 Computational Methods in Physics PHYS 3437

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Computational Methods in Physics PHYS 3437
Dr Rob Thacker Dept of Astronomy & Physics (MM-301C)

Today’s Lecture Interpolation & Approximation I Overview of ideas
Interpolation vs fitting Polynomial interpolation Lagrange interpolation Hermite interpolation Cubic splines (introduction)

General idea of interpolation
Suppose we are given data at discrete points only, and we wish to estimate values between these known points x1 x2 x3 x4 x5 f(x) X

Difference between interpolation & fitting
Interpolation passes through each datum Fitting seeks to produce a curve that approximates the data x1 x2 x3 x4 x5 f(x)

Global versus piecewise interpolation
We can fit a single polynomial of degree n to n+1 data points Alternatively we can use a series of polynomials to link points together – frequently called splines Splines will usually be cubic polynomials – I’ve used linear here to emphasize the piecewise nature x1 x2 x3 x4 x5 f(x)

Lagrange interpolation
To fit a polynomial of degree n we need n+1 points Consider first interpolating between two points, f(x2),f(x3) using a function F(x) The Taylor expansion about a point x gives f(x2)=F(x)+(x2-x)F’(x)+… f(x3)=F(x)+(x3-x)F’(x)+… (I’ve highlighted what we know) If we use a polynomial of degree 1, p(x), then the Taylor series truncates after p’(x) f(x2)=p(x)+(x2-x)p’(x) (1) f(x3)=p(x)+(x3-x)p’(x) (2)

Eliminate p’(x) After a short derivation (ex) we find that eqns (1) and (2) give On varying the expansion point x, but keeping fixed end points x2,x3 this corresponds to a straightline – as expected (check the following for yourself): If x=x3 => p(x)=f(x3) If x=x2 => p(x)=f(x2) Linear fits aren’t that helpful though, and bringing in more points allows us to up the order of the interpolation

Quadratic fit using three points
If we again Taylor expand put assume the function is a polymial, p(x) of degree 2, then f(x1)=p(x)+(x1-x)p’(x)+(x1-x)2p’’(x)/ (3) f(x2)=p(x)+(x2-x)p’(x)+(x2-x)2p’’(x)/ (4) f(x3)=p(x)+(x3-x)p’(x)+(x3-x)2p’’(x)/ (5) With a bit more algebra (fairly lengthy) we can eliminate both p’(x) and p’’(x) using (3),(4),(5) to get You should see a pattern emerging here…

General (Lagrange) Polynomial fit
If we have n points, a polynomial of degree n-1 can be constructed using the formula This is called the interpolation polynomial in Lagrange form, but it is frequently referred to as the Lagrange polynomial

Properties of the lj,n(x)
The lj,n(x) are clearly polynomials, and a called basis polynomials since they sum to find the interpolating polynomial If we let x=xi in the formula (so that we are picking one of the interpolation points) then

Problems with polynomial interpolation
As the number of points increases so does the degree of the polynomial This implies many turns and a lot of “structure” between interpolation points in high order polynomials High order polynomials will have a lot of “wiggles” and if points do not change smoothly they can “overshoot” in between datums Cubic fits tend to be about optimal 4 data points can surround x symmetrically (2 on each side) Increasing the number of points adds more data from further away from a given interpolation point This doesn’t necessarily improve the accuracy of the fit locally

Example of “wiggles” Piecewise linear fit 6th order polynomial fit
From

Segue - numerically implementing the general formula
Implementing the general formula is not exactly trivial It helps to notice that the lj,n(x) can be written as a product One can then think about using a loop to do the multiplication of the terms in the product However, the best way to calculate large interpolation polynomials is to use Neville’s Algorithm

Segue - Neville’s Algorithm
We won’t go into the details, however you can take a look (if you are interested) at Numerical Recipes section 3.1 Neville’s algorithm is better because it builds up the interpolation polynomial in a recursive fashion It uses a similar table idea to building up coefficients in a binomial expansion The employed recursion cuts down the total number of operations required and helps reduce concerns about rounding error

Hermite Interpolation
If we have both the function value, and the derivative we can fully constrain a higher order polynomial through two points f(x) f(x2) & f’(x2) f(x1) & f’(x1) x1 x x2 (x1<x<x2)

Interpolating a cubic to two points
Given the general form for a cubic So we have 4 equations with 4 unknowns to solve.

Solve for p(x) While again somewhat lengthy, we can solve for p(x) in terms of the known values (subscript denotes for x1,x2) One important property of Hermite interpolation is that if we fit a different polynomial p2,3(x) between x2 & x3 then it will have the same derivative at x2 as p1,2(x2) i.e. p’1,2(x2)=p’2,3(x2) Interpolation between successive pairs of points is continuous The derivatives are also continuous Hermite interpolation is thus considered to be smooth

Diagram of two Hermite interpolation polynomials
Second polynomial fit p2,3(x) First polynomial fit p1,2(x) f(x) f(x3) & f’(x3) f(x1) & f’(x1) f(x2) & f’(x2) Derivatives match at x2 x1 x2 x3

Advantages of Hermite interpolation over Lagrange interpolation
A series of H.I. polynomials will be continuous at the datums, as will the derivatives (i.e. smooth) While a series of L.I. polynomials is continuous at the datums the derivatives will not be continuous H.I. can fit a cubic to `local’ data (i.e. x1<x<x2) L.I. requires four points to fit a cubic (x1<x2<x<x3<x4) However, even if we don’t have derivatives it would be nice to fit a series of polynomials with continuous slopes at each datum

Cubic Splines Let’s examine how we can create a series of cubic polynomials with equal derivatives at the datums Choose x, such that xj<x<xj+1 where j=1,…,n Around x, we may write p(x) as

Next consider derivatives
We just derived two equations (3) & (2) for the value of the cubic fit at xj and xj+1 Taking the first & second derivative of p(x) we find If we again equate at xj and xj+1 but using 2nd deriv:

Next find the cj We’ve just shown

Substitute back to get p(x)
We now substitute (2),(4),(5) and (6) back into our first equation for p(x) to get To get any further we now need some information about p’’j+1 and p’’j

Setting p´´j+1 and p´´j f(x) f(x3) f(x1) f(x2) x1 x2 x3
We have no information about the derivative of f(x) though We just have the series of f(x) values What requirements can we impose? Recall we want the derivatives of neighbouring polynomials to match at the datums (xj) f(x) f(x3) f(x1) f(x2) Derivatives match at x2 x1 x2 x3

Apply continuity in the derivative
We now have to consider adjacent polynomials Label: pj,j+1 to be the fit between xj and xj+1 pj-1,j to be the fit between xj-1 and xj Continuity of the derivative implies that p´j,j+1 and p´j-1,j should be equal at xj, the general p’ forms are:

Equate at xj Setting p´j,j+1(xj)=p´j-1,j(xj) the first few terms of (8) vanish, while terms in (x-xj-1) in (9) cancel with hj-1: Where j=2,3,…,n-1

Finding the p´´j values
We now have a system of n-2 equations Recall limits of j=2,3,…,n-1 There are n unknown values though The p’’j values range through j=1,…,n To have enough information to solve for all the values we have to provide information about the end points – two cases Specify derivatives at end points “Clamped spline” Set p’’1=0 and p’’n=0 “Natural spline”

Summary Interpolation shouldn’t be confused with fitting, although the terms are frequently used interchangeably Lagrange interpolation will fit an order n polynomial to n+1 data points Hermite interpolation uses derivatives to solve for a cubic fit with only 2 datums Cubic spline interpolation gives a smooth piecewise interpolation scheme but requires conditions to be set for the end points

Next Lecture Matrix forms for spline fits Tridiagonal solvers
Clamped spline Natural spline Tridiagonal solvers

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