# Numerical Analysis/Loss of Significance

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

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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.

## Contents

## 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 .

Solution:

In (**bound**
), "x" is and "y" is 1; "q" = 1. We have . Solving for x, we have , which is our lower bound on x.

### 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 .

Solution:

In (**bound**
), "x" is and "y" is 1; "q" = 3. We have . Solving for x, we have , which is our lower bound on x.

### 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?

Solution:

We have

### 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.

Solution:

In (**bound**
), "x" is , "y" is , and "q" is 1.
We have .
Solving for x gives the interval .

### 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.

Solution:

We have

### Exercise 6[edit]

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

Solution:

We have

Comparing this equation to the general form,

and using the w:quadratic formula

we see that the variable in our equation is "h", a = 1, b = 2x, and c = x^2. We use the quadratic formula to solve for "h".

The quadratic formula itself can be a cause of w:loss of significance if the quantity "4ac" is very small. This can be remedied by not subtracting.

If "b" (in this case, "2x") is positive, subtraction can be avoided by using

Unfortunately, this gives a value for "h" that is always negative, which is unacceptable. Using one of w:Vieta's formulas,

which gives positive values for h. If "b" (here, "2x") is negative, we use

which, again, always gives a negative value for h. However, Vieta's formula gives

which returns positive values for h. The bounds on h are then:

### Exercise 7[edit]

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

Solution:

Rationalize the expression:

- .

Evaluating this expression at gives an answer of .

### Exercise 8[edit]

Rewrite so that loss of significance will be minimized.

Solution:

Use the Quotient Property of logarithms to rewrite as .

### 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.

Solution:

Giving both terms a common denominator and combining them into a single rational expression reduces loss of significance. We then have

### 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.

Solution:

The loss of significance occurs in the numerator, so rewrite the numerator using a trigonometric identity to get