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Computers in Civil Engineering 53:081 Spring 2003PowerPoint Presentation

Computers in Civil Engineering 53:081 Spring 2003

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Interpolation: Overview

- Objective: estimate intermediate values between precise data points using simple functions
- Solutions
- Newton Polynomials
- Lagrange Polynomials
- Spline Interpolation

Interpolation

Curve Fitting

Curve goes through data points

single value

Curve need not go through data points

multiple values

Example

High-precision data points

Newton’s Divided-Difference Interpolating Polynomials

- General comments
- Linear Interpolation
- Quadratic Interpolation
- General Form

The notation: means the first order interpolatingpolynomial

Linear Interpolation FormulaBy similar triangles:

Rearrange:

Example

Problem:

Estimate ln(2)(the true value is 0.69)

Solution:

We know that:

at x = 1 ln(x) =0

at x = e ln(x) =1 (e=2.718...)

Thus,

Quadratic Interpolation

General form:

Equivalent form:

(f2(x) means second-order interpolating polynomial)

To solve for ,three points are needed:

Substitute in (1) and evaluate at to find:

Substitute in (1) and evaluate at to find:

Quadratic InterpolationNote: this looks

like a second

derivative…

Example

Problem

Estimate ln(2)(the true value is 0.69)

Solution

We know that:

at x = x0 = 1 ln(x) =0

at x = x1 = e ln(x) =1 (e=2.718...)

at x = x2= e2 ln(x) = 2

How to Generalize This?

It would get pretty tedious to do this for third,

fourth, fifth, sixth, etc order polynominal

We need a plan:

Newton’s Interpolating Polynomials

General form of Newton’s Interpolating Polynomials

To solve for , n+1 points are needed:

Solution

What does this [ ]

notation mean?

Finite Divided Differences

First finite divided difference:

Second finite divided difference:

nth finite divided difference:

Finite Divided Differences

Finite divided difference table, case n = 3:

fdd(i,1)=f(i)

enddo

do j=2,n

do i=1,n-j+1

fdd(i,j)=(fdd(i+1,j-1)-fdd(i,j-1))/

& (x(i+j-1)-x(i))

enddo

enddo

Divided Differences Pseudo CodeNewton Interpolation Pseudo Code

See the textbook!

Features of Newton Divided-Differences to get Interpolating Polynomial

- Data need not be equally spaced
- Arrangement of data does not have to be ascending or descending, but it does influence error of interpolation
- Best case is when the base points are close to the unknown value
- Estimate of relative error:

Error estimate for nth-order polynomial is the difference between the (n+1)th and nth-order prediction.

Relative Error As a Function of Order Polynomial

Example 18.5 in text

Determine ln(2) using the following table

MATLAB function interp1 is very useful for this

Midterm 2 Polynomial

Tuesday 15 April

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