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Numerical method (curve fitting)

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Numerical method (curve fitting)

***TOPICS ARE****

Linear Regression
Multiple Linear Regression
Polynomial Regression
Example of Newton’s Interpolation Polynomial And example
Example of Newton’s Interpolation Polynomial And example

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Numerical method (curve fitting)

  1. 1. Name:Sujit Kumar Saha Lecturer at Varendra University Rajshahi Name: Istiaque Ahmed Shuvo Id: 141311057 5th batch, 7th Semester Sec-B Dept. Of Cse Varendra University, Rajshahi Submitted By: Submitted To 11-Apr-16 1
  2. 2. TOPICS ARE  Linear Regression  Multiple Linear Regression  Polynomial Regression  Example of Newton’s Interpolation Polynomial And example 11-Apr-16 3
  3. 3. Fitting a straight line to a set of paired observations: (x1, y1), (x2, y2),…,(xn, yn). y = a0+ a1 x + e a1 - slope a0 - intercept e - error, or residual, between the model and the observations Linear Regression 11-Apr-16 4
  4. 4.                 2 10 10 1 1 1 0 0 0)(2 0)(2 iiii ii iioi r ioi o r xaxaxy xaay xxaay a S xaay a S          2 10 10 00 iiii ii xaxaxy yaxna naa 2 equations with 2 unknowns, can be solved simultaneously Linear Regression: Determination of ao and a1 11-Apr-16 6
  5. 5.          221 ii iiii xxn yxyxn a xaya 10  Linear Regression: Determination of ao and a1 11-Apr-16 7
  6. 6. • Another useful extension of linear regression is the case where y is a linear function of two or more independent variables: • Again, the best fit is obtained by minimizing the sum of the squares of the estimate residuals: Multiple Linear Regression
  7. 7. • The least-squares procedure from Chapter 13 can be readily extended to fit data to a higher-order polynomial. Again, the idea is to minimize the sum of the squares of the estimate residuals. • The figure shows the same data fit with: a) A first order polynomial b) A second order polynomial Polynomial Regression 11-Apr-16 9
  8. 8. Many times, data is given only at discrete points such as (x0, y0), (x1, y1), ......, (xn−1, yn−1), (xn, yn). So, how then does one find the value of y at any other value of x ? Well, a continuous function f (x) may be used to represent the n +1 data values with f (x) passing through the n +1 points (Figure 1). Then one can find the value of y at any other value of x . This is called interpolation. Of course, if x falls outside the range of x for which the data is given, it is no longer interpolation but instead is called extrapolation. So what kind of function f (x) should one choose? A polynomial is a common choice for an interpolating function because polynomials are easy to (A) evaluate, (B) differentiate, and (C) integrate, relative to other choices such as a trigonometric and exponential series. Polynomial interpolation involves finding a polynomial of order n that passes through the n +1 points. One of the methods of interpolation is called Newton’s divided difference polynomial method. Other methods include the direct method and the Lagrangian interpolation method. We will discuss Newton’s divided difference polynomial method in this What is interpolation? 11-Apr-16 10
  9. 9. Newton’s Divided-Difference Interpolating Polynomials? 11-Apr-16 11
  10. 10. Example of Newton’s Interpolation Polynomial 11-Apr-16 12
  11. 11. 11-Apr-16 13

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