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 (STEP 7)

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 (STEP 8)

    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 (STEP 10)

    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 (STEP 11)

    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 (STEP 13)

    The system is:

    Points for study

    1. What is the purpose of the number s?
    2. What are the y1, y2 1, . . ., yn used for?
    3. Why is it possible to replace the yj in Line 13 by xj?
    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

  • Newton divided difference formula (STEP 24)

    You are to calculate for given data x0, x1, . . . , xn, f(x0), f(x1), . . ., f(xn), and for given the interpolating polynomial Pn(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 dii calculated are the divided differences f (x0, . . .,.xI).
    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), . . ., Pn-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(STEP 30)

    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 (STEP 32)

    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 (STEP 33)

      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.
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