Suppose you want to approximate a quantity *Q*, and you have available
approximations *Q*(*h*) for *h* > 0. Typically, you know that this
approximation is of a certain order, say
.
But often, more can be
said: for some
quantity *A* not depending on *h*,

The idea of Richardson extrapolation is to take two different values of

To eliminate the

The Richardson extrapolation value for

Its error, as an approximation to

For example, consider the approximation

Thus if we take

Another use of this technique is to get an estimate of the error in the
original approximation *Q*(*h*_{1}) or *Q*(*h*_{2}): this error should be approximately
or
respectively.
We have the (presumably) much more accurate approximation
,
but
we don't know how much more accurate! It won't hurt to use the
approximation
,
but, to be conservative, still use
or
as an indication of the error. So in the above example, we can estimate the errors
in *Q*(.1) and *Q*(.05) as about
*Q*_{R} - *Q*(.1) = .119110490 and
*Q*_{R} - *Q*(.05) =
.059555245 respectively (the actual errors are
.124539368 and
.064984123),
and we would expect the error in *Q*_{R} to be at most
.059555245 as well
(the actual error is
.005428878).

2002-01-29