numarial analysis presentation
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The document discusses the least squares method and cubic fitting method. [1] It explains that least squares finds the best fit curve to a set of data points by minimizing the sum of the squared residuals. [2] Cubic fitting finds the smoothest curve that exactly fits the data points using a cubic polynomial function. [3] An example applies the cubic fitting method to bacterial growth data to determine the parameters for the best fitting cubic curve.
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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](https://image.slidesharecdn.com/n-a-120923061958-phpapp02/85/numarial-analysis-presentation-10-320.jpg)
![ 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
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numarial analysis presentation
- 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