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Problems using InverseFunction

Posted 9 years ago

Hi all,

Do anyone know how Mathematica computes Inverse of a function using InverseFunction? The details of my issue are provided below:

First Issue:

I would like to evaluate the following:

Table[InverseFunction[(0.203*Log[#] - 0.218*Log[0.015 - 0.203*#])/0.015 &][i], {i, 1, 20, 1}]

There is no problem in computing the above. I get the answer to this almost instantly. But I couldn't evaluate the same function when my input is 201. i.e., When I do the following, Mathematica couldn't find an answer.

InverseFunction[(0.203Log[#] - 0.218Log[0.015 - 0.203*#])/ 0.015 &][201]

Second Issue:

Solving the Inverse function for 20 values takes almost no time. i.e., the following :

Table[InverseFunction[(0.203*Log[#] - 0.218*Log[0.015 - 0.203*#])/0.015 &][i], {i, 1, 20, 1}]

But if I try to evaluate the same using solve, it takes a long time even to Solve for a single value. i.e.,

Solve[-0.218Log[0.015 - 0.203 x] + 0.203*Log[x] == 3, x]

Please help me out what I'm missing and why the computations are taking time when I try evaluating the same using Solve and also why Mathematica couldn't evaluate the function value at certain points such as 201?

9 Replies


Thank you for your timely reply :)

Finally I could get correct answers provided I give the correct value for x to search (i.e., 0.1 in the above code you mentioned).

Can you please give me further tips on how to select the value for x to search for?

Having many orders of magnitude between the two results is fairly normal for a floating point error. Very often you have to come up with a technique specific to the problem. In this case rephrasing as a root finding problem seems sufficient to get a reasonable answer:

FindRoot[200/3 (-(109/500) Log[3/200 - (203 x)/1000] + (203 Log[x])/1000) - 201, {x, 0.1}]
POSTED BY: Sean Clarke

I think I was not clear in my previous post.

I tried to find the inverse function using 2 methods for the same expression. Directly solving using InverseFunction gave me a result which is 0.0101108 ; while solving it first by using Rationalize and then solving it using N gave me a result which is 4.26674*10^9.

Please note that there are many orders of magnitude difference between the both but not a floating point error.

The difference in value is probably floating point error. If you're comfortable knowing the result is close to 0, then the result might be fine. How much precision are you look for in your result? You may be able to get it using arbitrary precision arithmetic.

If InverseFunction isn't working out at all, then some serious numerical work will have to be done to get a good answer. Typically, computing the inverse of a function is viewed as a root finding problem. You may want to experiment with different Methods for FindRoot. The three statements below are essentially equivalent.

NSolve[f[x] == b,x]

x = InverseFunction[f][b]

FindRoot[f[x]-b, {x,0}]
POSTED BY: Sean Clarke

I would really appreciate if someone could help me deal with this issue.

Hi Clarke,

I tried what you said.

But doing so I got a completely different answer when I used the method you suggested me. Explanation as follows:

When I tried to compute the following expression directly in float, I got the following:

 InverseFunction[ (0.203*Log[#] - 0.218*Log[0.015 - 0.203*#])/0.015 &][1]

But when I did the same using the method you suggested me (first using rationalize and then solving it using N ), provided me altogether different solution and the solution doesn't change even when I change the input after the input is 24.

Rationalize[(0.203*Log[#] - 0.218*Log[0.015 - 0.203*#])/0.015]
200/3 (-(109/500) Log[3/200 - (203 #1)/1000] + (203 Log[#1])/1000)
N[InverseFunction[ 200/3 (-(109/500) Log[3/200 - (203 #1)/1000] + (203 Log[#1])/1000) &][1]]

Please tell me how to deal with this problem.

POSTED BY: Sean Clarke
POSTED BY: Sean Clarke
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