Interpolation Methods
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What is interpolation? How to interpolate a polynomial through a given set of data? General approach, Newton method, Lagrange method #WikiCourses https://wikicourses.wikispaces.com/Topic+Interpolation
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Interpolation Methods
- 2. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Understanding the difference between regression and interpolation • Knowing how to “best fit” a polynomial into a set of data • Knowing how to use a polynomial to interpolate data
- 7. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Curve Fitting • If the data measured is of high accuracy and it is required to estimate the values of the function between the given points, then, polynomial interpolation is the best choice. • If the measurements are expected to be of low accuracy (Real life data), or the number of measured points is too large, regression would be the best choice.
- 9. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Why Interpolation? • When the accuracy of your measurements are ensured • When you have discrete values for a function (numerical solutions, digital systems, etc …)
- 11. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com But, how to get the equation of a function that passes by all the data you have!
- 12. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Equation of a Line: Revision xaay 10 += If you have two points 1101 xaay += 2102 xaay += = 2 1 1 0 2 1 1 1 y y a a x x
- 13. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving for the constants! 12 12 1 12 2112 0 & xx yy a xx yxyx a − − = − − =
- 14. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com What if I have more than two points? • We may fit a polynomial of order one less that the number of points we have. i.e. four points give third order polynomial.
- 15. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Third-Order Polynomial 3 3 2 210 xaxaxaay +++= For the four points 3 13 2 121101 xaxaxaay +++= 3 23 2 222102 xaxaxaay +++= 3 33 2 323103 xaxaxaay +++= 3 43 2 424104 xaxaxaay +++=
- 16. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In Matrix Form = 4 3 2 1 3 2 1 0 3 4 2 24 3 3 2 23 3 2 2 22 3 1 2 11 1 1 1 1 y y y y a a a a xxx xxx xxx xxx Solve the above equation for the constants of the polynomial.
- 17. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • Find a 3rd order polynomial to interpolate the function described by the given points Yx 1-1 20 51 162
- 18. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution: System of equations • A third order polynomial is given by: ( ) 3 4 2 321 xaxaxaaxf +++= ( ) 11 4321 =−+−=− aaaaf ( ) 20 1 == af ( ) 51 4321 =+++= aaaaf ( ) 168422 4321 =+++= aaaaf
- 19. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In matrix form = −− 16 5 2 1 8421 1111 0001 1111 4 3 2 1 a a a a = 1 1 1 2 4 3 2 1 a a a a ( ) 32 2 xxxxf +++=
- 21. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton’s Method • In the previous procedure, we needed to solve a system of linear equations for the unknown constants. • This method suggests that we may just proceed with the values of x & y we have to get the constants without setting a set of equations • The method is similar to Taylor’s expansion without differentiation!
- 22. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For the two points ( )110 xxbby −+= ( ) 0111101 byxxbby =⇒−+= ( )1 12 12 1 xx xx yy yy − − − += ( )⇒−+= 12102 xxbby ( ) ( ) ( )12 12 112112 xx yy bxxbyy − − =⇒−+=
- 23. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Homework • Show that the polynomial obtained by solving a set of equations is equivalent to that obtained by Newton method
- 24. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For the three points ( ) ( ) ( )( )213 121 xxxxb xxbbxf −−+ −+= 10 yb = 12 12 1 xx yy b − − = 13 12 12 23 23 2 xx xx yy xx yy b − − − − − − =
- 25. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Using a table yixi y1x1 y2x2 y3x3 13 12 12 23 23 xx xx yy xx yy − − − − − − 12 12 xx yy − − 23 23 xx yy − −
- 26. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com In General • Newton’s Interpolation is performed for an nth order polynomial as follows ( ) ( ) ( )( ) ( ) ( )nn xxxxb xxxxbxxbbxf −−++ −−+−+= ...... 1 212110
- 27. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • Find a 3rd order polynomial to interpolate the function described by the given points using Newton’s method Yx 1-1 20 51 162
- 28. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton’s Method • Newton’s methods defines the polynomial in the form: ( ) ( ) ( )( ) ( )( )( )3213 212110 xxxxxxb xxxxbxxbbxf −−−+ −−+−+= ( ) ( ) ( )( ) ( )( )( )11 11 3 210 −++ ++++= xxxb xxbxbbxf
- 30. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Newton’s Method • Finally: ( ) ( ) ( )( ) ( )( )( )11 111 −++ ++++= xxx xxxxf ( ) ( ) ( ) ( )xxxxxxf −+++++= 32 11 ( ) 32 2 xxxxf +++=
- 31. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Advantage of Newton’s Method • The main advantage of Newton’s method is that you do not need to invert a matrix!
- 32. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Interpolation Lagrange Interpolation Polynomial
- 33. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Lagrange Method • First, we learned that a polynomial can pass by the points by using a simple polynomial with (n-1) terms. • Then, we learned a way that “looks like” the Taylor expansion (Newton’s method)
- 34. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Lagrange Method (cont’d) • Now, we will use polynomials that are zero at all points except the one we are evaluating at, but, in an easier form!
- 35. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For the two points ( ) ( )2211 xxcxxcy −+−= 2 12 1 1 21 2 y xx xx y xx xx y − − + − − = ( ) ( )2121111 xxcxxcy −+−= ( )21 1 2 xx y c − = ( ) ( )2221212 xxcxxcy −+−= ( )12 2 1 xx y c − =
- 36. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Two lines added! 2 12 1 1 21 2 y xx xx y xx xx y − − + − − =
- 37. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Homework • Show that the polynomial obtained by solving a set of equations is equivalent to that obtained by Lagrange method
- 38. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For the three points ( )( ) ( )( ) ( )( )133 322 211 xxxxc xxxxc xxxxcy −−+ −−+ −−= ( )( ) ( )( ) ( )( )11313 31212 211111 xxxxc xxxxc xxxxcy −−+ −−+ −−= ( )( )312121 xxxxcy −−= ( )( )3121 1 2 xxxx y c −− =
- 39. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Similarly ( )( )1323 3 1 xxxx y c −− = ( )( )3212 2 3 xxxx y c −− =
- 41. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Three parabolas added!!! 3 23 2 13 1 2 32 3 12 1 1 31 3 21 2 y xx xx xx xx y xx xx xx xx y xx xx xx xx y − − − − + − − − − + − − − − =
- 42. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • Find a 3rd order polynomial to interpolate the function described by the given points using Lagrange’s method Yx 1-1 20 51 162
- 45. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( ) 4 3 2 1 6 11 2 21 2 211 6 21 y xxx y xxx y xxx y xxx y −+ + − −+ + −−+ + − −− =
- 46. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution ( )( )( ) ( )( )( ) ( )( )( ) ( )( )( )16 6 11 5 2 21 2 2 211 1 6 21 −+ + − −+ + −−+ + − −− = xxx xxx xxx xxx y
- 47. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com • I will leave it for you to simplify the relation and obtain the same previous polynomial!