Interpolation handout

I NTRODUCTIONA census of the population of the USA is taken every 10 years.. The following table lists the population, in thousands of people, from 1950 to 2000, and the data are also re

INTERPOLATION AND POLYNOMIAL

APPROXIMATION

E LECTRONIC VERSION OF LECTURE

Dr Lê Xuân Đại

HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics

Email: ytkadai@hcmut.edu.vn

HCMC — 2019.

O UTLINE

2 L AGRANGE I NTERPOLATING P OLYNOMIALS

3 N EWTON ’ S D IVIDED D IFFERENCE M ETHOD

4 C UBIC S PLINE I NTERPOLATION

I NTRODUCTION

A census of the population of the USA is

taken every 10 years The following table

lists the population, in thousands of people, from 1950 to 2000, and the data are also

represented in the figure

In the reviewing these data, we might ask whether they could be used to provide a

reasonable estimate of the population, for instance, in 1975 or even in the year 2020 Predictions of this type can be obtained by using a function that fits the given data.

This process is called interpolation

One of the most useful and well-known

classes of functions mapping the set of real numbers into itself is the algebraic

form P n (x) = a n x n + a n−1 x n−1 + + a1x + a0,

where nis a non-negative integer and

a0, a1, , a n are real constants.

The polynomial which satisfies

P n (x i ) = y i , i = 0,1,2, ,n is called

Consider the construction of a polynomial

of degree at most n

y = P n (x) = a n x n + a n−1 x n−1 + + a1x + a0that passes through the (n + 1) points

M i (x i , y i ), i = 0,1,2, ,n.

THEOREM 1.1

The interpolation polynomial P n (x) for f (x),

is unique.

Construct a interpolation polynomial for

y = f (x), which is defined by the table

x 0 1 3

y 1 -1 2

Let y = f (x) be given by the table:

EXAMPLE 2.1

Construct the Lagrange interpolating

polynomial for y = sin(πx) using the numbers called nodesx0= 0, x1= 16, x2= 12

EXAMPLE 2.2

Let y be defined by x 0 1 3 4

y 1 1 2 -1 Use the Lagrange interpolating polynomial to

approximate the value of y when x = 2.

Let function f (x) be defined by the table

which is called the first divided difference of

f with respect to x k andx k+1

The second divided difference of f in

The first divided difference of f in [x, x0]is

Continuing the process to step n, we have

DEFINITION 3.2

The formula N (1)

n (x) is called the Newton forward divided-difference formula starting from x0 off and R n (x) is called the error of the interpolating polynomial.

Similarly, we can construct the Newton

forward divided-difference formula starting from x n of f as following

EXAMPLE 3.2

Let y = f (x) be defined by the table of data

x 0 2 3 5 6

y 1 3 2 5 6

1 Construct Newton forward

divided-difference formula, starting from

x0 of functiony = f (x)

2 Using the obtained interpolating

polynomial, estimate f (1.25)

Therefore, the Newton forward

DEFINITION 4.1

Given a function f defined on [a, b] and a set of nodes

a = x0< x1< x2= b Let y0= f (x0), y1= f (x1), y2= f (x2 ).A cubic spline interpolant for f on [a, b] is a function g that satisfies the following conditions:

1 g(x) has continuous derivatives to the second

Consider [x0, x1] Let h0= x1− x0 Sinceg0(x) is the cubic polynomial then

b0= y1− y0

h0 − c0h0− d0h20

Consider [x1, x2] Let h1= x2− x1 Sinceg1(x) is the cubic polynomial then

b1= y2− y1

h1 − c1h1− d1h21

Because the first and the second derivatives

Because the first and the second derivatives

N ATURAL C UBIC S PLINE

The boundary condition for natural cubic spline is

Then we can define the coefficients of g0

EXAMPLE 4.1

Construct a natural cubic spline for a

function defined by the table x 0 2 5

y 1 1 4

h0= 2, h1= 3 Because this is a natural cubic spline so c0= c2= 0 The coefficient c1 can be defined by the formula

Then we can define the coefficients of g0

Therefore, the required natural cubic spline is

C LAMPED C UBIC S PLINE

The boundary condition for clamped cubic spline is

Then we will have 2 equations

The coefficients of g k (x) (k = 0,1) can be

found by the formula

EXAMPLE 4.2

Construct the clamped cubic spline for a

function defined by the table x 0 1 2

y 1 2 1 which satisfies y0(0) = 0,y0(2) = 0

coefficients c0, c1, c2 can be found by solving the system

Then we can define the coefficients of g0

Therefore, the required clamped cubic

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