Unit-6

Unit-6

Unit-6

Numerical solution of Transcendental Equations, System of   Linear equations and Expansion of Functions

Question and answer

1.  Find a real root of the equation near, correct to three decimal place by the Regula Falsi method.

Let

Now,

And also

Hence the root of the equation lies between and and so,

By Regula Falsi Method

     Now,

So the root of the equation lies between 1 and 0.5 and so

By Regula Fasli Method

 Now,

So the root of the equation lies between 1 and 0.63637 and so

By Regula Fasli  Method

 Now,

So the root of the equation lies between 1 and 0.67112 and so

By Regula Fasli  Method

 Now,

So the root of the equation lies between 1 and 0.63636 and so

  By Regula Fasli  Method

 Now,

So the root of the equation lies between 1 and 0.68168 and so

By Regula Fasli  Method

 Now,

Hence the approximate root of the given equation near to 1 is 0.68217

2.     Apply Regula Falsi Method to solve the equation

Let 

By hit and trail

  And

So the root of the equation lies between and also

By Regula  Falsi Mehtod

     Now,

So, root of the equation   lies between 0.60709 and 0.61 and also

By Regula  Falsi Method

     Now,

So, root of the equation   lies between 0.60710 and 0.61 and also

By Regula Falsi Method

     Hence the root of the given equation correct to five decimal place is 0.60710

3.     Using Newton-Raphson method, find a root of the following equation correct to 3 decimal places:.

Given

By Newton  Raphson Method

                   = 

                  =

   The initial approximation is in radian.

      For n =0, the first approximation

For n =1, the second approximation

For n =2, the third approximation

For n =3, the fourth approximation

Hence the root of the given equation correct to five decimal place 2.79838.

4.     Using Newton-Raphson method, find a root of the following equation correct to 3 decimal places: near to 4.5

Let

  The initial approximation

     By Newton Raphson Method

     For n =0, the first approximation

  For n =1, the second approximation

For n =2, the third approximation

For n =3, the fourth approximation

     Hence the root of the equation correct to three decimal places is 4.5579

5.     Use Jacobi’s method to solve the system of equations:

Since

So, we express the unknown with large coefficient in terms of other coefficients.

                    (1)

Let the initial approximation be

                                                        2.35606

                                                        0.91666

                                                        1.932936

                                                        0.831912

                                                      3.016873

                                                        1.969654

                                                      3.010217

                                                        1.986010

                                                        1.988631

                                                        0.915055

                                                        1.986532

                                                        0.911609

                                                        1.985792

                                                        0.911547

                                                        1.98576

                                                        0.911698

Since the approximation in ninth and tenth iteration is same up to three decimal places, hence the solution of the given equations is

6.     Use Jacobi’s method to solve the system of the equations

Rewrite the given equations

                         (1)

Let the initial approximation be

                                                              1.2

                                                            1.3

                                       0.9

                             1.03

                                                  0.9946

                                                  0.9934

                                         1.0015

Hence the solution of the above equation correct to two decimal places is

7.     Use Gauss –Seidel Iteration method to solve the system of equations

Since

So, we express the unknown of larger coefficient in terms of the unknowns with smaller coefficients.

Rewrite the above system of equations

                       (1)

Let the initial approximation be

                        3.14814

                                                                      2.43217

                        2.42571

                        2.4260

Hence the solution correct to three decimal places is

8.     Solve the following equations by Gauss-Seidel Method

Rewrite the above system of equations

                     (1)

Let the initial approximation be

Hence the required solution is

9.     Expand in power of

Let

Also  

Differentiating f(x) with respect to x.

Again differentiating f(x) with respect to x.

Again differentiating f(x) with respect to x.

Also the value of above functions at x=1 will be

By Taylor’s theorem

On substituting above values we get

 =

10. If using Taylor’s theorem, show that for .

Deduce that

Let then

Differentiating with respect to x.

     .Then

Again differentiating with respect to x.

    Then  

Again differentiating with respect to x.

    Then  

By Maclaurin’s theorem

Substituting the above values we get

Since

Hence