Computers in Civil Engineering 53:081 Spring 2003 Lecture #14 Interpolation.

Computers in Civil Computers in Civil EngineeringEngineering

53:081 Spring 200353:081 Spring 2003

Lecture #14Lecture #14

InterpolationInterpolation

Interpolation: OverviewInterpolation: Overview Objective: estimate intermediate values

between precise data points using simple functions

Solutions– Newton Polynomials– Lagrange Polynomials– Spline Interpolation

Interpolation Curve Fitting

x

y

multiple values

Curve need not go through data

points

x

y

single value

Curve goes through data

points

?)( ixy

?)( xf

xix

y

High-precision data points

ExampleExample

BraidwoodBraidwoodLaSalleLaSalle

DresdenDresden

QuadQuadCitiesCities

Jain's TMC-1

y = -5.594406E-18x4 - 1.100233E-13x3 + 3.470629E-08x2 - 1.251748E-03x + 1.691667E+01R2 = 0.9999

0

1

2

3

4

5

6

7

8

9

0 5000 10000 15000 20000 25000 30000 35000

Miss. R. Discharge (cfs)

Del

ta T

(deg

F)

DT

Poly. (DT)

Poly. (DT)

Quad-Cities Nuke Station Diffuser CurveQuad-Cities Nuke Station Diffuser Curve

Examples of Simple Examples of Simple PolynomialsPolynomials

Fist-order (linear) Second-order (quadratic) Third-order (cubic)

Newton’s Divided-Difference Newton’s Divided-Difference Interpolating PolynomialsInterpolating Polynomials

General comments Linear Interpolation Quadratic Interpolation General Form

Linear Interpolation Linear Interpolation FormulaFormula

)( 1xf

)(1 xf

)( 0xf

0x x 1x

01

01

0

01 )()()()(

xx

xfxf

xx

xfxf

)()(

)()()(

)()(

00

001

0101

xxmxf

xxxx

xfxfxfxf

By similar triangles:

Rearrange:

)(1 xfThe notation: means the first order interpolating polynomial

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

We know that: at x = 1 ln(x) =0 at x = e ln(x) =1 (e=2.718...)

Thus,

ExampleExampleProblem:

Solution:

58.0)12(1178.2

010

)12(1

)1()()1(

)()()(

)()( 001

0101

e

feff

xxxx

xfxfxfxf

General form:

Equivalent form:

To solve for ,three points are needed:

22102 )( xaxaaxf

))(()()( 1020102 xxxxbxxbbxf

Quadratic InterpolationQuadratic Interpolation

))(,(),(,()),(,( 221100 xfxxfxxfx

210 ,, bbb and

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

02

01

01

12

12

2

)()()()(

xx

xx

xfxf

xx

xfxf

b

01

011

)()(

xx

xfxfb

(1) ))(()()( 1020102 xxxxbxxbbxf

)( 00 xfb Set in (1) to find0xx

0bSubstitute in (1) and evaluate at to find:1xx

10 ,bbSubstitute in (1) and evaluate at to find:2xx

Quadratic InterpolationQuadratic Interpolation

Note: this looksNote: this lookslike a second like a second derivative…derivative…

ExampleExample

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

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

ProblemSolution

05.0

)()()()(

02

01

01

12

12

2

xx

xx

xfxf

xx

xfxf

b

58.017183.2

01

1

)1ln()ln(1

e

eb0)1ln()( 00 xfb

62.0))(()()2( 1020102 xxxxbxxbbf

How to Generalize This?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

)())(()()( 110010 nnn xxxxxxbxxbbxf

To solve for , n+1 points are needed:

))(,(,),(,()),(,( 1100 nn xfxxfxxfx

nbbb ,, 10

Solution

],,,,[

],,[

],[

)(

011

0122

011

00

xxxxfb

xxxfb

xxfb

xfb

nnn

General form of Newton’s General form of Newton’s Interpolating PolynomialsInterpolating Polynomials

What does this [ ] notation mean?

First finite divided difference:

nth finite divided difference:

Finite Divided DifferencesFinite Divided Differences

ji

jiji xx

xfxfxxf

)()(],[

ki

kjjikji xx

xxfxxfxxxf

],[],[

],,[

0

02111011

],,,[],,,[],,,,[

xx

xxxfxxxfxxxxf

n

nnnnnn

Second finite divided difference:

],,,[)())((

],,[))((],[)()(

01110

012100100

xxxfxxxxxx

xxxfxxxxxxfxxbxf

nnn

n

)(3

],[)(2

],,[],[)(1

],,,[],,[],[)(0

thirdsecondfirst)(

33

2322

1231211

01230120100

xfx

xxfxfx

xxxfxxfxfx

xxxxfxxxfxxfxfx

xfxi ii

],,,[)())((

],,[))((],[)()(

01110

012100100

xxxfxxxxxx

xxxfxxxxxxfxxbxf

nnn

n

Finite divided difference table, case n = 3:

3210 ,,, bbbb

Finite Divided DifferencesFinite Divided Differences

Divided Differences Pseudo CodeDivided Differences Pseudo Code

do i=0,n-1 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

)(3

],[)(2

],,[],[)(1

],,,[],,[],[)(0

thirdsecondfirst)(

33

2322

1231211

01230120100

xfx

xxfxfx

xxxfxxfxfx

xxxxfxxxfxxfxfx

xfxi ii

3210 ,,, bbbb

Example – ln(2) againExample – ln(2) again

)(3

],[)(2

],,[],[)(1

],,,[],,[],[)(0

thirdsecondfirst)(

33

2322

1231211

01230120100

xfx

xxfxfx

xxxfxxfxfx

xxxxfxxxfxxfxfx

xfxi ii

6094.153

1823.07918.162

0204.02027.03863.141

0079.00518.04621.0010

thirdsecondfirst)(

ii xfxi

4621.014

)1ln()4ln(],[ 01

xxf

0079.015

)0518.0(204.0],,,[ 0123

xxxxf

3210 ,,, bbbb

],,,[))()((

],,[))((

],[)()(

0123210

01210

0100

xxxxfxxxxxx

xxxfxxxx

xxfxxbxfn

6094.153

1823.07918.162

0204.02027.03863.141

0079.00518.04621.0010

thirdsecondfirst)(

ii xfxi

)6)(4)(1(0079.0)4)(1(0518.0)1(4621.0)( xxxxxxxfn

6289.0

)62)(42)(12(0079.0)42)(12(0518.0)12(4621.0)2(

nf

Newton Interpolation Pseudo CodeNewton Interpolation Pseudo Code

See the textbook!

Features of Newton Divided-Features of Newton Divided-Differences to get Interpolating Differences to get Interpolating

PolynomialPolynomial 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:

)()(1 xfxfR nnn

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

Relative Error As a Function of Relative Error As a Function of OrderOrder

0.5

50

-0.5

Order

Estimated error

Estimated error (reversed)

True Error

Error Example 18.5 in text

x f(x)=ln(x) 1 4 6 5 3 1.5 2.5 3.5

0 1.3863 1.7918 1.6094 1.0986 0.4055 0.9163 1.2528

Determine ln(2) using the following table

MATLAB function interp1 is very MATLAB function interp1 is very useful for thisuseful for this

Tuesday 15 AprilTuesday 15 April

Midterm 2Midterm 2