Ch 10 Num Diff and Integrate

UTG

CPE 332

Computer Engineering

Mathematics II

Part III, Chapter 11

Numerical Differentiation and Integration

Today Topics

• Chapter 11 Numerical Differentiation and Integration

– Derivative Approximation

• Forward, Backward, Centered Difference

• High Order Derivative

• High Accuracy Approximation

– Integral Approximation

• Polynomial

– Zero Order

– First Order (Trapezoidal)

– Second Order (Simpson 1/3)

– Third Order (Simpson 3/8)

• More Accurate Method

– Richardson Extrapolation

Slope of Line

m=slope = tan θ

= (y2-y1)/ (x2-x1)

(x2,y2)

(y2-y1)

θ

θ

(x1,y1)

(x2-x1)

Definition of Derivative

• Derivative of f(x) at any point x is the slope of the tangent line at that point (x,f(x))

• Mathematically

y

f ( x + x

∆ ) − f ( x

Derivative of

( ) ≡ lim

=

)

f x

lim

x

∆ →0

x

x

∆ →0

x

• For function y=f(x), derivative of function is written in many forms

dy

' ( ),

',

(read '

dee -

d

y ee - x' or

)

f

x

y

y

dx

Approximation of slope at point x using secant line

• Slope at ‘ x’ ≅ [ f( x+∆ x) - f( x)] / ∆ x = y / x As ∆ x approaches 0,

we have ∆y/∆x

approach a true slope

of f at x.

( x+x, f( x+x)) f( x+x) - f( x) ( x, f( x))

=∆ y

x

Derivative(Forward) f ( x + ∆) − f x f '( x) =

( )

lim

∆→0

y

Derivative(Backward) f ( x) − f ( x − ∆

f '( x) =

)

lim

∆→0

y

Derivative(Central) f ( x + ∆) − f ( x − ∆

f '( x) =

)

lim

∆→0

2∆

y

x

x

Introduction

• ในทางปฎิบัติ เราได ้ Sample ของ Data เป็นจุด การหา

Derivative ก็คือการลบค่าของจุด Data ที่อยู่ติดกันและ

หารด ้วยระยะห่างระหว่างจุด นี่คือวิธีการของ Finite Divided-Difference h h

f(xi+1)

f(xi)

f(xi-1)

xi-1 x

i xi+1

Finite Divided-Difference

Finite Divided-Difference

Finite Divided-Difference

f '''( x ) 3

h

f ( x ) − f ( x ) = 2 f '( x ) h + 2

i

+...

i 1

+

i 1

i

!

3

Second Derivative

Second Derivative

High Accuracy Finite

Divided-Difference

High Accuracy Finite

Divided-Difference

Summary

Summary

Summary

Numerical Integration

• Newton-Cotes Integration Formula

• Zero-Order Approximation

• First-Order Approximation

– Trapezoidal Rule

• Second-Order Approximation

– Simpson 1/3 rule

• Third-Order Approximation

– Simpson 3/8 rule

• Romberg Integration

– Richardson Extrapolation

– Romberg Integration Algorithm

Integral Approximation

Integral Approximation x0 x1 x2 x3 x4

xn-1 xn

Zero-Order Approximation

x0 x1 x2 x3 x4

xn-1 xn

Zero-Order Approximation

• a0=f(a)

x0 x1 x2 x3 x4

xn-1 xn

b

b

I =

f ( x) dx a dx = a [ b a] = f ( a)[ b a]

∫ 0

0

a

a

Zero-Order Approximation

• a0=f((a+b)/2)

x0 x1 x2 x3 x4

xn-1 xn

b

b

a + b

I =

f ( x) dx a dx = a [ b a] = f (

)[ b a]

∫ 0

0

2

a

a

Zero-Order

App

roximation

First-Order Approximation x0 x1 x2 x3 x4

xn-1 xn

First-Order Approximation

First-Order Approximation

First-Order Approximation

Trapezoidal Rule

Trapezoidal Rule

Second-Order Approximation x0 x1 x2 x3 x4

xn-1 xn

Simpson’s 1/3 Rule

Second-Order Approximation

Second-Order Approximation

Second-Order Approximation

3n

Second-Order Approximation

Third-Order Approximation x0 x1 x2 x3 x4

xn-1 xn

Simpson’s 3/8 Rule

Third Degree Approximation

Simson’s 3/8 Rule

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Romberg Integration

Summary

Homework 10: Ch 11

• Download HW

• Next Week

– ส่งการบ ้าน HW 10

– Chapter 11 Solutions of ODE

– HW 11 (Option)

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