Numerical Analysis/Loss of Significance

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Loss of significance occurs in numerical calculations when too many significant digits cancel. As a result of the floating point arithmetic used by computers, when a number is subtracted from another number that is almost exactly the same, catastrophic cancellation may occur and an erroneous value returned. As an example, consider the behavior of

as x approaches 0. Evaluating this function at using Matlab incorrectly returns the answer 0, which shows that too many significant digits have cancelled.

If x and y are positive, normalized floating point binary numbers such that andString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(bound)

String Module Error: function rep expects a number as second parameter, received "

"

then at most and at least significant binary bits are lost in the subtraction .

On this page we will consider several exercises/examples of using this formula and show how sometimes we can rearrange the calculation to reduce loss of significance. Please try the exercise yourself before revealing the solution.

Exercises[edit]

Exercise 1[edit]

Use to find a lower bound on the input x if one desires to lose no more than 1 significant binary bit in the calculation of .

Exercise 2[edit]

Use to find a lower bound on the input x if one desires to lose at most 3 significant binary bits in the calculation of .

Exercise 3[edit]

Consider

As x gets very large, loss of significance can occur. What is the bound on x if we want to lose no more than one binary digit?

Exercise 4[edit]

Now consider the function

For large values of x, loss of significance may occur.

Use (bound ) to find a bound on the input so that at most 1 significant binary bit will be lost in the calculation.

Exercise 5[edit]

In w:Numerical differentiation, the w:Finite difference

is often used to approximate derivatives. w:Truncation error can be reduced by decreasing "h", the step size, but if h becomes too small, loss of significance can become a factor. For , find a bound on h such that at most 1 binary bit will be lost in the calculation.

Exercise 6[edit]

For , find a bound on h such that at most 1 binary bit will be lost in the calculation, if .

Exercise 7[edit]

Rewrite so that there is no loss of significance, then evaluate it at .

Exercise 8[edit]

Rewrite so that loss of significance will be minimized.

Exercise 9[edit]

For very small values of x, loss of significance can occur in . Rewrite this function in a way that will minimize loss of significance.

Exercise 10[edit]

As x gets close to zero, loss of significance can occur in . Rewrite this function in a way that will minimize loss of significance.