id: 06575083
dt: j
an: 2016c.00971
au: Gordon, Sheldon P.; Yang, Yajun
ti: Deriving Simpson’s rule using Newton interpolation.
so: Math. Comput. Educ. 50, No. 1, 34-41 (2016).
py: 2016
pu: MATYC Journal, Old Bethpage, NY
la: EN
cc: N40 N50
ut: numerical analysis; numerical integration; Simpson’s rule; Newton
interpolation polynomial; approximation; quadratic polynomials; area
under a curve; error functions; intermediate value theorem
ci:
li:
ab: From the text: Almost every numerical integration technique is based on the
idea of fitting a series of polynomials to a function using successive
subsets of (usually horizontally uniformly spaced) points,
approximating the area under each portion of the graph of the function
with the area under the corresponding polynomial, and summing the
results. For instance, Simpson’s rule fits one quadratic function to
the first group of three points, then another quadratic to the next
group of three points, and so forth. Simpson’s rule is typically
derived in one of two ways, either by using the Lagrange interpolation
formula (particularly in numerical analysis courses) or, more
frequently (in calculus courses), by writing the quadratic in the form
$Q(x)=A+B(x-x_0)+C(x-x_0)^2$ for the first polynomial and then setting
up a system of linear equations in the three unknowns based on the
three points to solve for $A$, $B$, and $C$. Some relatively simple
algebra lets one solve this system of equations for the three
coefficients and then a simple integration of the resulting quadratic
function from $x=x_0$ to $x=x_2$ gives an approximation to the area
under the original curve from $x=x_0$ to$x=x_2$. When repeated over
successive sets of points, Simpson’s rule results. We now look at an
alternative approach to deriving Simpson’s rule, one that can be
easily extended to derive numerical integration methods of higher
degree and that seems to be more direct, more in the spirit of a
calculus course, and perhaps more elegant. It is based on Newton’s
forward difference interpolating formula.
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