lecture 7 - CE 30125 - Lecture 7 LECTURE 7 NEWTON FORWARD...

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CE 30125 - Lecture 7 p. 7.1 LECTURE 7 NEWTON FORWARD INTERPOLATION ON EQUISPACED POINTS • Lagrange Interpolation has a number of disadvantages The amount of computation required is large Interpolation for additional values of requires the same amount of effort as the first value (i.e. no part of the previous calculation can be used) • When the number of interpolation points are changed (increased/decreased), the results of the previous computations can not be used Error estimation is difficult (at least may not be convenient) • Use Newton Interpolation which is based on developing difference tables for a given set of data points • The degree interpolating polynomial obtained by fitting data points will be identical to that obtained using Lagrange formulae! • Newton interpolation is simply another technique for obtaining the same interpo- lating polynomial as was obtained using the Lagrange formulae x N th N 1 +
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CE 30125 - Lecture 7 p. 7.2 Forward Difference Tables • We assume equi-spaced points (not necessary) • Forward differences are now defined as follows: (Zero th order forward difference) (First order forward difference) x 0 f 1 = f(x 1 ) f f 2 = f(x 2 ) f 3 = f(x 3 ) f 0 = f(x 0 ) f N = f(x N ) x 1 x 2 x 3 x N x 0123 h = interval size N (i) 0 f i f i f i f i 1 + f i
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CE 30125 - Lecture 7 p. 7.3 (Second order forward difference) (Third order forward difference) ( k th order forward difference) • Typically we set up a difference table 2 f i f i 1 + f i 2 f i f i 2 + f i 1 +  f i 1 + f i = 2 f i f i 2 + 2 f i 1 + f i + = 3 f i 2 f i 1 + 2 f i 3 f i f i 3 + 2 f i 2 + f i 1 + + f i 2 + 2 f i 1 + f i + = 3 f i f i 3 + 3 f i 2 + 3 f i 1 + f i + = k f i k 1 f i 1 + k 1 f i =
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CE 30125 - Lecture 7 p. 7.4 • Note that to compute higher order differences in the tables, we take forward differences of previous order differences instead of using expanded formulae. • The order of the differences that can be computed depends on how many total data points, , are available data points can develop up to order forward differences 0 1 2 3 4 i f i f i 2 f i 3 f i 4 f i f o f o f 1 f o = 2 f o f 1 f o = 3 f o 2 f 1 2 f o = 4 f o 3 f 1 3 f o = f 1 f 1 f 2 f 1 = 2 f 1 f 2 f 1 = 3 f 1 2 f 2 2 f 1 = f 2 f 2 f 3 f 2 = 2 f 2 f 3 f 2 = f 3 f 3 f 4 f 3 = f 4 x o x N  N 1 + N th
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CE 30125 - Lecture 7 p. 7.5 Example 1 • Develop a forward difference table for the data given 02 - 7 45531 14 - 3 9 1 0 84 266 1 9 1 8 1 2 3 8 25 37 30 4 1 06 26 7 51 2 1 2 9 i x i f i f i 2 f i 3 f i 4 f i 5 f i
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CE 30125 - Lecture 7 p. 7.6 Deriving Newton Forward Interpolation on Equi-spaced Points •Summ a ry o f S t ep s • Step 1: Develop a general Taylor series expansion for about .
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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lecture 7 - CE 30125 - Lecture 7 LECTURE 7 NEWTON FORWARD...

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