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All errors are equal, but some errors are more equal than others...

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• Sources of Errors in Numerical Computational Computer Programs
  • There are errors and errors - different kinds of errors in numerical analysis computer programs

  • The following is an abbreviate treatise of various kinds of errors and some techniques in tracking down the kind of errors and fix it.

• Types of Errors
  • Syntax errors
  • Program logic errors
  • Rounding errors
  • Discretization errors (in numerical methods)

• Syntax Errors

  • What are they:
    • Syntax errors are errors in statements in a computer program that do not conform to the language structure of the programming language.

  • Example:

    X := 5;

  • What are the symptoms:
    • Program will not compile - the compiler will output error messages

  • How to solve the problem:
    • Get a manual on the programming language and learn the correct syntax.

• Program Logic Errors

  • What are they:
    • Program logic errors are errors in statements in a computer program that are the result of a muddy/unclear thought process...

  • Example:

    Program segment that determine a letter grade:

    Incorrect: Correct: if ( NumGrade >= 60 ) if ( NumGrade >= 90 ) LetterGrade = 'D'; LetterGrade = 'A'; else if ( NumGrade >= 70 ) else if ( NumGrade >= 80 ) LetterGrade = 'C'; LetterGrade = 'B'; else if ( NumGrade >= 80 ) else if ( NumGrade >= 70 ) LetterGrade = 'B'; LetterGrade = 'C'; else if ( NumGrade >= 90 ) else if ( NumGrade >= 60 ) LetterGrade = 'A'; LetterGrade = 'D';

  • What are the symptoms:
    • Program compiles, but does not produce the correct result.

  • How to solve the problem:
    • The most difficult part of the solution is to pinpoint the location of the error.

    • You can try put print-statements in key locations of program to assertain that the values are computed correctly

    • Or use a debugger to set break points at the suspected error location and display the values of key variables when program stops at the pre-defined break points

• Rounding error

  • What are they:
    • Rounding errors are errors that are caused by the computer's inability to represent floating number perfectly

      Computation with inaccurate values may resulting in small and even very large errors.

  • Example:
    • Try run this program and see for yourself: click here -----

  • What are the symptoms:
    • Program compiles, but does not produce the correct result.

  • How to solve the problem:
    • It is usually cause by very large or very small values stored in variables.

    • Use a debugger and set breakpoint on suspected variables becoming too large or too small.

    • The problem can be lessen by using a higher precision representation

    • If that does not help, you need to find a better method (in the above example, you need to find a better pivot).

• Discretization error

  • What are they:
    • Discretization errors are errors that are caused by the discretization method's inability to solve the mathematical problem perfectly

      Each discretization method has an inherent error that is usually dependent on the step size

  • Example:
    • The rectangle rule method to approximate a definite integral using step size h has an inherent error of O(h)

  • What are the symptoms:
    • Program outputs different results. for different step size.

  • How to solve the problem:
    • There is nothing you can do; just be aware of how accurate your solution is.