Polynomial Approximation. PSCI 702 October 05, 2005. What is a Polynomial?. Functions of the form: Polynomial of degree n, having n+1 terms. Will take n(n+1)n/2 multiplications and n additions. Can be re-written to take n additions and n multiplications. Factored form: N roots.
October 05, 2005
Newton’s method - tangent line
x2 = x1 - f(x1)/f `(x1)
or abs(x2 - x1) < tolerance,
If not, x1 x2, repeat the iteration.
Examples of poor convergence
y = ln x
fzero(function, [x0 x1])
Interpolation uses the data to approximate a function, which will fit all of the data points. All of the data is used to approximate the values of the function inside the bounds of the data.
All interpolation theory is based on polynomial approximation.
Solution: for each i=1,...,k
find a polynomial pi(x) that takes on the value yi at xi, and is zero for all other instances of
Hermite Polynomials produce a smooth interpolation, they have a disadvantage that the slope of the input function must be specified at each breakpoint.
Cubic Splines interpolation use only the data points used to maintaining the desired smoothness of the function and is piecewise continuous.
Polynomial are not always the best match of data. A rational function can be used to represent the steps. A rational function is a ratio of two polynomials. This is useful when you deal with fitting imaginary functions z=x + iy. The Bulirsch-Stoer algorithm creates a function where the numerator is of the same order as the denominator or 1 less.
The Rational Function interpolation are required for the location and function value need to be known.
The Legendre polynomials are a set of orthogonal functions, which can be used to represent a function as components of a function.
These function are orthogonal over a range [-1, 1 ]. This range can be scaled to fit the function. The orthogonal functions are defined as: