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Ecs 555 - Interpolation: Learning Outcomes

This document provides an overview of interpolation techniques. It discusses linear and polynomial interpolation methods. Specifically, it covers: - Linear and polynomial interpolation can be used to estimate values between known data points. - Polynomial interpolation fits a curve that passes through all given points. - Common polynomial interpolation methods include Newton's and Lagrange's polynomials. - Polynomials are commonly used for interpolation because they are easy to evaluate, differentiate, and integrate. - Examples are provided to demonstrate linear and quadratic interpolation.

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Siti Maimunah
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267 views16 pages

Ecs 555 - Interpolation: Learning Outcomes

This document provides an overview of interpolation techniques. It discusses linear and polynomial interpolation methods. Specifically, it covers: - Linear and polynomial interpolation can be used to estimate values between known data points. - Polynomial interpolation fits a curve that passes through all given points. - Common polynomial interpolation methods include Newton's and Lagrange's polynomials. - Polynomials are commonly used for interpolation because they are easy to evaluate, differentiate, and integrate. - Examples are provided to demonstrate linear and quadratic interpolation.

Uploaded by

Siti Maimunah
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ECS 555 - INTERPOLATION

LEARNING OUTCOMES;

At the end of the week,


students will be able:

 To evaluate the intermediate value between


precise data points using linear and polynomial
interpolation.
Adiza Jamadin
INTRODUCTION

• Interpolation is a procedure for


Interpolation estimating a value between known
values of data points.

The most common method is polynomial interpolation:


i) The Newton polynomial
ii) The Lagrange polynomial

f ( x )= a 0 +a1 x+a 2 x 2 +⋯+a n x n

 Polynomial interpolation is used when


the point determined are very precise.
The curve representing the behavior has
to pass through every point.
INTRODUCTION

Polynomials are the most common choice of interpolation because


they are easy to:
Evaluate
Differentiate, and
Integrate.
Newton’s Divided-Difference
Interpolating Polynomials
Linear Interpolation – Example 18.1
Linear Interpolation
Quadratic Interpolation
Quadratic Interpolation
- Example 18.2
Quadratic Interpolation - Example 18.2

f2(x) = b0 + b1(x- x0) + b2(x- x0)(x- x1)


= 0 + 0.4620981(2-1) + (-0.0518731)(2-4)
= 0.56584
Newton’s Divided-Difference
Interpolating Polynomials
Quadratic Interpolation
- Example 18.3

x0 = 1 f(x0) = 0
x1 = 4 f(x1) = 1.386294
x2 = 6 f(x2) = 1.791759
x3 = 5 f(x3) = 1.609438
Quadratic Interpolation
- Example 18.3 x0 = 1
x1 = 4
f(x0) =
f(x1) =
0
1.386294
x2 = 6 f(x2) = 1.791759
x3 = 5 f(x3) = 1.609438

f[x1, x0]= 0.4620981


f[x2, x1]= 0.2027326
f[x3, x2]= 0.1823216
Quadratic Interpolation
- Example 18.3
Exercise

The following data defines the sea-level


concentration of dissolved oxygen for fresh water
as a function of temperature:

Estimate o(27) using;


(a) Linear interpolation
(b) Newton’s interpolating polynomial

Note the exact result is 7.986 mg/L


Answer;
(a) Linear Interpolation

(b) Polynomial Interpolation

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