Numerical methods. Note 7
|
[ Home ]
|
Interpolation and Extrapolation. |
Numerical Methods. Note
« 7
»
|
In engineering and science one often has a number of data points, as
obtained by sampling or some experiment, and tries to construct a function
which closely fits those data points. This is called curve fitting
(cf linear least-squares fit). Interpolation is a specific case of curve
fitting, in which the function must go exactly through the data points.
Polynomial interpolation.
Lagrange interpolating polynomial Pn-1(x) is the polynomial of the
order n-1 that passes through the n given points (xi,yi),
|
P(x) = ∑i yi ∏k≠i
(x-xk)/(xi-xk) .
|
Might show erratic behaviour for higher degree polynomials. Not useful
for n larger than, say, 5.
Rational function interpolation. The rational function R(x) is a
ratio,
R(x)=Pm(x)/Qk(x) ,
of two polynomials Pm(x) and Qk(x).
It has potentially larger range of conversion than polynomial. Still dangerous
for higher orders.
Spline interpolation.
At the interval between the tabulated points i and i+1 the function is
approximated by a polynomial:
|
y[i](x) = yi + bi (x-xi) + ci (x-xi)2
+ di (x-xi)3 + ... .
|
The coefficients bi, ci, di,... are calculated from the
continuity equations for the function and its derivatives at the
boundaries of the intervals,
|
y[i-1](xi)=y[i](xi),
y[i-1]'(xi)=y[i]'(xi),
y[i-1]''(xi)=y[i]''(xi), ...
|
For the quadratic spline the equations become (hi=xi+1-xi)
yi+bihi+cihi2 = yi+1 (continuity of the function at
the (i+1)st point),
bi + 2cihi = bi+1 (continuity of the derivative)
|
The most popular spline is the cubic spline. Here are the fortran routines
from netlib.org: spline.f - for constructing the
spline (calculating coefficients b,c,d) and
seval.f - for the very interpolation.
Integration and differentiation of tabulated functions.
First interpolate the tabulated data and then integrate or differentiate
the intepolating function.
Problems
-
Make a function locate(x,xt[]) which, given an ordered array xt[]
and a number x, finds index j such that xt[j] <= x <= xt[j+1].
-
Make a function polinterp(x,xt[],yt[],k) which does a polynomial
interpolation of a table (xt[],yt[]) at a point x using k nearest
tabulated points.
- Make a function for the quadratic spline interpolation.
-
Make a Lagrange polynomial interpolation (using all points) and a spline
interpolation of the following tabulated data:
|
{sinh(1.5 i/5), 1/cosh2(1.5 i/5)}, i=-5..5
|
Make a plot of the results.