LECTURE 3: Polynomial interpolation and numerical differentiation

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6 x 10-10 5 4 3 2 1 0 0 5 10 15 20 25 30 35 Figure 2: The absolute error| f(x)− p(x)| between the interpolating polynomial p and the underlying function f at the 37 interpolation points.
2.9 Inverse interpolation A process called inverse interpolation can be used to approximate an inverse function. Suppose that the values y i = f(x i ) have been computed at x 0 ,x 1 ,...,x n . We can form the interpolation polynomial as p(y)= n i−1 ∑ c i ∏ i=0 j=0 (y−y j ) The original relationship y = f(x) has an inverse under certain conditions. This can be approximated by x= p(y). The procedures Coefficients and Evaluate can be used to carry out the inverse interpolation by reversing the arguments x and y. Inverse interpolation can be used to locate the root of a given function f ; i.e. we want to solve f(x)=0. First create a set of function values { f(x i ),x i } and find the interpolating polynomial p(y i )=x i . The approximate root is now given by r≈ p(y=0). Example. Having the following set of values y x -0.57892000 1.00000000 -0.36263700 2.00000000 -0.18491600 3.00000000 -0.03406420 4.00000000 0.09698580 5.00000000 We can use the interpolation program with x←y and y←x. This gives us the coefficients for the polynomial p(y). Evaluating p at point y=0, gives p(0)=4.24747001.
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 and 10: Example. Determine the quantities f
Page 11 and 12: 2.7 Recursive property of divided d
Page 13 and 14: The table with a completed second c
Page 15 and 16: The following procedure Coefficient
Page 17: Figure 1 shows a plot of the data p
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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