NUMERICAL METHOD

NUMERICAL METHOD

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Introduction

Numerical methods can be used to find roots of a function

→We find roots of a function by;

1. Direct method

2. Iterative method

ERROR

An error can be defined as the deviation from accuracy or correctness

Error

Where;

X = is the exact value of a number

X₀ = is an approximate value of a number

Example

If X₀ = 3.14 and x=  then

.

=

=0.001592654

TYPES OF ERROR

A)   Systematic error

This is a predictable error or constant caused by imperfect calibration of measurements instruments or something is wrong from the measuring instrument

B)   Random error

Unpredictable error caused either by weather or anything else.

Sources of errors

1. Experimentation error/modeling error

2. Truncation error/terminating error

E.g.

You can see that the series is terminated at power of 3

3. Rounding off numbers

4. Mistakes and blunder

ABSOLUTE AND RELATIVE ERROR

Absolute error

Is the difference between the measured value of a quantity X₀ and its value

Absolute error

Relative error

i.e. Relative error =

A relative error gives an indication of how good measurement is relative to the size measurement is relative to the size of the thing that measured.

Example

i)   Absolute error

ii)  Relative error

iii)Percentage error

Solution

i. Absolute error = /X – X₀/

=  X – X₀

∆_x = 0.001592654

ii.  Relative error

=

=  0.000050696(9dp)

iii. Percentage error

= 0.000050696 x 100%

= 5.0696 x 10^-2%

Roots by iterative methods

Iterative method is used to find a root of function by approximations repeatedly.

If f(x₁), f(x₂) < 0 the root lies between X₁and X₂

Newton’s Raphson formula

The formula is based on the tangent lines drawn to the curves through x-axis

Consider the graph below

Suppose f(x) = 0 has a root  that x is an approximation for

Choosing a point which is very close to  let be X₁

X₁→

A line AB is drawn tangent to the curve at a point A where A(X₁, f(X₁))

X₂ is the point which is very near to ₁

This slope is equal to the tangent of the curve at X=X₁

i.e.

But At B    f(X₂) = 0

X₃ will be the best approximation

In general N-R formula can be written as

 

Example

Show that the equation X³ – X² + 10x – 2 = 0 has a root between x = 0 and x = 1 and find the approximation for this root by carrying out 3 iterations

Solution

Application of N-R Formula

1.   Find approximation for roots of numbers suppose we want to approximate

 

 

 

 

 

Example

B y using 2 iteration only and starting with an initial value 2, find the square root of 5 correct to four decimal place

Solution

Let x=

X²=5

X² – 5 = 0

Let f(x) = x² – 5 by N.R formula

f’(x)= 2x

Given X₁=2

Then

First iteration

Example

Apply the N.R formula to establish the root of a number A

Solution

 

 

Finding approximations for reciprocals of numbers
Suppose we want to approximate

Example
Use N.R formula to find the inverse of 7 to 4, and perform 3 iteration only starting with X_o=0.1

Solution

 

 

SECANT METHOD
The secant method requires two initials values X₀ and X₁

Line AB is a secant line on the curve f(x)

We find the roots of this, the value of x such as that y=0

In general secant formula is given

Comparison with Newton’s method
-Newton’s converges faster (order 2 against   ≈1.6)
– Newton’s requires the evaluation of f and f₁ at every step
-Secant method only requires the evaluation of f

Example
Calculate in 3 iteration the root of the function f(x)= x²-4x+2 which his between
X₀=0 and X ₁=1
Solution

 

NUMERICAL INTEGRATION

Definite integral    is used to determine the area between y= f(x), the x -axis and the ordinates x = a and x = b

An approximate value for the integral can be found by estimating this area by another two methods

A.   Trapezium rule

Example
Estimate to 4 decimal places
Using five ordinates by the trapezium rule

Solution

Taking five ordinates from X = 0 to X = 1
5 – 1 = 4 number of strips

X	0	0.25	0.5	0.75	1
Y	1	0.9412	0.8	0.64	0.5

 

 

Simpson’s rule

Simpson’s rule is another method which can be used to find the area under the curve y= f(x) between x = a and x = b

A quadratic equation is fitted (parabola) passing through the three points i.e through A,B,C

Then

Example

 

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