All root finding algorithms with pseudo code - Simple guide to easily understand

PSEUDO-CODE

We present basic pseudo-code for some of the algorithms, discussed in the Steps. In our experience, students do benefit by studying the pseudo-code of a method at the same time as they learn it in a Step. If they are familiar with a programming language, they should attempt to convert at least some of the pseudo-codes into computer programs, and apply them to the set Exercises.

Bisection Method 

The equation is f (x ) = 0.

Points for study:

1. What is the input used for?
   
2. Explain the purpose of Lines 8 - 12.
3. Amend the speudo-code, so that the process will always stop after preset M iterations.
4. Amend the pseudo-code so that the process will stop as soon as 
5. Write a computer program, based on this speudo-code.
6. Use your program to solve Exercises 1 and 2 in the Applied Exercises.

Method of False position 

The equation is f (x ) = 0.

Points for study

1. What are the input values used for?
2. Under what circumstances may the process stop with a large error in x?
3. Amend the pseudo-code so that the process will stop after M iterations, if the condition in Line 13 is not satisfied.
4. Write a computer Program based on the pseudo-code.
5. Use your program to solve Exercises 1 and 2 in the Applied Exercises.

Newton-Raphson iterative method 

The equation is f (x ) = 0.

Points for study

1. 8
2. How are the input values used?
3. Why is M given in the output of Line 10?
4. What happens if f'(a) is very small?
5. Amend the pseudo-code to take suitable action if f'(a) is very small.
6. Write a computer program based on the pseudo-code.
7. Use your program to solve Exercises 1 and 2 in the Applied Exercises.

Gauss Elimination

The system is:

*

Points for study

1. Explain what happens in Lines 2 - 10.
2. What process is implemented in Lines 11 - 18`?
3. Amend the pseudo-code so that the program terminates with an informative message when a zero pivot element is found.
4. Write a program based on the pseudo-code.
5. Use your program to solve Exercises 3 and 4 in the Applied Exercises.

Gauss-Seidel Iteration 

The system is:

Points for study

1. What is the purpose of the number s?
2. What are the y₁, y_{2 1}, . . ., y_n used for?
3. Why is it possible to replace the y_j in Line 13 by x_j?
4. Amend the pseudo-code to allow a maximum of M iterations.
5. Write a program based on the pseudo-code.
6. Use thc computer program to solve the system:
7. Use your program to solve Exercises 3 and 4 in the Applied Exercises
8. 

Newton divided difference formula 

You are to calculate for given data x₀, x₁, . . . , x_n, f(x₀), f(x₁), . . ., f(x_n), and for given  the interpolating polynomial P_n(x> of degree n. (The algorithm is based on divided differences.)

Points for study

1. Follow the pseudo-code through with the data n = 2, .x = 1.5, x₀ = 0, f (.x₀) = 2.5,.x₁ = 1, f (x₁) = 4.7, xSS₂ = 3, and f x₂) = 3.1. Verify that the values d_iicalculated are the divided differences f (x₀, . . .,.x_I).
2. What quantity (in algebraic terms) is calculated in Lines 10 - 15?
3. Amend the pseudo-code so that the values P₁(x), P₂(x)P&127;(x), . . ., P_n-1(x)are also printed out.
4. Write a computer program based on the pseudo-code.
5. Use your program to estimate f(2) for the data given in 1 above.
6. For the data, given in Exercise 6 of the Applied Exercises, use the program to obtain an estimate of J₀(0.25).

Trapezoidal Rule

The integral is:

Points for study

1. What are the input values used for?
2. What value (in algebraic terms) does T have after Line 11?
3. What is the purpose of Lines 12-17?
4. Write a program based on the pseudo-code.
5. Apply your program to Exercises 7 and 8 of the Applied Exercises.

Gauss integration formula 

The integral is:

Use the Gauss two-point formula.

Points for study

1. What is the purpose of Lines 2 and 3?
2. What changes are required to produce an algorithm based on the Gauss three-point formula?
3. Write a computer program based on this pseudo-code.
4. Use your program to solve Exercises 7 and 8 of the Applied Exercises.

5. Runge-Kutta method 

   Process the equation y' = f (x,y) and use the usual fourth-order method.
   
   

   Points for study

   1. What are the input values used for?
   2. How many times is the function f evaluated between Lines 4 and 17?
   3. Amend the pseudo-code for use with the second-order Runge-Kutta method.
   4. Write a computer program based on the pseudo-code.
   5. Use the computer program to solve Exercises 9 and 10 of the Applied Exercises.