revision for final examination 2015-2016

7/24/2019 Revision for Final Examination 2015-2016

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Revision for Final ExaminationComputational Mathematics

Interpolationi) Polynomial Interpolation

2

1 2 3( )f x p x p x p

ii) Newtons Interpolating Polynomial

2nd

Order (need 3 points)

2 1 2 1 3 1 2

1 1

2 2 1

3 3 2 1

( ) ( ) ( )( )

( )

[ , ]

[ , , ]

f x b b x x b x x x x

b f x

b f x x

b f x x x

3rd


3 1 2 1 3 1 2 4 1 2 3

1 1

2 2 1

3 3 2 1

4 4 3 2 1

( ) ( ) ( )( ) ( )( )( )

( )

[ , ]

[ , , ]

[ , , , ]

f x b b x x b x x x x b x x x x x x

b f x

b f x x

b f x x x

b f x x x x

iii) Lagrange Polynomial

1st


2 11 1 2

1 2 2 1

( ) ( )( ) ( ) ( )

( ) ( )

x x x xf x f x f x

x x x x

2nd


2 3 1 3 1 22 1 2 3

1 2 1 3 2 1 2 3 3 1 3 2

( )( ) ( )( ) ( )( )( ) ( ) ( ) ( )

( )( ) ( )( ) ( )( )

x x x x x x x x x x x xf x f x f x f x

x x x x x x x x x x x x


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Example Tutorial 4 Question 1:

Estimate the logarithm of 5 to the base 10 (log 5) using Newtons interpolation:

i) Linear interpolation between log 4 and log 6.

ii) Linear interpolation between log 4.5 and 5.5.

iii) 2nd

order (use log 4, log 4.5, log 5.5)iv) 2ndorder (use log 4.5, log 5.5, log 6)

v) 3rdorder

vi) 1storder Lagrange

vii) 2ndorder Lagrange

Numerical Integration

i) Trapezoidal rule

2 points => 1 22

hI f f

3 points => 1 2 322

hI f f f

4 points => 1 2 3 42( )2

hI f f f f

ii) Simpsons 1/3 rule

Single Application (3 points) => 1 2 343

hI f f f

Composite (Odd no points) => 1 2 4 3 54( ) 23

hI f f f f f

iii) Simpsons 3/8 rule(4 points) => 1 2 3 43

3 38

hI f f f f

Example Tutorial 6 Question 1:

Evaluate the following integral:

4

0(1 )xI e dx

a) Analytically

b) Single application of the trapezoidal rule,

c) Composite trapezoidal rule with n = 2 and 4,

d) Single application of Simpsons 1/3 rule,

e) Composite Simpsons 1/3 rule with n = 4,

f) Simpsons 3/8 rule

g) Composite Simpsons rule, with n = 5. For each of the numerical estimates (b) through (g),

determine the true percent relative error based on (a).


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Ordinary Differential Equations

i) Eulers method

1 ( , )i i i iy y f x y h

ii) Midpoints method

1 1/2 1/2( , )i i i iy y f x y h

iii) Heuns method0

1 11

( , ) ( , )

2

i i i ii i

f x y f x yy y h

iv) Ralston method

1 1 2

1 2

3 3i iy y k k h

1

2 1

( , )

3 3( , )

4 4

i i

i i

k f x y

k f x h y k h

v) Runge-Kutta method

1

2 1

3 2

4 3

1 1 2 3 4

( , )

1( , )

2

1( , )

2( , )

1( 2 2 )

6

i i

i i

i i

i i

i i

K f x y

K f x h y K h

K f x h y K h

K f x h y K h

y y K K K K h

Example (Question 1: Tutorial 7)

Solve the following problem over the interval fromx = 0 to 1 using a step size of 0.25 where y(0) = 1.

Display all your results on the same graph.

(1 4 )y

x yx

i) Analytically.

ii) Using Eulers method.

iii) Using Heuns method without iteration.

iv) Using the fourth-order RK method.


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Gauss Elimination

i) Cramers rule

1 12 13

2 22 23

3 32 33

1

b a a

b a a

b a ax

D

11 1 13

21 2 23

31 3 33

2

a b a

a b a

a b ax

D

11 12 1

21 22 2

31 32 3

3

a a b

a a b

a a bx

D

ii) Nave Gauss elimination

iii) Pivotingpartial and complete

Example (Question 2 Tutorial 8)

Solve the following equations by using Gauss elimination with partial pivoting:

1 2 3

1 2 3

1 2 3

2 6 38

3 7 34

8 2 20

x x x

x x x

x x x


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Linear Algebraic Equation

i) Jacobi

ii) Gauss-Seidel

iii) SOR

Example (Question 2: Tutorial 8)

For the system below, solve using Jacobi and Gauss-Seidel methods with error of below 5%.

1

2

3

3 6 2 61.5

10 2 1 27

1 1 5 21.5

x

x

x

Finite Difference Method

i) Forward in time - first derivative

1n n

i iT TT

t t

ii) Central in space - second derivative2

1 1

2 2

2

( )

n n n

i i iT T TT

x x

Example (Assignment 9)

revision for final examination 2015-2016

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