Numerical differentiation and integration handout

NUMERICAL DIFFERENTIATION ANDINTEGRATION Dr.. Lê Xuân Đại HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics Email: ytkadai@hcmut.edu.v

NUMERICAL DIFFERENTIATION AND

INTEGRATION

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

1 N UMERICAL D IFFERENTIATION

2 N UMERICAL I NTEGRATION

Consider the table of data x x0 x1

f0(x)≈ y1− y0

h = f (x0+ h) − f (x0)

h

Consider the table x x0 x1 x2

which is called centripetal difference

formula and can be written in the form

f0(x0)≈ f (x0+ h) − f (x0− h)

At x2 we also have

f0(x2)≈L0(x2) = y0− 4y1+ 3y2

2h which is called the backward-difference formula

f0(x0)≈ f (x0− 2h) − 4f (x0− h) + 3f (x0)

2 × 5(−3×1.6990+4×1.1704−1.7782) = −0.21936

However, if the function y = f (x) is defined

by the table of data then we can not apply this formula.

In order to approximate the definite integral

f on [a, b], we can replace f (x) by the

interpolating polynomial P n (x) and we

by Newton forward divided-difference

formula which passes through 2 points

(a, f (a)) and (b, f (b))starting from (a, f (a)).

Therefore, P1(x) = f (a) + f [a,b](x − a) =

= f (a) + f (b) − f (a)

b − a (x − a)

Divide the interval [a, b] into n subinterval

of the equal width h = b − a

1 + x using composite trapezoidal rule

and dividing [0, 1] into n = 10subintervals of equal width.

In order to approximate the integral

Z b

a

f (x)dx we divide[a, b] into 2 subinterval of

equal width by a, x1= a + h, b whereh = b − a

2

and replace the integrand f (x)by the second order forward Newton divided-difference , which passes through 3 points

(a, f (a)), (x1, f (x1))and (b, f (b))starting from

(a, f (a)).

Therefore,

P2(x) = f (a)+f [a,x1](x−a)+f [a,x1, b](x−a)(x−x1)

On the other hand, we have

We divide [a, b] into 2n subintervals of equal width h = b − a

1 + x using composite Simpson’s Rule

and dividing the interval [0, 1]into 2n = 20

subintervals of equal width.

THANK YOU FOR YOUR ATTENTION