# 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

X_{0} = is an approximate value of a number

Example

If X_{0} = 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 X_{0} 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 – X_{0}/

= X – X_{0}

âˆ†_{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(x_{1}), f(x_{2}) < 0 the root lies between X_{1}and X_{2 }

**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 ï„ƒ let be X_{1}

X_{1}→

A line AB is drawn tangent to the curve at a point A where A(X_{1}, f(X_{1}))

X_{2} is the point which is very near to _{1}

This slope is equal to the tangent of the curve at X=X_{1}

i.e.

But At B f(X_{2}) = 0

X_{3} will be the best approximation

In general N-R formula can be written as

Example

Show that the equation X^{3 }– X^{2 }+ 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=

X^{2}=5

X^{2 }– 5 = 0

Let f(x) = x^{2 }– 5 by N.R formula

f’(x)= 2x

Given X_{1}=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 X_{0} and X_{1}

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 f_{1} at every step

-Secant method only requires the evaluation of f

Example

Calculate in 3 iteration the root of the function f(x)= x^{2}-4x+2 which his between

X_{0}=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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