bvpset (MATLAB Functions)

Create or alter boundary value problem (BVP) options structure

Syntax

options = bvpset('name1',value1,'name2',value2,...)
options = bvpset(oldopts'name1',value1,...)
options = bvpset(oldopts,newopts)
bvpset

Description

options = bvpset('name1',value1,'name2',value2,...) creates a structure options that you can supply to the boundary value problem solver bvp4c, in which the named properties have the specified values. Any unspecified properties retain their default values. For all properties, it is sufficient to type only the leading characters that uniquely identify the property. bvpset ignores case for property names.

options = bvpset(oldopts,'name1',value1,...) alters an existing options structure oldopts. This overwrites any values in oldopts that are specified using name/value pairs and returns the modified structure as the output argument.

options = bvpset(oldopts,newopts) combines an existing options structure oldopts with a new options structure newopts. Any values set in newopts overwrite the corresponding values in oldopts.

bvpset with no input arguments displays all property names and their possible values, indicating defaults with braces {}.

You can use the function bvpget to query the options structure for the value of a specific property.

BVP Properties

bvpset enables you to specify properties for the boundary value problem solver bvp4c. There are several categories of properties that you can set:

Error Tolerance Properties

Because bvp4c uses a collocation formula, the numerical solution is based on a mesh of points at which the collocation equations are satisfied. Mesh selection and error control are based on the residual of this solution, such that the computed solution is the exact solution of a perturbed problem . On each subinterval of the mesh, a norm of the residual in the ith component of the solution, res(i), is estimated and is required to be less than or equal to a tolerance. This tolerance is a function of the relative and absolute tolerances, RelTol and AbsTol, defined by the user.

The following table describes the error tolerance properties.

BVP Error Tolerance Properties
Property
Value
Description

RelTol
Positive scalar {1e-3}
A relative error tolerance that applies to all components of the residual vector. It is a measure of the residual relative to the size of . The default, 1e-3, corresponds to 0.1% accuracy.
The computed solution is the exact solution of . On each subinterval of the mesh, the residual satisfies

AbsTol
Positive scalar or vector {1e-6}
Absolute error tolerances that apply to the corresponding components of the residual vector. AbsTol(i) is a threshold below which the values of the corresponding components are unimportant. If a scalar value is specified, it applies to all components.

Vectorization

The following table describes the BVP vectorization property. Vectorization of the ODE function used by bvp4c differs from the vectorization used by the ODE solvers:

For bvp4c, the ODE function must be vectorized with respect to the first argument as well as the second one, so that F([x1 x2 ...],[y1 y2 ...]) returns [F(x1,y1) F(x2,y2) ...].

bvp4c benefits from vectorization even when analytical Jacobians are provided. For stiff ODE solvers, vectorization is ignored when analytical Jacobians are used.

**Vectorization Properties**
Property	Value	Description
`Vectorized`	`on` \| {`off`}	Set `on` to inform `bvp4c` that you have coded the ODE function `F` so that `F([x1 x2 ...],[y1 y2 ...])` returns `[F(x1,y1) F(x2,y2) ...]`. That is, your ODE function can pass to the solver a whole array of column vectors at once. This enables the solver to reduce the number of function evaluations and may significantly reduce solution time. With the MATLAB array notation, it is typically an easy matter to vectorize an ODE function. In the `shockbvp` example shown previously, the `shockODE` function has been vectorized using colon notation into the subscripts and by using the array multiplication (`.`) operator. function dydx = shockODE(x,y,e) pix = pix; dydx = [ y(2,:) -x/e.y(2,:)-pi^2cos(pix)-pix/e.*sin(pix) ];

Analytical Partial Derivatives

By default, the bvp4c solver approximates all partial derivatives with finite differences. bvp4c can be more efficient if you provide analytical partial derivatives of the differential equations, and analytical partial derivatives, and , of the boundary conditions. If the problem involves unknown parameters, you must also provide partial derivatives, and , with respect to the parameters.

The following table describes the analytical partial derivatives properties.

BVP Analytical Partial Derivative Properties
Property
Value
Description

FJacobian
Function handle
Handle to a function that computes the analytical partial derivatives of . When solving , set this property to @fjac if dfdy = fjac(x,y) evaluates the Jacobian . If the problem involves unknown parameters , [dfdy,dfdp] = fjac(x,y,p) must also return the partial derivative . For problems with constant partial derivatives, set this property to the value of dfdy or to a cell array {dfdy,dfdp}.
See Function Handles in the MATLAB Programming documentation for more information.

BCJacobian
Function handle
Handle to a function that computes the analytical partial derivatives of . For boundary conditions , set this property to @bcjac if [dbcdya,dbcdyb] = bcjac(ya,yb) evaluates the partial derivatives , and . If the problem involves unknown parameters , [dbcdya,dbcdyb,dbcdp] = bcjac(ya,yb,p) must also return the partial derivative . For problems with constant partial derivatives, set this property to a cell array {dbcdya,dbcdyb} or {dbcdya,dbcdyb,dbcdp}.

Singular BVPs

bvp4c can solve singular problems of the form

posed on the interval where . For such problems, specify the constant matrix as the value of SingularTerm. For equations of this form, odefun evaluates only the term, where represents unknown parameters, if any.

Singular BVP Property
Property
Value
Description

SingularTerm
Constant matrix
Singular term of singular BVPs. Set to the constant matrix for equations of the form

posed on the interval where .

Mesh Size Property

bvp4c solves a system of algebraic equations to determine the numerical solution to a BVP at each of the mesh points. The size of the algebraic system depends on the number of differential equations (n) and the number of mesh points in the current mesh (N). When the allowed number of mesh points is exhausted, the computation stops, bvp4c displays a warning message and returns the solution it found so far. This solution does not satisfy the error tolerance, but it may provide an excellent initial guess for computations restarted with relaxed error tolerances or an increased value of NMax.

The following table describes the mesh size property.

BVP Mesh Size Property
Property
Value
Description

NMax
positive integer {floor(1000/n)}
Maximum number of mesh points allowed when solving the BVP, where n is the number of differential equations in the problem. The default value of NMax limits the size of the algebraic system to about 1000 equations. For systems of a few differential equations, the default value of NMax should be sufficient to obtain an accurate solution.

Solution Statistic Property

The Stats property lets you view solution statistics.

The following table describes the solution statistics property.

BVP Solution Statistic Property
Property
Value
Description

Stats
on | {off}

Specifies whether statistics about the computations are displayed. If the stats property is on, after solving the problem, bvp4c displays:

The number of points in the mesh

The maximum residual of the solution

The number of times it called the differential equation function odefun to evaluate

The number of times it called the boundary condition function bcfun to evaluate

Example

To create an options structure that changes the relative error tolerance of bvp4c from the default value of 1e-3 to 1e-4, enter

```
options = bvpset('RelTol', 1e-4);
```

To recover the value of 'RelTol' from options, enter

bvpget(options, 'RelTol')

ans =

  1.0000e-004

See Also

@ (function_handle), bvp4c, bvpget, bvpinit, deval

bvpinit bvpval