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Finite Difference Weights Using The Modified Lagrange Interpolant | | Burhan Sadiq
; Divakar Viswanath
; | | Date: |
16 Feb 2011 | | Abstract: | Let $z_{1},z_{2},ldots,z_{N}$ be a sequence of distinct grid points. A
finite difference formula approximates the $m$-th derivative $f^{(m)}(0)$ as
$sum w_{i}fleft(z_{i}
ight)$, with $w_{i}$ being the weights. We give two
algorithms for finding the weights $w_{i}$ either of which is an improvement of
an algorithm of Fornberg (emph{Mathematics of Computation}, vol. 51 (1988), p.
699-706). The first algorithm, which we call the direct method, uses fewer
arithmetic operations than that of Fornberg by a factor of $4/(5m+5)$.
The order of accuracy of the finite difference formula for $f^{(m)}(0)$ with
grid points $hz_{i}$, $1leq ileq N$, is typically
$mathcal{O}left(h^{N-m}
ight)$. However, the most commonly used finite
difference formulas have an order of accuracy that is higher than the typical.
For instance, the centered difference approximation
$left(f(h)-2f(0)+f(-h)
ight)/h^{2}$ to $f’’(0)$ has an order of accuracy
equal to $2$ not $1$ . Even unsymmetric finite difference formulas can have
such boosted order of accuracy, as shown by the explicit algebraic condition
that we derive. If the grid points are real, we prove a basic result stating
that the order of accuracy can never be boosted by more than $1$. | | Source: | arXiv, 1102.3203 | | Services: | Forum | Review | PDF | Favorites | |
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