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Numerical Analysis Lecture 25
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Chapter 5 Interpolation
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Finite Difference Operators Newton’s Forward Difference
Finite Difference Operators Newton’s Forward Difference Interpolation Formula Newton’s Backward Difference Interpolation Formula Lagrange’s Interpolation Formula Divided Differences Interpolation in Two Dimensions Cubic Spline Interpolation
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Newton’s Forward Difference Interpolation Formula
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An alternate expression is
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NEWTON’S BACKWARD DIFFERENCE INTERPOLATION FORMULA
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The formula is,
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Alternatively form Here
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LAGRANGE’S INTERPOLATION FORMULA
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The Lagrange Formula for Interpolation
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Alternatively compact form
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Also
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Finally, the Lagrange’s interpolation polynomial of degree n can be written as
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DIVIDED DIFFERENCES
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Let us assume that the function y = f (x) is known for several values of x, (xi, yi), for i=0,1,..n. The divided differences of orders 0, 1, 2, …, n are now defined recursively as:
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The zero-th order divided difference
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The first order divided difference is defined as
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Second order divided difference
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Generally
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Standard format of the Divided Differences
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NEWTON’S DIVIDED DIFFERENCE INTERPOLATION FORMULA
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Let y = f (x) be a function which takes values y0, y1, …, yn corresponding to x = xi, i = 0, 1,…, n. We choose an interpolating polynomial, interpolating at x = xi, i = 0, 1, …, n in the following form
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Here, the coefficients ak are so chosen as to satisfy above equation by the (n + 1) pairs (xi, yi). Thus, we have
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The first equation gives
The second equation gives
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Third equation yields which can be rewritten as that is
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Thus, in terms of second order divided differences, we have
Similarly, we can show that
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Newton’s divided difference interpolation formula
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Newton’s divided differences can also be expressed in terms of forward, backward and central differences.
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Assuming equi-spaced values of abscissa, we have
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By induction, we can in general arrive at the result
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Similarly
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In general, we have
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Also, in terms of central differences, we have
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In general, we have the following pattern
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Example Find the interpolating polynomial by (i) Lagrange’s formula and (ii) Newton’s divided difference formula for the following data. Hence show that they represent the same interpolating polynomial. X 1 2 4 Y 5
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Solution The divided difference table for the given data is constructed as follows:
X Y 1st divided difference 2nd divided difference 3rd divided difference 1 2 1/2 -1/2 4 5 3/2 1/6
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(i) Lagrange’s interpolation formula gives
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(ii) Newton’s divided difference formula gives
We observe that the interpolating polynomial by both Lagrange’s and Newton’s divided difference formulae is one and the same.
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Note! Newton’s formula involves less number of arithmetic operations than that of Lagrange’s.
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Example Using Newton’s divided difference formula, find the quadratic equation for the following data. Hence find y (2). X 1 4 y 2
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Solution: The divided difference table for the given data is constructed as:
x y 1st divided difference 2nd divided difference 2 1 -1 1/2 4
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Now, using Newton’s divided difference formula, we have
Hence, y (2) = 1.
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Example A function y = f (x) is given at the sample points x = x0, x1 and x2. Show that the Newton’s divided difference interpolation formula and the corresponding Lagrange’s interpolation formula are identical.
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Solution For the function y = f (x), we have the data
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The interpolation polynomial using Newton’ divided difference formula is given as
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Using the definition of divided differences, we can rewrite the equation in the form
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On simplification, it reduces to
which is the Lagrange’s form of interpolation polynomial. Hence two forms are identical.
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Newton’s Divided Difference Formula with Error Term
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Following the basic definition of divided differences, we have for any x
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Multiplying the second Equation by (x – x0), third by (x – x0)(x – x1) and so on, and the last by (x – x0)(x – x1) … (x – xn-1) and adding the resulting equations, we obtain
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Please note that for x = x0, x1, …, xn, the error term vanishes
Where Please note that for x = x0, x1, …, xn, the error term vanishes
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Numerical Analysis Lecture 25
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