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From: Cheng Cosine on 5 May 2010 13:07
Hi:

Interpolation is a common technique to determine the value of
dependent variable for a given independent variable when we only have
discrete data set.

However, sometimes we want to know the value of the independent
variable for a particular value of dependent variable.

Suppose again we only have discrete data set and the value of
dependent variable we are interested in does not fall in the data set
we have. How do we do then?

Must we first determine the approximate formula of the data set using
interpolation and then use root-finding technique to realize our
attempt?

Or can we simply switch the role of the dependent and independent
variables and then use interpolation technique?

Are these two approaches always equivalent and why?

Thanks,


From: William Elliot on 6 May 2010 06:57
On Wed, 5 May 2010, Cheng Cosine wrote:

> Interpolation is a common technique to determine the value of
> dependent variable for a given independent variable when we only have
> discrete data set.
>
> However, sometimes we want to know the value of the independent
> variable for a particular value of dependent variable.
>
> Suppose again we only have discrete data set and the value of
> dependent variable we are interested in does not fall in the data set
> we have. How do we do then?
>
> Must we first determine the approximate formula of the data set using
> interpolation and then use root-finding technique to realize our
> attempt?
>
> Or can we simply switch the role of the dependent and independent
> variables and then use interpolation technique?
>
> Are these two approaches always equivalent and why?
>
No if you're not using linear interpolation between two
adjacent data points both ways.

Even than, a problem can arise wit finding a data point with value 3.
with the data: (0,2), (1,4), (2,0).

Reversed data is (0,2), (2,0), (4,1).
Extrapolating at 3, one gets 1/2.

Using the original data, extrapolating at 1/2 gives 3
and extrapolating at 1.25 also gives 3. A root is missed.

If the data is increasing or decreasing, then I consider the
reverse extrapolation to be equivalent to approximately finding roots.

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