NUMERICAL ANALYSIS

Lecture -3

Muhammad Rafiq

Assistant Professor University of Central Punjab

Lahore Pakistan

ERROR IN MULTIPLICATION

Suppose exact value of a number be= x1

Suppose exact value of another number be= x2

Suppose approximate value of a number be= x1

Suppose approximate value of another number be= x2

Error in first number:e1=x1-x1

Error in first number:e2=x2-x2

Let Z be the product of two numbers=Z=x1.x2

Let Z be the product of approximate values=Z=x1.x2

Error=ez=Z-Z

= (x1.x2 )- (x1.x2)

= x1.x2- (x1- e1)(x2-e2)

=x1x2-( x1 x2- e1x2-e2x1-e1e2)

= x1x2-( x1x2- e1x2-e2x1)

Neglecting e1e2 as it is very small

=x1x2-x1x2+e1x2+e2x1

So

ez =e1x2+e2x1

Absolute error=|ez|=|e1x2+e2x1||e1x2|+|e2x1|

And =|ez||e1x2|+|e2x1|

Relative error= R.E=Absolute Error/|Z|

=|e1x2+e2x1|/|x1x2|

=| e1x2 +e2x1/x1x2 |

=| e1x2/x1x2 + e2x1/x1x2 |

R.E=| e1/x1 + e2/x2|| e1/x1|+| e2/x2|

So

R.E| e1/x1|+| e2/x2|

ERROR IN QUOTIENT

Suppose exact value of a number be= x1

Suppose exact value of another number be= x2

Suppose approximate value of 1st number be= x1

Suppose approximate value of 2nd

number be= x2

Error in first number = e1=x1-x1

Error in second number=e2=x2-x2

Let Z be the quotient of two numbers=Z=x1/x2

Let Z be the quotient of approximate values: Z=x1/x2

Error=ez=Z-Z

=x1/x2-x1/x2

=x1/x2 - x1-e1/x2-e2

= x1/x2 - x1(1-e1/x1)/x2(1-e2/x2)

= x1/x2 -x1(1-e1/x1) x2(1-e2/x2)-1

= x1/x2 - x1(1-e1/x1) x2(1- (-1 )e2/x2)

[As (1+x)n=1+nx+]

= x1/x2-x1(1-e1/x1)x2(1+e2/x2)

= x1/x2-x1x2 (1-e1/x1+e2/x2)

= x1/x2-x1/x2+e1/x2-x1/x2e2

= e1/x2 - x1/x22

e2

R.E=A.E/|Z|

| e1/x2-x1/x22. e2|/|x1/x2|

|(e1/x2-x1/x22. e2 )/ x1/x2|

|e1/x1 - e2/x2||e1/x1| + |e2/x2|

| | | | | |

ERROR IN POWER

Let a number be=x

Such that Z=xn

Appraximate Value of x=x

Error in x: e=x-x

Error in Z: e=Z-Z

=xn-(x)

n

=x

n-(x-e)

n

=xn-x

n(1 - e/x)

n

=xn-x

n(1+n(-e/x))

=xn-x

n+nex

n-1

So

ez = nexn-1

R.E=|nexn-1

|/|xn|

=| nexn-1

/xn|

=|ne/x|

=|n||e/x|

EXAMPLE # 1

Given the data

then estimate the relative error, maximum

absolute error, the range in which true answers lie.

Solution:

Let x1=4.0643 e1=1/210-4

Let x2=37.487 e=1/210

-3

Z=x1/x2

= 4.0643/37.487

= 0.1084.

R.E e1/x1 + e2/x2

1/210-3/4.0643 + 1/210-3/37.487

2.5640 10-5

A.E R.E Z

0.10842.564010-5

2.77937610-6

Max. Absolute Error=2.77937610-6

The range in which true answer lies

Z-A.E Z Z+A.E

0.1084-2.779410-6Z0.1084+2.779410-6

0.108397 Z 0.108402

(Correct to 3 d.p)

EXAMPLE # 2

Given the data the estimate the relative error ,

maximum Absolute error and the range in which answer lies.

Solution:

Z=(48.425)1/2

=6.9588.

LET x=48.425 e=1/210-3

R.E=n | |

=1/2(110/248.425)

=

5.162610

-6

Maximum Absolute error = R.E Z

=5.1626106.9588

= 3.592510-5

The range in which true answers lie:

Z-A.E Z Z+A.E

6.9588- 3.5925 10

-5 Z 6.9588- 3.5925 10-5

6.9518 Z 6.9588

(Correct to 2 d.p)

EXAMPLE # 3

If given numbers are rounded estimate the relative error and

absolute error in the product 4.06430.37487

Solution:

Let x1=4.0643 e1=1/210-4

Let x2=0.37487 e2=1/210-5

Let Z= x1 x2

= 4.06430.377487

= 1.52358.

A.E : ez x1 . e2 + x2 . e1

(4.06431/210-3

)+(0.374871/210-4

)

3.906510-5

R.E e1/x1 + e2/x2 |

110-4

/ 24.0643 + 110-5

/ 20.37487

2.5640 10-5

EXAMPLE # 4

Given

find relative error, absolute error and the

range in which true answer lies.

Solution:

Z =

=

= (1.8778)1/2

Z = 1.37034 n=1/2 e=1/210-4

R.E1/21/210-4/1.8778

A.E 1.3313410-5

R.E|Z |

1.3313410-5 1.37034

1.8243810-5

Z-A.E Z Z+A.E

1.37034-1.8243810-5Z 1.37034+1.8243810-5

1.37032 Z 1.370358

(correct to 4 d.p)

EXAMPLE # 5.Let 65.43 and 17.0591 be correctly rounded to

the number of digits shown. What is the smallest interval in

which the exact sum of numbers lie.

SOL: x1=65.43 e1=1/210-2

x2=17.0591 e2= 1/210-4

Let Z=x1+x2

=65.43+17.0591

=82.4891.

A.Eeze1+e2

510-3+510-5

510-3

R.E A.E/|Z|

510-3/|82.4891|

6.0610-5

Z-A.E Z Z+A.E

82.4891-510-3

Z 82.4891+510-3

82.4841 Z 82.4941

Exercise

Q # 1:Given the data .Then estimate the

relative error, maximum absolute error, the range in which true

answers lies.

Q #2:Given the data

then estimate the relative error,

maximum absolute error, the range in which true answers lie.

Q # 3:Given

find relative error, absolute error

and the range in which true answer lies.

Q #4:Given

find relative error, absolute error

and the range in which true answer lies.