numarial analysis presentation BY OSAMA TAHIR UET TAXILA

2. PRESENTED BY
OSAMA TAHIR
09-EE-88
SECTION (A)

4. 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

5. 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.

6. GRAPH

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

9.  Generalizing from a straight line the ith
fitting function for a cubic fitting can be
written as:
2 3
si x ai bi x ci x di x

10. 2 3
yx a0 a1 x a2 x a3 x
The residual of above equation is
n
R2 [ y (a0 a1 x a 2 x 2 a3 x 3 )] 2 0
i 1

11.  Now we take its partial derivatives
 The partial derivatives are
n
2
R / a0 2 [y (a 0 a1 x ..... a 3 x 3 )] 0
i 1
n
2
R / a1 2 [y ( a0 a1 x ..... a 3 x 3 )] x 0
i 1
n
R 2 / a2 2 [y (a0 a1 x ..... a 3 x 3 )] x 2 0
i 1
n
2
R / a3 2 [y (a0 a1 x ..... a 3 x 3 )] x 3 0
i 1

12. n n n
a 0 n a1 xi ...... a3 xi3 y
i 1 i i 1
n n n n
2 3
a0 xi a1 x i ...... a3 x i xi y
i 1 i 1 i i 1

13. Cont...
 Now to write this least square equation
In matrix form
y0 1 x0 x0 x0 2 3 a0
y1 1 x1 x12 x13 a1
y2 1 x2 x2 x2 2 3
a2
y3 1 x x2 x3 a
3 3 3 3
In Matrix notation the equation for a Polynomial is
given by
Y=XA

14. ,
Cont ...
This can be solved by premultiplying by the transpose
xt y x t xa
t 1 t
a ( x x) x y

15. EXAMPLES

16. 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 method
ax3 + bx2 + cx + d = y

17. Cont …
 This gives six equations with four
unknowns
a + b + c + d = 2.1
8a + 4b + 2c + d = 3.5
27a + 9b + 3c + d = 4.2
64a + 16b + 4c + d = 3.1
125a + 25b + 5c + d = 4.4
216a + 36b + 6c + d = 6.8

18. Cont...
 The corresponding matrix equation is
2 .1
a 3 .5
b 4 .2
c 3 .1
d 4 .4
6 .8

19. Cont ...
0 .2
a
b t 1 t
2 .0
( x x) x y
c 6 .1
d 2 .3
So that the best fitting cubic is
y = 0.2x3 - 2.0x2 + 6.1x - 2.3

21.  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.

22.  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

23. 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

24. REFERENCES...
 Wikipedia.com
 Advanced Engineering Mathematics
 Nocedal J. & Wright, S. (1999). Numerical
optimization. New youk
 NUMARICAL ANALYSIS

26. THANKYOU
FOR YOUR ATTENTION