LECTURE 3: Polynomial interpolation and numerical differentiation

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This can be easily programmed; the algorithm is capable of computing the divided differences at the cost of 1 2n(3n+1) additions/subtractions,(n−1)(n−2) multiplications, <strong>and</strong> n divisions. There is, however, a more refined method for calculating the divided differences.
2.7 Recursive property of divided differences The efficient method for calculating the divided differences is based on two properties of the divided differences. First, the divided differences obey the formula (the recursive property of divided differences) f[x 0 ,x 1 ,...,x k ]= f[x 1,x 2 ,...,x k ]− f[x 0 ,x 1 ,...,x k−1 ] x k − x 0 The proof is left to the student (see e.g., Cheney <strong>and</strong> Kincaid, pp. 134, Theorem 2). Second, because of the uniqueness of the interpolating polynomials, the divided differences are not changed if the nodes are permuted. This is expressed more formally by the following theorem. Invariance theorem. The divided difference f[x 0 ,x 1 ,...,x k ] is invariant under all permutations of the arguments x 0 ,x 1 ,...,x k . Thus, the recursive formula can also be written as f[x i ,x i+1 ,...,x j−1 ,x j ]= f[x i+1,x i+2 ,...,x j ]− f[x i ,x i+1 ,...,x j−1 ] x j − x i We can now determine the divided differences as follows: f[x i ]= f(x i ) f[x i+1 ]= f(x i+1 ) f[x i+2 ]= f(x i+2 ) f[x i ,x i+1 ]= f[x i+1]− f[x i ] x i+1 − x i f[x i+1 ,x i+2 ]= f[x i+2]− f[x i+1 ] x i+2 − x i+1 f[x i ,x i+1 ,x i+2 ] = f[x i+1,x i+2 ]− f[x i ,x i+1 ] x i+2 − x i
Page 1 and 2: LECTURE 3: Polynomial interpolation
Page 3 and 4: Note that when l i (x) is evaluated
Page 5 and 6: 2.4 Newton form The inductive proof
Page 7 and 8: 2.5 Nested multiplication For effic
Page 9: Example. Determine the quantities f
Page 13 and 14: The table with a completed second c
Page 15 and 16: The following procedure Coefficient
Page 17 and 18: Figure 1 shows a plot of the data p
Page 19 and 20: 2.9 Inverse interpolation A process
Page 21 and 22: We can simplify the notation by S i
Page 23 and 24: 3.1 Choice of nodes Contrary to int
Page 25 and 26: Example. In a previous example, we
Page 27 and 28: 4 Numerical differentiation We now
Page 29 and 30: 4.3 Richardson extrapolation theore
Page 31 and 32: 4.5 First-derivative formulas via i
Page 33 and 34: This allows us to write out the for

LECTURE 3: Polynomial interpolation and numerical differentiation

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