naginterfaces.library.ode.bvp_​ps_​lin_​grid_​vals

naginterfaces.library.ode.bvp_ps_lin_grid_vals(nip, x, f)

bvp_ps_lin_grid_vals interpolates from a set of function values on a supplied grid onto a set of values for a uniform grid on the same range. The interpolation is performed using barycentric Lagrange interpolation. bvp_ps_lin_grid_vals is primarily a utility function to map a set of function values specified on a Chebyshev Gauss–Lobatto grid onto a uniform grid.

For full information please refer to the NAG Library document for d02uw

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d02/d02uwf.html

Parameters

nipint

The number, $m$, of grid points in the uniform mesh $\hat{x}$ onto which function values are interpolated. If $nip=1$ then on successful exit from bvp_ps_lin_grid_vals, $fip\left[0\right]$ will contain the value $f\left(x_n\right)$.

xfloat, array-like, shape $(n+1)$

The grid points, $x_{\text{i}}$, for $\text{i}=0,1,\dots ,n$, at which the function is specified.

Usually this should be the array of Chebyshev Gauss–Lobatto points returned in bvp_ps_lin_cgl_grid().

ffloat, array-like, shape $(n+1)$

The function values, $f\left(x_{\text{i}}\right)$, for $\text{i}=0,1,\dots ,n$.

Returns

xipfloat, ndarray, shape $\left(nip\right)$

The evenly-spaced grid points, ${\hat{x}}_{\text{j}}$, for $\text{j}=1,2,\dots ,m$.

fipfloat, ndarray, shape $\left(nip\right)$

The set of interpolated values $\hat{f}\left({\hat{x}}_{\text{j}}\right)$, for $\text{j}=1,2,\dots ,m$. Here $\hat{f}\left({\hat{x}}_{\text{j}}\right)\approx f(x={\hat{x}}_{\text{j}})$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $\text{n}$ is even.

(errno $2$)

On entry, $nip=⟨value⟩$.

Constraint: $nip>0$.

Notes

bvp_ps_lin_grid_vals interpolates from a set of $n+1$ function values, $f\left(x_{\text{i}}\right)$, on a supplied grid, $x_{\text{i}}$, for $\text{i}=0,1,\dots ,n$, onto a set of $m$ values, $\hat{f}\left({\hat{x}}_{\text{j}}\right)$, on a uniform grid, ${\hat{x}}_{\text{j}}$, for $\text{j}=1,2,\dots ,m$. The image $\hat{x}$ has the same range as $x$, so that ${\hat{x}}_{\text{j}}=x_{\text{min}}+\left((\text{j}-1)/(m-1)\right)\times (x_{\text{max}}-x_{\text{min}})$, for $\text{j}=1,2,\dots ,m$. The interpolation is performed using barycentric Lagrange interpolation as described in Berrut and Trefethen (2004).

bvp_ps_lin_grid_vals is primarily a utility function to map a set of function values specified on a Chebyshev Gauss–Lobatto grid computed by bvp_ps_lin_cgl_grid() onto an evenly-spaced grid with the same range as the original grid.

References

Berrut, J P and Trefethen, L N, 2004, Barycentric lagrange interpolation, SIAM Rev. (46(3)), 501–517