numarial analysis presentation

PRESENTED BY

OSAMA TAHIR

09-EE-88

SECTION (A)

LEAST SQUARE

The least squares technique is the simplest and most commonly applied form and provides a solution to the problem through a set of points.

The term least squares describes a frequently used approach to solving overdeter-mined or inexactly specied systems of equations in an approximate sense. Instead of solving the equations exactly, we seek only to minimize the sum of the squares of the residuals

LEAST SQUARE

least square method is widely used to find or estimate the numerical values of the parameters to fit a function to a set of data and to characterize the statistical properties of estimates.

GRAPH

A cubic fitting is defined as the smoothest curve that exactly fits a set of data points.

Generalizing from a straight line the ith fitting function for a cubic fitting can be written as:

32xdxcxbaxs iiiii

0)]([ 23

3

2

210

1

2 xaxaxaayRn

i

3

3

2

210 xaxaxaaxy

The residual of above equation is

Now we take its partial derivatives

The partial derivatives are

0)].....([2/ 3

10

1

0

2 3 xaxaayaRn

i

0)].....([2/ 3

10

1

1

2 3 xxaxaayaRn

i

0)].....([2/ 23

10

1

2

2 3 xxaxaayaRn

i

0)].....([2/ 33

10

1

3

2 3 xxaxaayaRn

i

n

i

n

i

i

n

i

i yxaxana1

3

3

1

10 ......

n

i

i

n

i

i

n

i

i

n

i

i yxxaxaxa1

3

3

1

2

1

1

0 ......

Cont...

Now to write this least square equation In matrix form

In Matrix notation the equation for a Polynomial is given byY=XA

3

3

2

33

3

2

2

22

3

1

2

11

3

0

2

00

1

1

1

1

xxx

xxx

xxx

xxx

3

2

1

0

a

a

a

a

3

2

1

0

y

y

y

y

Cont ...

yxxxa tt 1)(

xaxyx tt

This can be solved by premultiplying by the transpose

,

EXAMPLES

Example…

A bioengineer is studying the growth of a genetically engineered bacteria culture and suspects that is it approximately follows a cubic model. He collects six data points listed below

Time in Days 1 2 3 4 5 6

Grams 2.1 3.5 4.2 3.1 4.4 6.8

Let we solve it by cubic fitting methodax3 + bx2 + cx + d = y

Cont …

This gives six equations with four unknowns

a + b + c + d = 2.18a + 4b + 2c + d = 3.5

27a + 9b + 3c + d = 4.264a + 16b + 4c + d = 3.1

125a + 25b + 5c + d = 4.4216a + 36b + 6c + d = 6.8

Cont...

The corresponding matrix equation is

d

c

b

a

8.6

4.4

1.3

2.4

5.3

1.2

Cont ...

d

c

b

a

yxxx tt 1)(

3.2

1.6

0.2

2.0

So that the best fitting cubic is

y = 0.2x3 - 2.0x2 + 6.1x - 2.3

cubic fittings are preferred over other methods because they provide the simplest representation that exhibits the desired appearance of smoothness

Cubic fittings provide a great deal of flexibility in creating a continuous smooth curve both between and at tenor points.

If we have damped curves or very humped curves then we can not obtain usefull results from Cubic Fitting Method..because they are used for smooth curves

Applications in

“The Real World”???

The cubic fitting method (CFM) is probably the most popular technique in statistics.

In mathematics it is used to solve different curved spaces.

It is used in the manufacturing of plumbing materials.

It is much important in mechanical and Electrical and Civil Engineering

REFERENCES...

Wikipedia.com

Advanced Engineering Mathematics

Nocedal J. & Wright, S. (1999). Numerical optimization. New youk

NUMARICAL ANALYSIS

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