MATH 174: Numerical Analysis Lecturer: Jomar F. Rabajante 1st Sem AY 2018-2019

INTERPOLATION THEORY We want to select a function p(x) from a given class of functions in such a way that the graph of y=p(x) passes through a finite set of given data points (nodes). We are going to start with POLYNOMIAL INTERPOLATION.

POLYNOMIAL INTERPOLATION Given a set of  n+1 data points (xi ,yi) where no two  xi  are the same, we want to find a polynomial  p  of degree at most  n  with the property p(xi)=yi , i=0,1,…,n

POLYNOMIAL INTERPOLATION UNISOLVENCE THEOREM (Existence and Uniqueness Theorem) Polynomial interpolation defines a linear bijection where {(xi ,yi) , i=0,1,…,n} ϵ 𝕂n+1. Пn is the vector space of polynomials (defined on any interval containing the nodes) of degree at most n.

POLYNOMIAL INTERPOLATION UNISOLVENCE THEOREM (Existence and Uniqueness Theorem) In other words, Given n+1 interpolation nodes (xi ,yi), where xi’s are distinct, there is a polynomial p(x) of degree less than or equal to n that interpolates yi at xi , i=0,1,…,n. This polynomial p(x) is unique among the set of all polynomials of degree at most n.

POLYNOMIAL INTERPOLATION Proof of existence and uniqueness of the interpolating polynomial: We can write p(x) as There are m+1 coefficients. By hypothesis, there are n+1 conditions on p(x), set m=n. We want to find the values of the coefficients: (This is a system of n+1 linear equations in n+1 unknowns. Solving it is equivalent to solving the polynomial interpolation problem.)

POLYNOMIAL INTERPOLATION Proof of existence and uniqueness of the interpolating polynomial We can write the system of linear equations in matrix form: Vandermonde matrix

POLYNOMIAL INTERPOLATION Proof of existence and uniqueness of the interpolating polynomial Denote the Vandermonde matrix as X. It can be shown using cofactor expansion that Since the points xi are distinct, Thus, X is nonsingular and the system of linear equation has a unique solution (for ai’s). This proves the existence and uniqueness of an interpolating polynomial of degree less than or equal to n.

POLYNOMIAL INTERPOLATION SOME WAYS OF CONSTRUCTING THE INTERPOLATING POLYNOMIAL UNIVARIATE General Case (can be used for unequally spaced data points) Lagrange Interpolation Newton’s Divided Differences Neville’s Method Special Case (for equally spaced data points) Forward Differences Backward Differences Centered Differences MULTIVARIATE OSCULATING POLYNOMIALS The outputs here are the same (using same set of nodes).

LAGRANGE INTERPOLATION Given the finite data points below, the Lagrange Formula is as follows: i xi yi x0 y0 1 x1 y1 . n-1 xn-1 yn-1 n xn yn

LAGRANGE INTERPOLATION Given the finite data points below, the Lagrange Formula is as follows: pn, being a linear combination of the polynomials li, is itself a polynomial. The formula is invariant under different order of the data set. i xi yi x0 y0 1 x1 y1 . n-1 xn-1 yn-1 n xn yn

LAGRANGE INTERPOLATION Example 1: Function Interpolation Representing continuous functions by interpolating polynomials Interpolate ln(2) using 2 equally-spaced points Assume that the given points are (rounded off up to 6 decimal places)

LAGRANGE INTERPOLATION Example 1: This is called linear interpolation. Try y-0 = 1.098612(x-1)/2 The interpolating polynomial

LAGRANGE INTERPOLATION Does this polynomial really passes through the given points? Assume xj is one of the given abscissas. The li functions obey the Kronecker delta equation: Try it here:

LAGRANGE INTERPOLATION Example 2: Data Interpolation Interpolating discrete data sets The viscosity of water has been experimentally determined at different temperatures, as indicated in the following table: From this table, how can we estimate a reasonable value for the viscosity at temperature 8 degrees? Use Polynomial Interpolation. Temp (deg) 5 10 15 Viscosity 1.792 1.519 1.308 1.140 Let’s try this on the board!