Contents
- 1 What is catastrophic cancellation and what is caused by?
- 2 What’s the connection between subtraction and loss of significance?
- 3 How is catastrophic cancellation used in numerical analysis?
- 4 When is it possible to avoid catastrophic cancellation?
- 5 How is catastrophic cancellation avoided in binary64 arithmetic?
What is catastrophic cancellation and what is caused by?
It occurs when an operation on two numbers increases relative error substantially more than it increases absolute error, for example in subtracting two nearly equal numbers (known as catastrophic cancellation). The effect is that the number of significant digits in the result is reduced unacceptably.
What’s the connection between subtraction and loss of significance?
Loss of significance occurs when two nearly equal numbers are subtracted to produce a result much smaller than either of the original numbers. The effect is that the number of accurate (significant) digits in the result may be reduced unacceptably.
How do you cancel out division?
We can solve the equations below by “getting rid of” or “cancelling out” the division by multiplying (doing the opposite operation.) The same idea is used to solve the equation below: You can “get rid of” or “cancel” multiplication by dividing and you can “get rid of” or “cancel” division by multiplying.
How is catastrophic cancellation used in numerical analysis?
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. long. If you measure them with a ruler that is good only to the centimeter, you might get approximations
When is it possible to avoid catastrophic cancellation?
This phenomenon makes some problems very difficult to solve numerically. In some cases, catastrophic cancellation can be avoided. For example, when solving the quadratic equation x = 1000.001 ± 1000.001 2 − 4 2 = 100.001 ± 999.999 2. where the second root is very very inaccurate.
When does catastrophic cancellation happen in floating point arithmetic?
Catastrophic cancellation may happen even if the difference is computed exactly, as in the example above—it is not a property of any particular kind of arithmetic like floating-point arithmetic; rather, it is inherent to subtraction, when the inputs are approximations themselves.
How is catastrophic cancellation avoided in binary64 arithmetic?
, avoids catastrophic cancellation because it avoids introducing rounding error leading into the subtraction. . In IEEE 754 binary64 arithmetic, evaluating the alternative factoring , of which only half the digits are correct and the other half (underlined) are garbage.