EE3561_Unit 5(c)AL-DHAIFALLAH14361 EE 3561 : Computational Methods Unit 5 : Interpolation Dr. Mujahed AlDhaifallah (Term 351) Read Chapter 18, Sections 1-5

EE3561_Unit 5(c)AL-DHAIFALLAH14362 Introduction to Interpolation Introduction Interpolation problem Existence and uniqueness Linear and Quadratic interpolation Newton’s Divided Difference method Properties of Divided Differences

EE3561_Unit 5(c)AL-DHAIFALLAH14363 Introduction Interpolation was used for long time to provide an estimate of a tabulated function at values that are not available in the table. What is sin (0.15)? xsin(x) Using Linear Interpolation sin (0.15) ≈ True value( 4 decimal digits) sin (0.15)=

EE3561_Unit 5(c)AL-DHAIFALLAH14364 The interpolation Problem Given a set of n+1 points, find an n th order polynomial that passes through all points

EE3561_Unit 5(c)AL-DHAIFALLAH14365 Example An experiment is used to determine the viscosity of water as a function of temperature. The following table is generated Problem: Estimate the viscosity when the temperature is 8 degrees. Temperature (degree) Viscosity

EE3561_Unit 5(c)AL-DHAIFALLAH14366 Interpolation Problem Find a polynomial that fit the data points exactly. Linear Interpolation V(T)= 1.73 − T V(8)=

EE3561_Unit 5(c)AL-DHAIFALLAH14367 Existence and Uniqueness Given a set of n+1 points Assumption are distinct Theorem: There is a unique polynomial f n (x) of order ≤ n such that

EE3561_Unit 5(c)AL-DHAIFALLAH14368 Examples of Polynomial Interpolation Linear Interpolation  Given any two points, there is one polynomial of order ≤ 1 that passes through the two points. Quadratic Interpolation Given any three points there is one polynomial of order ≤ 2 that passes through the three points.

EE3561_Unit 5(c)AL-DHAIFALLAH14369 Linear Interpolation Given any two points, The line that interpolates the two points is Example : find a polynomial that interpolates (1,2) and (2,4)

EE3561_Unit 5(c)AL-DHAIFALLAH Quadratic Interpolation  Given any three points,  The polynomial that interpolates the three points is

EE3561_Unit 5(c)AL-DHAIFALLAH General n th order Interpolation Given any n+1 points, The polynomial that interpolates all points is

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Differences

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Table xF[ ]F[, ]F[,, ]F[,,,] x0x0 F[x0]F[x 0,x 1 ]F[x 0,x 1, x 2 ]F[x 0,x 1,x 2,x 3 ] x1x1 F[x 1 ]F[x 1,x 2 ]F[x 1,x 2,x 3 ] x2x2 F[x 2 ]F[x 2,x 3 ] x3x3 F[x 3 ]

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Table f(x i ) xF[ ]F[, ]F[,, ] Entries of the divided difference table are obtained from the data table using simple operations

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Table f(x i ) xF[ ]F[, ]F[,, ] The first two column of the table are the data columns Third column : first order differences Fourth column: Second order differences

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Table xF[ ]F[, ]F[,, ]

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Table xF[ ]F[, ]F[,, ]

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Table xF[ ]F[, ]F[,, ]

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Table xF[ ]F[, ]F[,, ] f 2 (x)= F[x 0 ]+F[x 0,x 1 ] (x-x 0 )+F[x 0,x 1,x 2 ] (x-x 0 )(x-x 1 )

EE3561_Unit 5(c)AL-DHAIFALLAH Two Examples xy Obtain the interpolating polynomials for the two examples xy What do you observe?

EE3561_Unit 5(c)AL-DHAIFALLAH Two Examples xY xY Ordering the points should not affect the interpolating polynomial

EE3561_Unit 5(c)AL-DHAIFALLAH Properties of divided Difference Ordering the points should not affect the divided difference

EE3561_Unit 5(c)AL-DHAIFALLAH Example  Find a polynomial to interpolate the data. xf(x)

EE3561_Unit 5(c)AL-DHAIFALLAH Example xf(x)f[, ]f[,, ]f[,,, ]f[,,,, ]

EE3561_Unit 5(c)AL-DHAIFALLAH Summary

EE3561_Unit 5(c)AL-DHAIFALLAH Inverse interpolation Error in polynomial interpolation Dr. Mujahed Al-Dhaifallah (Term 351)

EE3561_Unit 5(c)AL-DHAIFALLAH Inverse Interpolation …. One approach: Use polynomial interpolation to obtain f n (x) to interpolate the data then use Newton’s method to find a solution to

EE3561_Unit 5(c)AL-DHAIFALLAH Inverse Interpolation Alternative Approach …. Inverse interpolation: 1. Exchange the roles of x and y 2. Perform polynomial Interpolation on the new table. 3. Evaluate ….

EE3561_Unit 5(c)AL-DHAIFALLAH Inverse Interpolation x y y x

EE3561_Unit 5(c)AL-DHAIFALLAH Inverse Interpolation Question: What is the limitation of inverse interpolation The original function has an inverse y 1,y 2,…,y n must be distinct.

EE3561_Unit 5(c)AL-DHAIFALLAH Inverse Interpolation Example x123 y

EE3561_Unit 5(c)AL-DHAIFALLAH Errors in polynomial Interpolation  Polynomial interpolations may lead to large error (especially when high order polynomials are used) BE CAREFUL  When an nth order interpolating polynomial is used, the error is related to the (n+1) th order derivative

EE3561_Unit 5(c)AL-DHAIFALLAH th order Polynomial Interpolation

EE3561_Unit 5(c)AL-DHAIFALLAH Error in polynomial interpolation Dr. Mujahed Al-Dhaifallah (Term 351)

EE3561_Unit 5(c)AL-DHAIFALLAH Errors in polynomial Interpolation Theorem

EE3561_Unit 5(c)AL-DHAIFALLAH Example 1

EE3561_Unit 5(c)AL-DHAIFALLAH Interpolation Error Not useful. Why?

EE3561_Unit 5(c)AL-DHAIFALLAH Approximation of the Interpolation Error

Example  Given the following table of data EE3561_Unit 5(c)AL-DHAIFALLAH143639

EE3561_Unit 5(c)AL-DHAIFALLAH143640

EE3561_Unit 5(c)AL-DHAIFALLAH Divided Difference Theorem

EE3561_Unit 5(c)AL-DHAIFALLAH Summary  The interpolating polynomial is unique  Different methods can be used to obtain it Newton’s Divided difference Lagrange interpolation others  Polynomial interpolation can be sensitive to data.  BE CAREFUL when high order polynomials are used