Machine Arithmetic and Related Matters

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Consider an irrational number like root (8) = 2.82842712474619009760…, which is represented by infinite non-repeating digits.

Practically it would require infinite digits to be represented much a computer store in the form of a finite digit which leads to error.

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There are multiple ways to represent a number and floating-point representation and fixed representation are few of them. The computer uses these to store a number (i.e. a finite collection of decimals to represent a number).

Since representing a number with finite digits lead to the generation of error also if the output is used further then the error propagates. Thus, we must use a well-conditioned algorithm as well as the problem.

The condition number is used to define how sensitive the output is concerning changes in the input. Thus, a problem with a high condition will yield results out of huge errors.

We can manipulate our problem in such a way that the condition number is close to one. Consider a number such as (x-a) where x is a variable and a is a constant.

The condition number for this is given by |x/(x-a)|. You can see that as limit x->a the value of the condition number tends to infinity thus you need to manipulate the expression to convert it into a well-conditioned problem.

Look for a few mechanical problems and try to find the condition number of those problems. If they have a high condition number, then manipulate the expression so that the condition number comes close to unity.

You’ll notice that that minor changes in the input do not affect the output of Newtonian systems much but If you were to consider quantum mechanics you’ll find out that even minor changes or inaccuracy in the value leads to high relative error which means the condition number was high.

Since you aim to use machines to reduce your calculation time you should always make sure to analyze the problem you’re feeding into a specific algorithm to avoid getting erroneous results.

Back to: Numerical Analysis

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