AML702 Applied Computational Methods
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Lecture 26
Newton's Polynomial
Interpolation
Quadratic Interpolation
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When we approximate a curve using a straight line it
leads to large errors
Therefore one of the possible ways to improve this is
to introduce second order polynomial to approximate
the curve. A convenient form to do this is given below
To determine the coefficients bi, we start substituting
x=x1, we get b1=f(x1). Evaluating this at x=x2, we
obtain
This is now evaluated
at x=x3 to obtain
Quadratic Interpolation
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We can make these observations:
b2 is actually representing the slope between x1 and
x2
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Thus the first 2 terms correspond to linear
interpolation between x1 and x2 in the quadratic
interpolation eqn.
The last term i.e.,
introduces
second order curvature into the formula.
The expression for b3 is similar to the finite difference
approximation for the second derivative
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General form of Newton’s Interpolating
Polynomial
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Now we can generalize to fit an (n-1)th order
polynomial to n data points as follows:
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To evaluate coefficients of (n-1)th order polynomial
we need n data points
Quadratic Newton Interpolation Polynomial
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The final form of higher order interpolation
polynomial is as follows
Where [ ] bracketed function evaluation are the divided
differences, like