7/26/2019 Curve Fitting Notes

1/12

Q. 4.1. What is curve fitting?

Sol. Suppose there be two variables x and y which give us a set of n pairs of values (x1,y1) (xv, yr). To find an approximate idea about relationship of these two variables, weplot these n paired points on graph like scatter or dot diagram. From scatter diagram,

we get only non-mathematical relation between two variables. Thus an exactmathematical relation between two variables is called curve fitting. Curve fitting meansto form an equation of curve from given data. We get a curve of Best fit.

The following methods are used for curve fitting:

1. Graphical method2. Method of group averages3. Principle of least squares

4. Method of moments

Q. 4.2. State two differences between curve fi tting and Interpolation.

Sol. 1. Curve fitting is an exact relationship between two variables where asinterpolation is process of estimating the value of dependent variable y for a given valueof independent variable x in given range.2. In curve fitting, we get a curve of best fit. In interpolation, we attempt to find a simplefunction say (x) such that f (x) and (x) agrees at set of specified values.

Q. 4.3. Explain method of least squares for curve fi tting.

Sol.The principle of least squares provides an elegant procedure of fitting a unique

The- curve of best fit is that for which e. are small i.e. sum of squares of error isminimum.From (2)

For E to be minimum

On solving, we get

The equations (3), (4) and (5) are known as normal equations.

7/26/2019 Curve Fitting Notes

2/12

Q. 4.4. Give working procedure for fit ting a straight line.

Sol.To fit a straight line y = a + bx1. Substitute the observed set of n values in this equation.2. -Write normal equations.

3. Solve these normal equations as simultaneous equations for a and b (use Cramers rule). -4. Substitute value of a and b in y = a + bx, which is required line of best fit.

Applications:1. It gives best data fit.2. It is applicable to both linear and non-linear curves.3. It gives accurate result.

Q. 4.6. Give working procedure for f itting a parabola.

Ans. To fit a parabola

7/26/2019 Curve Fitting Notes

3/12

Q. 4.7. Fit a polynomial second degree by using the following data points.

7/26/2019 Curve Fitting Notes

4/12

7/26/2019 Curve Fitting Notes

5/12

7/26/2019 Curve Fitting Notes

6/12

Step 3 Table

Substitut ing value from table and solve normal equations.Step 4.

On solving

7/26/2019 Curve Fitting Notes

7/12

Step 3. Prepare table

Step 4. Substituting values from table and solve normal equations.

7/26/2019 Curve Fitting Notes

8/12

Step 3.

7/26/2019 Curve Fitting Notes

9/12

Q.4.16 Explain curve fi tting by sum of exponentials.

Sol.consider sum of exponential of form

7/26/2019 Curve Fitting Notes

10/12

Q. 4.17 For the following for y measured for a set of values of x, fit a seconddegree

polynomial:

7/26/2019 Curve Fitting Notes

11/12

Put values in normal equations

Q. 4.18. Give principle of least square curve fit ting.

Sol.Let the curve y = a + bx + cx2 + tke m-be fitted to set of data points (x, y.), (2 Y2) (xv, y). At x = x1, the observed(experimental) value is y1 and corresponding value on the fitting curve (1) which is -expected (calculated) value is Y1 (say). The difference of observed and expectedvalues is called error (or Residual) at x = x,

The curve of best fit is that for which es are as small as possible i.e. sum of the squareof the errors is a minimum. This is known as principle of least square for curve fitting.

Q. 4.19. Using method of least squares, fit a relation of the form y = au to thefollowing data. Also estimate y(3.5)

7/26/2019 Curve Fitting Notes

12/12