Top Banner
CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier
94

CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Dec 20, 2015

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

CURVE FITTING

ENGR 351 Numerical Methods for Engineers

Southern Illinois University Carbondale

College of EngineeringDr. L.R. Chevalier

Page 2: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Copyright © 2000 by Lizette R. Chevalier

Permission is granted to students at Southern Illinois University at Carbondaleto make one copy of this material for use in the class ENGR 351, NumericalMethods for Engineers. No other permission is granted.

All other rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,electronic, mechanical, photocopying, recording, or otherwise, withoutthe prior written permission of the copyright owner.

Page 3: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Applications

Food Available, S

Spe

cifi

c G

row

th R

ate,

Need to determine parameters for saturation-growth rate model to characterize microbial kinetics

SKS

s max

Page 4: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Applications

0

5

10

15

20

25

30

0 10 20 30

T (o C)

z (m

)

Epilimnion

Thermocline

Hypolimnion

Page 5: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

0

0.5

1

1.5

2

0 10 20 30

T(oC)

v, 1

0-2 c

m2/s

Applications

Interpolation of dataWhat is kinematic viscosity at 7.5º C?

T (oC) v, 10-2 (cm2/s)

0 1.79234 1.56158 1.387412 1.239616 1.116820 1.010524 0.9186

Page 6: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f(x)

x

We want to find the best “fit” of a curve through the data.

Page 7: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Mathematical Background

• The prerequisite mathematical background for interpolation is found in the material on the Taylor series expansion and finite divided differences

• Simple statisticsaverage

standard deviation

normal distribution

Page 8: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Normal Distribution

A histogram usedto depict the distributionsof an exam grade.

Page 9: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

x

x 2

95%

x 68%

Page 10: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Material to be Covered in Curve Fitting

• Linear RegressionPolynomial Regression

Multiple Regression

General linear least squares

Nonlinear regression

• InterpolationNewton’s Polynomial

Lagrange polynomial

Coefficients of polynomials

Page 11: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Specific Study Objectives

• Understand the fundamental difference between regression and interpolation and realize why confusing the two could lead to serious problems

• Understand the derivation of linear least squares regression and be able to assess the reliability of the fit using graphical and quantitative assessments.

Page 12: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Specific Study Objectives

• Know how to linearize data by transformation• Understand situations where polynomial, multiple

and nonlinear regression are appropriate• Understand the general matrix formulation of

linear least squares• Understand that there is one and only one

polynomial of degree n or less that passes exactly through n+1 points

Page 13: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Specific Study Objectives

• Realize that more accurate results are obtained if data used for interpolation is centered around and close to the unknown point

• Recognize the liabilities and risks associated with extrapolation

• Understand why spline functions have utility for data with local areas of abrupt change

Page 14: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Least Squares Regression

• Simplest is fitting a straight line to a set of paired observations(x1,y1), (x2, y2).....(xn, yn)

• The resulting mathematical expression isy = ao + a1x + e

• We will consider the error introduced at each data point to develop a strategy for determining the “best fit” equations

Page 15: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

S e y a a xri

n

i o ii

n

i

2

11

1

2

f(x)

x

y a a xi o i 1

Page 16: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

S e y a a xri

n

i o ii

n

i

2

11

1

2

To determine the values for ao and a1, differentiatewith respect to each coefficient

S

ay a a x

S

ay a a x x

r

oi o i

ri o i i

2

2

1

11

Note: we have simplified the summation symbols.What mathematics technique will minimize Sr?

Page 17: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

S

ay a a x

S

ay a a x x

r

oi o i

ri o i i

2

2

1

11

Setting the derivative equal to zero will minimizing Sr.If this is done, the equations can be expressed as:

0

0

1

12

y a a x

y x a x a x

i o i

i i o i i

Page 18: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

0

0

1

12

y a a x

y x a x a x

i o i

i i o i i

Note:

We have two simultaneous equations, with two unknowns, ao and a1.

What are these equations? (hint: only place terms with ao and a1 on the LHS of the equations)

What are the final equations for ao and a1?

a nao o

Page 19: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

na x a y

x a x a x y

an x y x y

n x x

a y a x

o i i

i o i i i

i i i i

i i

o

1

21

1 2 2

1

These first twoequations are calledthe normal equations

Page 20: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Error

S e y a a xri

n

i o ii

n

i

2

11

1

2

Recall:f(x)

x

S y y

sS

n

t i

yt

2

1

The most common measure of the “spread” of a sample is the standard deviation about the mean:

Page 21: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Introduce a term to measure the standard error of the estimate:

sS

nyx

r 2

Coefficient of determination r2:

rS S

St r

t

2

r is the correlation coefficient

Page 22: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

rS S

St r

t

2

The following signifies that the line explains 100 percent of the variability of the data:

Sr = 0 r = r2 = 1

If r = r2 = 0, then Sr = St and the fit is invalid.

Page 23: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Data 1 Data 2 Data 3 Data 410 8.04 10 9.14 10 7.46 8 6.588 6.95 8 8.14 8 6.77 8 5.7613 7.58 13 8.74 13 12.74 8 7.719 8.81 9 8.77 9 7.11 8 8.8411 8.33 11 9.26 11 7.81 8 8.4714 9.96 14 8.10 14 8.84 8 7.046 7.24 6 6.13 6 6.08 8 5.254 4.26 4 3.10 4 5.39 19 12.5012 10.84 12 9.13 12 8.15 8 5.567 4.82 7 7.26 7 6.42 8 7.915 5.68 5 4.74 5 5.73 8 6.89

Consider the following four sets of data

Page 24: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

y = 0.5001x + 3.0001

R2 = 0.66650

2

4

6

8

10

12

14

0 5 10 15

x

y

y = 0.5x + 3.0009

R2 = 0.66620

2

4

6

8

10

12

14

0 5 10 15

x

y

y = 0.4999x + 3.0017

R2 = 0.6667

0

2

4

6

8

10

12

14

0 5 10 15 20

x

y

y = 0.4997x + 3.0025

R2 = 0.6663

0

2

4

6

8

10

12

14

0 5 10 15

x

y

Page 25: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Linearization of non-linear relationships

Some data is simply ill-suited for linear least squares regression....

or so it appears.

f(x)

x

Page 26: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

P

t

ln P

t

slope = r

intercept = ln P0

P P eort

Lin

eari

ze

why?

EXPONENTIALEQUATIONS

Page 27: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

P P e

P P e

P e

P rt

rt

rt

rt

0

0

0

0

ln ln

ln ln

ln

slope = rintercept = ln Po

Can you see the similaritywith the equation for a line:

y = b + mx

where b is the y-intercept and m is the slope?lnP

t

Page 28: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

P P e

P P e

P e

P rt

rt

rt

rt

0

0

0

0

ln ln

ln ln

ln

ln P0

t

slope = rintercept = ln P0

After taking the natural logof the y-data, perform linearregression.From this regression:

The value of b will give usln (P0). Hence, P0 = eb

The value of m will give us rdirectly.

Page 29: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

POWER EQUATIONS

log Q

log H

Q

H

acHQ

Here we linearizethe equation bytaking the log ofH and Q data.What is the resultingintercept and slope?

(Flow over a weir)

Page 30: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Q cH

Q cH

c H

c a H

a

a

a

log log

log log

log log

log Q

log H

slope = a

intercept = log c

Page 31: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

So how do we getc and a fromperforming regressionon the log H vs log Qdata?From : y = mx + b

b = log c

c = 10b

m = a

log Q

log H

slope = a

intercept = log c

Q cH

Q cH

c H

c a H

a

a

a

log log

log log

log log

Page 32: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

S1/

1/ S

SATURATION-GROWTHRATE EQUATION

SKS

s max

slope = Ks/max

intercept = 1/max

Here, is the growth rate of a microbial population,max is the maximum growth rate, S is the substrate or food concentration, Ks is the substrate concentration at a value of = max/2

Page 33: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

General Comments of Linear Regression

• You should be cognizant of the fact that there are theoretical aspects of regression that are of practical importance but are beyond the scope of this book

• Statistical assumptions are inherent in the linear least squares procedure

Page 34: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

General Comments of Linear Regression

• x has a fixed value; it is not random and is measured without error

• The y values are independent random variable and all have the same variance

• The y values for a given x must be normally distributed

Page 35: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

General Comments of Linear Regression

• The regression of y versus x is not the same as x versus y

• The error of y versus x is not the same as x versus y f(x)

x

y-di

rect

ion

x-direction

Page 36: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Polynomial Regression

• One of the reasons you were presented with the theory behind linear regression was to allow you the insight behind similar procedures for higher order polynomials

• y = a0 + a1x

• mth - degree polynomialy = a0 + a1x + a2x2 +....amxm + e

Page 37: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Based on the sum of the squares of the residuals

S y a a x a x a xr i o i i m im 1 2

2 2......

1. Take the derivative of the above equation with respect to each of the unknown coefficients: i.e. the partial with respect to a2

S

ax y a a x a x a xr

i i o i i m im

2

21 2

22 .....

Page 38: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

2. These equations are set to zero to minimize Sr., i.e. minimize the error.

3. Set all unknowns values on the LHS of the equation. Again, using the partial of Sr. wrt a2

a x a x a x a x x yo i i i m im

i i2

13

24 2 2 .....

4. This set of normal equations result in m+1 simultaneous equations which can be solved using matrix methods to determine a0, a1, a2......am

Page 39: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Multiple Linear Regression

• A useful extension of linear regression is the case where y is a linear function of two or more variablesy = ao + a1x1 + a2x2

• We follow the same procedurey = ao + a1x1 + a2x2 + e

Page 40: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Multiple Linear Regression

For two variables, we would solve a 3 x 3 matrixin the following form:

n x x

x x x x

x x x x

a

a

a

y

x y

x y

i i

i i i i

i i i i

i

i i

i i

1 2

12

1 1 2

2 1 22

2

0

1

2

1

2

[A] and {c}are clearly based on data given for x1, x2 and y to solve for the unknowns in {x}.

Page 41: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Interpolation• General formula for an n-th order

polynomialy = a0 + a1x + a2x2 +....amxm

• For m+1 data points, there is one, and only one polynomial of order m or less that passes through all points

• Example: y = a0 + a1x

– fits between 2 points– 1st order

Page 42: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Interpolation

• We will explore two mathematical methods well suited for computer implementation

• Newton’s Divided Difference Interpolating Polynomials

• Lagrange Interpolating Polynomial

Page 43: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Newton’s Divided Difference Interpolating Polynomials

• Linear Interpolation

• Quadratic Interpolation

• General Form

• Errors

Page 44: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Linear InterpolationTemperature, C Density, kg/m3

0 999.9

5 1000.0

10 999.7

15 999.1

20 998.2

How would you approach estimating the density at 17 C?

Page 45: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Temperature, C Density, kg/m3

0 999.9

5 1000.0

10 999.7

15 999.1

20 998.2 ???999.1 > > 998.2

15 20

998.

2

999.

1

Page 46: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Assume a straight line between the known data.Then calculate the slope.T1520

1.9992.998

15 20

998.

2

999.

1

Page 47: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Assuming this linear relationship is constant,the slope is the same between the unknown point and a known point.

T1517

1.999

15 20

998.

2

999.

1

17

Page 48: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Therefore, the slope of one interval will equal theslope of the other interval.

T1517

1.999

1520

1.9992.998

Solve for

15 20

998.

2

999.

1

17

Page 49: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x f xf x f x

x xx xo1

1 0

1 00

998 2 999 1

20 15

998 2

20 17

. . .

Note: The notation f1(x) designates that this is a first orderinterpolating polynomial

Page 50: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f(x)

x

true solution

smaller intervalsprovide a better estimate

1

2

Page 51: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f(x)

x

true solution1

2

Alternative approach would be to include a third point and estimate f(x) from a 2nd order polynomial.

Page 52: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f(x)

x

true solution

Alternative approach would be to include a third point and estimate f(x) from a 2nd order polynomial.

Page 53: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Quadratic Interpolation

f x b b x x b x x x x2 0 1 0 2 0 1

Prove that this a 2nd order polynomial ofthe form:

f x a a x a x 0 1 22

Page 54: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x b b x b x b x b x x b xx b xx2 0 1 1 0 22

2 0 1 2 0 2 1

First, multiply the terms

f x a a x a x 0 1 22

f x b b x x b x x x x2 0 1 0 2 0 1

Collect terms and recognize that:

a b b x b x x

a b b x b x

a b

0 0 1 0 2 0 1

1 1 2 0 2 1

2 2

Page 55: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f(x)

x

x2, f(x2)

x1, f(x1)

x0, f(x0)

x, f(x)

b f x

bf x f x

x x

b

f x f x

x x

f x f x

x x

x x

0 0

11 0

1 0

2

2 1

2 1

1 0

1 0

2 0

Procedure for Quadratic Interpolation

Page 56: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Procedure for Quadratic Interpolation

b f x

bf x f x

x x

b

f x f x

x x

f x f x

x x

x x

0 0

11 0

1 0

2

2 1

2 1

1 0

1 0

2 0

Page 57: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Procedure for Quadratic Interpolation

b f x

bf x f x

x x

b

f x f x

x x

f x f x

x x

x x

0 0

11 0

1 0

2

2 1

2 1

1 0

1 0

2 0

f x b b x x b x x x x2 0 1 0 2 0 1

Page 58: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Example

998

998.5

999

999.5

1000

0 5 10 15 20

Temp

Den

sity

Temperature, C Density, kg/m3

0 999.9

5 1000.0

10 999.7

15 999.1

20 998.2

Include 10 degrees inyour calculation of thedensity at 17 degrees.

Page 59: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

General Form of Newton’s Interpolating Polynomials

for the nth-order polynomial

f x b b x x b x x x x x xn n n 0 1 0 0 1 1....

To establish a methodical approach to a solution definethe first finite divided difference as:

f x x

f x f x

x xi ji j

i j

,

Page 60: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x x

f x f x

x xi ji j

i j

,

if we let i=1 and j=0 then this is b1

b

f x f x

x x11 0

1 0

Similarly, we can define the second finite divided difference, which expresses both b2 and the difference of the first two divided difference

Page 61: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Similarly, we can define the second finite divided difference, which expresses both b2 and the difference of the first two divided difference

b

f x f x

x x

f x f x

x xx x

f x x xf x x f x x

x xi j ki j j k

i k

2

2 1

2 1

1 0

1 0

2 0

, ,, ,

Following the same scheme, the third divided difference is the difference of two second finite divided difference.

Page 62: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

This leads to a scheme that can easily lead to theuse of spreadsheets

i xi f(xi) first second third

0 x0 f(x0) f[x1,x0] f[x2,x1,x0] f[x3,x2,x1,x0]

1 x1 f(x1) f[x2,x1] f[x3,x2,x0]

2 x2 f(x2) f[x2,x3] 3 x3 f(x3)

Page 63: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x b b x x b x x x x x xn n n 0 1 0 0 1 1....

These difference can be used to evaluate the b-coefficients.

The result is the following interpolation polynomial calledthe Newton’s Divided Difference Interpolating Polynomial

f x f x f x x x x

f x x x x x x x x xn

n n n

0 1 0 0

1 0 0 1 1

, ....

, , ,

To determine the error we need an extra point.The error would follow a relationship analogous to the errorin the Taylor Series.

Page 64: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Lagrange Interpolating Polynomial

f x L x f x

L xx x

x x

n i ii

n

ij

i jjj i

n

0

0

wheredesignates the “product of”The linear version of this expression is at n=1

Page 65: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x L x f x

L xx x

x x

fx x

x xf x

x x

x xf x

n i ii

n

ij

i jjj i

n

0

0

11

0 10

0

1 01

Linear version: n=1

Your text shows you how to do n=2 (second order).What would third order be?

Page 66: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x L x f x

L xx x

x x

fx x x x x x

x x x x x xf x

n i ii

n

ij

i jjj i

n

0

0

31 2 3

0 1 0 2 0 30

.......

Page 67: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x L x f x

L xx x

x x

fx x x x x x

x x x x x xf x

x x x x x x

x x x x x xf x

n i ii

n

ij

i jjj i

n

0

0

31 2 3

0 1 0 2 0 30

0 2 3

1 0 1 2 1 31

.......

Note:x1 isnot being subtractedfrom the constantterm x or xi = x1 inthe numeratoror the denominatorj= 1

Page 68: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x L x f x

L xx x

x x

fx x x x x x

x x x x x xf x

x x x x x x

x x x x x xf x

x x x x x x

x x x x x xf x

n i ii

n

ij

i jjj i

n

0

0

31 2 3

0 1 0 2 0 30

0 2 3

1 0 1 2 1 31

0 1 3

2 0 2 1 2 32

......

Note:x2 isnot being subtractedfrom the constantterm x or xi = x2 inthe numeratoror the denominatorj= 2

Page 69: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x L x f x

L xx x

x x

fx x x x x x

x x x x x xf x

x x x x x x

x x x x x xf x

x x x x x x

x x x x x xf x

x x x x x x

x x x x x xf x

n i ii

n

ij

i jjj i

n

0

0

31 2 3

0 1 0 2 0 30

0 2 3

1 0 1 2 1 31

0 1 3

2 0 2 1 2 32

0 1 2

3 0 3 1 3 23

Note:x3 isnot being subtractedfrom the constantterm x or xi = x3 inthe numeratoror the denominatorj= 3

Page 70: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Example

998

998.5

999

999.5

1000

0 5 10 15 20

Temp

Den

sity

Temperature, C Density, kg/m3

0 999.9

5 1000.0

10 999.7

15 999.1

20 998.2

Determine the densityat 17 degrees.

Page 71: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

In fact, you can derive Lagrange directly fromNewton’s Interpolating Polynomial

f 2 17 119.964 839.244 279.496 998 776 .

f

f2

1

17 998 776

17 998 74

.

.Using Newton’sInterpolating Polynomial

Page 72: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Coefficients of an Interpolating Polynomial

f x b b x x b x x x x x xn n n 0 1 0 0 1 1....

y = a0 + a1x + a2x2 +....amxm

HOW CAN WE BE MORE STRAIGHT FORWARD IN GETTING VALUES?

Page 73: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

f x a a x a x

f x a a x a x

f x a a x a x

0 0 1 0 2 02

1 0 1 1 2 12

2 0 1 2 2 32

This is a 2nd order polynomial.

We need three data points.

Plug the value of xi and f(xi)directly into equations.

This gives three simultaneous equationsto solve for a0 , a1 , and a2

Page 74: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Example

998

998.5

999

999.5

1000

0 5 10 15 20

Temp

Den

sity

Temperature, C Density, kg/m3

0 999.9

5 1000.0

10 999.7

15 999.1

20 998.2

Determine the densityat 17 degrees.

Page 75: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Spline Interpolation

• Our previous approach was to derive an nth order polynomial for n+1 data points.

• An alternative approach is to apply lower-order polynomials to subset of data points

• Such connecting polynomials are called spline functions

• Adaptation of drafting techniques

Page 76: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Spline interpolation is an adaptation of thedrafting technique of using a spline to draw smooth curvesthrough a series of points

Page 77: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Linear Splines

f x f x m x x x x x

f x f x m x x x x x

f x f x m x x x x x

where

mf x f x

x x

n n n n n

ii i

i i

0 0 0 0 1

1 1 1 1 2

1 1 1 1

1

1

Page 78: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Quadratic Spline

112

1 cxbxa

222

2 cxbxa

332

3 cxbxa

442

4 cxbxa

Page 79: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

ExampleA well pumping at 250 gallons per minute has observationwells located at 15, 42, 128, 317 and 433 ft awayalong a straight line from the well.After three hours of pumping, the following drawdownsin the five wells were observed: 14.6, 10.7, 4.81.7 and 0.3 ft respectively.Derive equations of each quadratic spline.

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500

Distance from well

Dra

wdo

wn

Page 80: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Splines• To ensure that the mth derivatives are continuous

at the “knots”, a spline of at least m+1 order must be used

• 3rd order polynomials or cubic splines that ensure continuous first and second derivatives are most frequently used in practice

• Although third and higher derivatives may be discontinuous when using cubic splines, they usually cannot be detected visually and consequently are ignored.

Page 81: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Splines

• The derivation of cubic splines is somewhat involved

• First illustrate the concepts of spline interpolation using second order polynomials.

• These “quadratic splines” have continuous first derivatives at the “knots”

• Note: This does not ensure equal second derivatives at the “knots”

Page 82: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Quadratic Spline

1.The function must be equal at the interior knots. This condition can be represented as:

a x b x c f x

a x b x c f x

i i i i i i

i i i i i i

1 12

1 1 1 1

12

1 1

note: we are referencing the same x and f(x)

Page 83: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

a x b x c f x

a x b x c f x

i i i i i i

i i i i i i

1 1

21 1 1 1

12

1 1

This occurs between i = 2, n

Using the interior knots (n-1) this will provide2n -2 equations.

Page 84: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

2. The first and last functions must pass through the end points.

This will add two more equations.

a x b x c f x

a x b x c f xn n n n n n

1 02

1 0 1 0

2

We now have 2n - 2 +2 = 2n equations.

How many do we need?

Page 85: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

3. The first derivative at the interior knots must be equal.

This provides another n-1 equations for 2n + n-1 =3n -1.

We need 3n

2 21 1 1 1a x b a x bi i i i i

Page 86: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

4. Unless we have some additional information regarding the functions or their derivatives, we must make an arbitrary choice in order to successfully compute the constants.

5. Assume the second derivative is zero at the first point. The visual interpretation of this condition is that the first two points will be connected by a straight line. a1 = 0

Page 87: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Cubic Splines

• Third order polynomial

• Need n+1 = 3+1 = 4 intervals

• Consequently there are 4n unknown constants to evaluate

• What are these equations?

f x a x b x c x di i i i i 3 2

Page 88: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Cubic Splines

• The function values must be equal at the interior knots (2n -2)

• The first and last functions must pass through the end points (2)

• The first derivatives at the interior knots must be equal (n-1)

• The second derivatives at the interior knots must be equal (n-1)

• The second derivative at the end knots are zero (2)

Page 89: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Cubic Splines

• The function values must be equal at the interior knots (2n -2)

• The first and last functions must pass through the end points (2)

• The first derivatives at the interior knots must be equal (n-1)

• The second derivatives at the interior knots must be equal (n-1)

• The second derivative at the end knots are zero (2)

Page 90: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

SPECIAL NOTE

On the surface it may appear that a third order approximationusing splines would be inferior to higher order polynomials.

Consider a situation where a spline may perform better:

A generally smooth function undergoes an abrupt change in a region of interest.

Page 91: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

The abrupt changeinduces oscillationsin interpolating polynomials.

In contrast,the cubic splineprovides amuch moreacceptableapproximation

Page 92: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Previous Exam Question

Given the following data, develop the simultaneous equations for a quadratic spline. Express your final answers in matrix form.

x f(x)1 0.504 4.606 1.507 3.00

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 1 2 3 4 5 6 7 8

x

f(x)

Page 93: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0 1 2 3 4 5 6 7 8

x

f(x)

Interior knots:16a1 + 4b1 + c1 = 4.616a2 + 4b2 + c2 = 4.6

36a2 + 6b2 + c2 = 1.536a3 + 6b3 + c3 = 1.5

End conditionsa1 + b1 + c1 = 0.549a3 + 7b3 + c3 = 3.0

First derivative cot. at interior knots8a1 + b1 = 8a2 + b2

12a2 + b2 = 12a3 + b3

Extra equationa1 =0

x f(x)1 0.504 4.606 1.507 3.00

(4, 4.6)

Page 94: CURVE FITTING ENGR 351 Numerical Methods for Engineers Southern Illinois University Carbondale College of Engineering Dr. L.R. Chevalier.

Interior knots:16a1 + 4b1 + c1 = 4.6 16a2 + 4b2 + c2 = 4.636a2 + 6b2 + c2 = 1.5 36a3 + 6b3 + c3 = 1.5

4 1 0 0 0 0 0 0

0 0 16 4 1 0 0 0

0 0 36 6 1 0 0 0

0 0 0 0 0 36 6 1

1 1 0 0 0 0 0 0

0 0 0 0 0 49 7 1

1 0 8 1 0 0 0 0

0 0 12 1 0 12 1 0

4.6

4.6

15

15

0 5

3 0

0

0

1

1

2

2

2

3

3

3

b

c

a

b

c

a

b

c

.

.

.

.

End conditionsa1 + b1 + c1 = 0.5 49a3 + 7b3 + c3 = 3.0First derivative cont. at interior knots8a1 + b1 = 8a2 + b2 12a2 + b2 = 12a3 + b3

Extra equationa1 =0