup and Parameters for bvp_coll_nlin_setup.

If bvp_coll_nlin_solve() returned with $errno$ = 5, $ermx$ will be greater than $\text{tols}[ijermx-1]$ in bvp_coll_nlin_setup().

If bvp_coll_nlin_solve() returned any other value for $\text{errno}$ then an error estimate is not available and $ermx$ is initialized to $0.0$.

iermxint

Indicates the mesh sub-interval where the value of $ermx$ has been computed, that is $\left[mesh\right[iermx-1],mesh[iermx+1-1]]$.

If an estimate of the error is not available then $iermx$ is initialized to $0$.

ijermxint

Indicates the component $i$ ($=ijermx$) of the solution for which $ermx$ has been computed, that is the approximation of $y_i$ on $[mesh[iermx-1],mesh\left[iermx\right]]$ is estimated to have the largest error of all components $y_i$ over mesh sub-intervals defined by $mesh$.

If an estimate of the error is not available then $ijermx$ is initialized to $0$.

Raises

NagValueError

(errno $1$)

The solver function did not produce any results suitable for interpolation.

(errno $1$)

The solver function does not appear to have been called.

(errno $1$)

On entry, $mxmesh=⟨value⟩$ and $\text{mxmesh}=⟨value⟩$ in bvp_coll_nlin_setup().

Constraint: $mxmesh=\text{mxmesh}$ in bvp_coll_nlin_setup().

Warns

NagAlgorithmicWarning

(errno $2$)

The solver function did not satisfy the error requirements.

Information has been supplied on the last mesh used.

(errno $2$)

The solver function did not converge to a suitable solution.

A converged intermediate solution has been used.

Error estimate information is not available.

Notes

bvp_coll_nlin_diag and its associated functions (bvp_coll_nlin_setup(), bvp_coll_nlin_solve(), bvp_coll_nlin_contin() and bvp_coll_nlin_interp()) solve the two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations

$$\begin{array}{c}\begin{array}{ccc}{y}_1^{\left(m_1\right)}\left(x\right)& =& f_1(x,y_1,y_1^{\left(1\right)},\dots ,y_1^{(m_1-1)},y_2,\dots ,y_n^{(m_n-1)})\\ y_2^{\left(m_2\right)}\left(x\right)& =& f_2(x,y_1,y_1^{\left(1\right)},\dots ,y_1^{(m_1-1)},y_2,\dots ,y_n^{(m_n-1)})\\ & ⋮& \\ y_n^{\left(m_n\right)}\left(x\right)& =& f_n(x,y_1,y_1^{\left(1\right)},\dots ,y_1^{(m_1-1)},y_2,\dots ,y_n^{(m_n-1)})\end{array}\end{array}$$

over an interval $[a,b]$ subject to $p$ ($>0$) nonlinear boundary conditions at $a$ and $q$ ($>0$) nonlinear boundary conditions at $b$, where $p+q={\sum }_{i=1}^n{m}_i$. Note that $y_i^{\left(m\right)}\left(x\right)$ is the $m$th derivative of the $i$th solution component. Hence $y_i^{\left(0\right)}\left(x\right)=y_i\left(x\right)$. The left boundary conditions at $a$ are defined as

$$g_i\left(z\left(y\left(a\right)\right)\right)=0\text{,}i=1,2,\dots ,p\text{,}$$

and the right boundary conditions at $b$ as

$${\overline{g}}_j\left(z\left(y\left(b\right)\right)\right)=0\text{,}j=1,2,\dots ,q\text{,}$$

where $y=(y_1,y_2,\dots ,y_n)$ and

$$z\left(y\left(x\right)\right)=(y_1\left(x\right),y_1^{\left(1\right)}\left(x\right),\dots ,y_1^{(m_1-1)}\left(x\right),y_2\left(x\right),\dots ,y_n^{(m_n-1)}\left(x\right))\text{.}$$

First, bvp_coll_nlin_setup() must be called to specify the initial mesh, error requirements and other details. Then, bvp_coll_nlin_solve() can be used to solve the boundary value problem. After successful computation, bvp_coll_nlin_diag can be used to ascertain details about the final mesh. bvp_coll_nlin_interp() can be used to compute the approximate solution anywhere on the interval $[a,b]$ 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).

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

Cole, J D, 1968, Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, Mass.

Keller, H B, 1992, Numerical Methods for Two-point Boundary-value Problems, Dover, New York