naginterfaces.library.quad.dim1_​gauss_​recm

naginterfaces.library.quad.dim1_gauss_recm(n, mu)

Given the $2n+l$ moments of the weight function, dim1_gauss_recm generates the recursion coefficients needed by dim1_gauss_wrec() to calculate a Gaussian quadrature rule.

For full information please refer to the NAG Library document for d01te

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d01/d01tef.html

Parameters

nint

$n$, the number of weights and abscissae required.

mufloat, array-like, shape $(2\times n+1)$

$mu\left[i\right]$ must contain the value of the moment with respect to $x^i$ i.e., $\int w\left(x\right)x^{\text{i}}dx$, for $\text{i}=0,1,\dots ,2n$.

Returns

afloat, ndarray, shape $\left(n\right)$

Values helping define the three term recurrence used by dim1_gauss_wrec().

bfloat, ndarray, shape $\left(n\right)$

Values helping define the three term recurrence used by dim1_gauss_wrec().

cfloat, ndarray, shape $\left(n\right)$

Values helping define the three term recurrence used by dim1_gauss_wrec().

Raises

NagValueError

(errno $1$)

The number of weights and abscissae requested ($n$) is less than $1$: $n=⟨value⟩$.

(errno $2$)

The problem is too ill conditioned, it breaks down at row $⟨value⟩$.

Notes

dim1_gauss_recm should only be used if the three-term recurrence cannot be determined analytically. A system of equations are formed, using the moments provided. This set of equations becomes ill-conditioned for moderate values of $n$, the number of abscissae and weights required. In most implementations quadruple precision calculation is used to maintain as much accuracy as possible.

References

Golub, G H and Welsch, J H, 1969, Calculation of Gauss quadrature rules, Math. Comput. (23), 221–230