naginterfaces.library.ode.bvp_​coll_​nth_​comp

naginterfaces.library.ode.bvp_coll_nth_comp(m, l, x0, x1, k1, kp, coeff, bdyc, data=None, spiked_sorder='C')

bvp_coll_nth_comp solves a system of linear ordinary differential equations by least squares fitting of a series of Chebyshev polynomials using collocation.

For full information please refer to the NAG Library document for d02tg

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d02/d02tgf.html

Parameters

mint, array-like, shape $\left(n\right)$

$m\left[\text{i}\right]$ must be set to the highest order derivative occurring in the $\text{i}$th equation, for $\text{i}=1,2,\dots ,n$.

lint, array-like, shape $\left(n\right)$

$l\left[\text{i}\right]$ must be set to the number of boundary conditions associated with the $\text{i}$th equation, for $\text{i}=1,2,\dots ,n$.

x0float

The left-hand boundary, $x_0$.

x1float

The right-hand boundary, $x_1$.

k1int

The number of coefficients, $k_1$, to be returned in the Chebyshev series representation of the solution (hence, the degree of the polynomial approximation is $k1-1$).

kpint

The number of collocation points to be used, $k_p$.

coeffcallable (a, rhs) = coeff(x, i, a, rhs, data=None)

$coeff$ defines the system of differential equations (see Notes).

It must evaluate the coefficient functions $f_{kj}^i\left(x\right)$ and the right-hand side function $r^i\left(x\right)$ of the $i$th equation at a given point.

Only nonzero entries of the array $a$ and $rhs$ need be specifically assigned, since all elements are set to zero by bvp_coll_nth_comp before calling $coeff$.

Parameters

xfloat

$x$, the point at which the functions must be evaluated.

iint

The equation for which the coefficients and right-hand side are to be evaluated.

afloat, ndarray, shape $(\text{ia},\text{ia1})$

All elements of $a$ are set to zero.

rhsfloat

Is set to zero.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

afloat, array-like, shape $(\text{ia},\text{ia1})$

$a[k-1,j-1]$ must contain the value $f_{kj}^i\left(x\right)$, for $1\le k\le n$, $1\le j\le m_i+1$.

rhsfloat

It must contain the value $r^i\left(x\right)$.

bdyccallable (x, a, rhs) = bdyc(i, j, a, rhs, data=None)

$bdyc$ defines the boundary conditions (see Notes).

It must evaluate the coefficient functions $f_{kj}^{ij}$ and right-hand side function $r^{ij}$ in the $j$th boundary condition associated with the $i$th equation, at the point $x^{ij}$ at which the boundary condition is applied.

Only nonzero entries of the array $a$ and $rhs$ need be specifically assigned, since all elements are set to zero by bvp_coll_nth_comp before calling $bdyc$.

Parameters

iint

The differential equation with which the condition is associated.

jint

The boundary condition for which the coefficients and right-hand side are to be evaluated.

afloat, ndarray, shape $(\text{ia},\text{ia1})$

All elements of $a$ are set to zero.

rhsfloat

Is set to zero.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

xfloat

$x^{ij}$, the value at which the boundary condition is applied.

afloat, array-like, shape $(\text{ia},\text{ia1})$

The value $f_{kj}^{ij}\left(x^{ij}\right)$, for $1\le k\le n$, $1\le j\le m_i+1$.

rhsfloat

The value $r^{ij}\left(x^{ij}\right)$.

dataarbitrary, optional

User-communication data for callback functions.

spiked_sorderstr, optional

If $a$ in $bdyc$ or $a$ in $coeff$ are spiked (i.e., have unit extent in all but one dimension, or have size $1$), $spiked\_sorder$ selects the storage order to associate with them in the NAG Engine:

spiked_sorder = $\text{'C'}$

row-major storage will be used;

spiked_sorder = $\text{'F'}$

column-major storage will be used.

Returns

cfloat, ndarray, shape $(k1,n)$

The $k$th column of $c$ contains the computed Chebyshev coefficients of the $k$th component of the solution, $y_k$; that is, the computed solution is:

$$y_k={\sum ^{\prime }}_{i=1}^{k_1}c[i-1,k-1]T_{i-1}\left(x\right)\text{,}1\le k\le n\text{,}$$

where $T_i\left(x\right)$ is the Chebyshev polynomial of the first kind and ${\sum }^{\prime }$ denotes that the first coefficient, $c[0,k-1]$, is halved.

Raises

NagValueError

(errno $1$)

On entry, $kp=⟨value⟩$ and $k1=⟨value⟩$.

Constraint: $kp+1\ge k1$.

(errno $1$)

On entry, $k1=⟨value⟩$ and $max\left(m\right[i-1\left]\right)=⟨value⟩$.

Constraint: $k1\ge 1+max\left(m\right[i-1\left]\right)$.

(errno $1$)

On entry, one of the equations is of order less than $1$.

(errno $1$)

On entry, $x1=⟨value⟩$ and $x0=⟨value⟩$.

Constraint: $x1>x0$.

(errno $1$)

On entry, a negative number of boundary conditions was set for one of the system equations.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $3$)

Either the boundary conditions are not linearly independent, or the coefficient matrix is rank deficient. Increasing the number of collocation points may overcome this latter problem.

(errno $4$)

Iterative refinement in the least squares solution has failed to converge. The coefficient matrix is too ill-conditioned.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

bvp_coll_nth_comp calculates an approximate solution of a linear or linearized system of ordinary differential equations as a Chebyshev series. Suppose there are $n$ differential equations for $n$ variables $y_1,y_2,\dots ,y_n$, over the range $(x_0,x_1)$. Let the $i$th equation be

$$\sum _{j=1}^{m_i+1}\sum _{k=1}^n{f}_{kj}^i\left(x\right)y_k^{(j-1)}\left(x\right)=r^i\left(x\right)$$

where $y_k^{\left(j\right)}\left(x\right)=\frac{d^j{y}_k\left(x\right)}{d{x}^j}$. $coeff$ evaluates the coefficients $f_{kj}^i\left(x\right)$ and the right-hand side $r^i\left(x\right)$ for each $i$, $1\le i\le n$, at any point $x$. The boundary conditions may be applied either at the end points or at intermediate points; they are written in the same form as the differential equations, and specified by $bdyc$. For example the $j$th boundary condition out of those associated with the $i$th differential equation takes the form

$$\sum _{j=1}^{l_i+1}\sum _{k=1}^n{f}_{kj}^{ij}\left(x^{ij}\right)y_k^{(j-1)}\left(x^{ij}\right)=r^{ij}\left(x^{ij}\right)\text{,}$$

where $x^{ij}$ lies between $x_0$ and $x_1$. It is assumed in this function that certain of the boundary conditions are associated with each differential equation. This is for your convenience; the grouping does not affect the results.

The degree of the polynomial solution must be the same for all variables. You specify the degree required, $k_1-1$, and the number of collocation points, $k_p$, in the range. The function sets up a system of linear equations for the Chebyshev coefficients, with $n$ equations for each collocation point and one for each boundary condition. The collocation points are chosen at the extrema of a shifted Chebyshev polynomial of degree $k_p-1$. The boundary conditions are satisfied exactly, and the remaining equations are solved by a least squares method. The result produced is a set of Chebyshev coefficients for the $n$ functions $y_1,y_2,\dots ,y_n$, with the range normalized to $[-1,1]$.

fit.dim1_cheb_eval2 can be used to evaluate the components of the solution at any point on the range $[x_0,x_1]$. fit.dim1_cheb_deriv and fit.dim1_cheb_integ may be used to obtain Chebyshev series representations of derivatives and integrals (respectively) of the components of the solution.

References

Picken, S M, 1970, Algorithms for the solution of differential equations in Chebyshev-series by the selected points method, Report Math. 94, National Physical Laboratory