Re: parabolic interpolation
greg wrote:
> Hello,
>
> sorry if this is a brain-dead question... I want to do second order
> polynomial (i.e. parabolic) interpolation, but after reading the help
> file for 'inter1' it appears that the only options are nearest
> nearest neighbor, linear, cubic spline and cubic. I assumed there
> would be a way to specify the order of the polynomial fit (i.e. 2).
> Am I missing something?
Cubics work nicely for interpolation because by specifying
continuity (each segment passes through the data points at
either end) and smoothness (where two segments meet, their
slopes match), you get just enough conditions to pin down
the 4 different coefficients you need in each segment. (Well,
almost: you end up with two free parameters, so you need
two extra conditions, usually something to do with the
beginning and end of the data).
If you wanted a different curve between every pair of
datapoints, higher-order polynomials have too many
free parameters, and quadratics too few. What properties
did you want an interpolating quadratic to have? Did
you want to give up smoothness? Group the data
in groups of 3 instead of 2 (3 points specify a unique
quadratic)? Something else?
If you do a search for "quadratic splines" you'll find a
little bit on the web, but they're rare.
- Randy
.
Relevant Pages
- Re: Secant method vs Newtons method
... It is found that the Secant Method is obtained using zero order NIH and higher order methods with increasingly higher order error terms are obtained by using First or Second Order NIH. ... The simple innovation which I am using in NIH, and which is indicated by the word Normalized, is that I am using for interpolation the two point Hermite Interpolating Polynomial interpolating the function and some of its derivatives on a normalized interval by simply mapping the bracketing interval to the normalized interval each iteration. ... This allows use of the same interpolation formula every step of the iterative search for the root by evaluating the root of the Hermite polynomial interpolating the inverse function and its first or even second derivatives on two points within the normalized interval. ... I have described this on my website www.numerical-algorithms.com under the section "Root Finding Using Inverse Normalized Hermite Interpolation" and present an example where I compute the roots of the Legendre polynomials. ...
(sci.math.num-analysis) - Re: Secant method vs Newtons method
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(comp.dsp) - Re: irreducible polynomial in Z_7[t] roots of which are primitive in GF(49)
... get a polynomial in Z_pwhose roots are primitives in GF, ... Then I suppose those quadratic factors are the irreducible polynomials ... It looks like Derek's list includes those quadratics whose roots are ...
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