![]() |
|||||
|
|||||
![]() |
|||||
![]() |
![]() |
How can d being too small cause problems? Usually computations on a calculator or computer or by hand are not performed to perfect accuracy. There are very small errors. Normally, these very small errors (called round off errors) can be ignored because the "noise" they represent in your evaluation is extremely small compared to the signal, which consists of the value of f itself. (A notable exception occurs when your answer is 0; then the machine's answer will be only the error it has created.) In general, if you take two very similar numbers, like f(x0 + d) and f(x0) and take their difference, that difference will be very much smaller than either term and the information in the signal represented by the difference will therefore be much smaller than the signal represented by either, while the noise level usually remains about the same for the terms and the difference. Taking the result of the subtraction and dividing by a very
small d (which is the same as multiplying by a huge If you make d smaller than the accuracy of your machine's computation, your answer will typically be off by more than 1, or your program will accuse you of dividing by 0 when you divide by d. The spreadsheet allows you to perform a very large number of calculations of this kind for a wide choice of d values with essentially no more work than is involved in one such calculation. This usually gives you the power to look for yourself and see where round off error is causing significant error. You will then be troubled by this effect only when the answer
you are computing is too far from the correct answer for d
values at which this effect becomes noticeable. In consequence
we try to make use of techniques that will allow us to get
accurate estimates for as large d values as possible. How? Set up a computation using one d value on one line of the
spreadsheet, then on the next line set d = the old If your estimate of the derivative were to home in on a value
and stay there, that would probably be the derivative you
seek. Alas this does not always happen. The estimates tend
to home in then start to move away again, as the effects of
round off error make themselves felt. However there is something much better that generally does
home in on a value that is recognizable as the derivative
you seek, and it takes no more work! Instead of computing
|
![]() |