naginterfaces.library.ode.bvp_​ps_​lin_​quad_​weights

naginterfaces.library.ode.bvp_ps_lin_quad_weights(n)

bvp_ps_lin_quad_weights obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points. This allows for fast approximations of integrals for functions specified on Chebyshev Gauss–Lobatto points on $[-1,1]$.

For full information please refer to the NAG Library document for d02uy

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d02/d02uyf.html

Parameters

nint

$n$, where the number of grid points is $n+1$.

Returns

wfloat, ndarray, shape $(n+1)$

The Clenshaw–Curtis quadrature weights, $w_{\text{i}}$, for $\text{i}=0,1,\dots ,n$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>0$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n$ is even.

Notes

bvp_ps_lin_quad_weights obtains the weights for Clenshaw–Curtis quadrature at Chebyshev points.

Given the (Clenshaw–Curtis) weights $w_{\text{i}}$, for $\text{i}=0,1,\dots ,n$, and function values $f_{\text{i}}=f\left(t_{\text{i}}\right)$ (where $t_{\text{i}}=-\cos (\text{i}\times \pi /n)$, for $\text{i}=0,1,\dots ,n$, are the Chebyshev Gauss–Lobatto points), then ${\int }_{-1}^{1}f\left(x\right)dx\approx {\sum }_0^n{w}_i{f}_i$.

For a function discretized on a Chebyshev Gauss–Lobatto grid on $[a,b]$ the resultant summation must be multiplied by the factor $(b-a)/2$.

References

Trefethen, L N, 2000, Spectral Methods in MATLAB, SIAM