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Interpolation Methods
The document discusses interpolation and curve fitting, highlighting the differences between regression and interpolation, with guidance on how to fit polynomials to data sets. It includes methods such as Newton's and Lagrange's for deriving polynomial equations that pass through given data points. Additionally, it emphasizes when to use polynomial interpolation versus regression based on the accuracy of measurements and the number of data points.
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Interpolation Methods
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- 2. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Understandingthe 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
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- 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.
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- 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 …)
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- 11. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com But, howto get the equation of a function that passes by all the data you have!
- 12. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Equation ofa 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 forthe constants! 12 12 1 12 2112 0 & xx yy a xx yxyx a − − = − − =
- 14. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com What ifI 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 210xaxaxaay +++= 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 MatrixForm = 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 • Finda 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: Systemof 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 matrixform = −− 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 +++=
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- 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 thetwo 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 • Showthat 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 thethree 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 atable 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 • Finda 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
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- 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 ofNewton’s Method • The main advantage of Newton’s method is that you do not need to invert a matrix!
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- 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 thetwo 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 linesadded! 2 12 1 1 21 2 y xx xx y xx xx y − − + − − =
- 37. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Homework • Showthat 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 thethree 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 −− =
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- 41. Interpolation Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Three parabolasadded!!! 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 • Finda 3rd order polynomial to interpolate the function described by the given points using Lagrange’s method Yx 1-1 20 51 162
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- 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 • Iwill leave it for you to simplify the relation and obtain the same previous polynomial!