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