SoatDev IT Consulting
SoatDev IT Consulting
  • About us
  • Expertise
  • Services
  • How it works
  • Contact Us
  • News
  • June 3, 2025
  • Rss Fetcher

“The inverse of an approximation is an approximation of the inverse.” This statement is either obvious or really clever. Maybe both.

Logs and exponents

Let’s start with an example. For moderately small x,

10^x approx frac{1 + x}{1 - x}

That means that near 1 (i.e. near 10 raised to a small number,

log_{10}x approx frac{1-x}{1 + x}

The inverse of a good approximation is a good approximation of the inverse. But the measure of goodness might not be the same in both uses of the word “good.” The inverse of a very good approximation might be a mediocre approximation of the inverse, depending on the sensitivity of the problem. See, for example, Wilkinson’s polynomial.

Inverse function theorem

Now let’s look at derivatives and inverse functions. The derivative of a function is the best linear approximation to that function at a point. And it’s true that the best linear approximation to the inverse of a function at a point is the inverse of the best linear approximation to the function. In symbols,

frac{dx}{dy} = frac{1}{dfrac{dy}{dx}}

This is a good example of a bell curve meme, or coming full circle. The novice naively treats the symbols above as fractions and says “of course.” The sophomore says “Wait a minute. These aren’t fractions. You can’t just do that.” And they would be correct, except the symbols are limits of fractions, and in this context naive symbol manipulation in fact leads to the correct result.

However, the position on the right side of the meme is nuanced. Treating derivatives as fractions, especially partial derivatives, can get you into trouble.

The more rigorous statement of the inverse function theorem is harder to parse visually, and so the version above is a good mnemonic. If f(x) = y then

bigl(f^{-1}bigr)^prime (y) = frac{1}{f^prime(x)} = frac{1}{f^prime(f^{-1}(y))}

The same is true in multiple variables. If f is a smooth function from ℝn to ℝn then the best linear approximation to f at a point is a linear transformation, which can be represented by a matrix M. And the best linear approximation to the inverse function at that point is the linear transformation represented by the inverse of M. In symbols,

Dbigl(f^{-1}bigr)(y) = left[Dfbigl(f^{-1}(y)bigr)right]^{-1}

There’s fine print, such as saying a function has to be differentiable in a neighborhood of x and the derivative has to be non-singular, but that’s the gist of the inverse function theorem.

Note that unlike in the first section, there’s no discussion of how good the approximations are. The approximations are exact, at a point. But it might still be the case that the linear approximation on one side is good over a much larger neighborhood than the linear approximation on the other side.

The post Approximation of Inverse, Inverse of Approximation first appeared on John D. Cook.

Previous Post
Next Post

Recent Posts

  • OpenAI’s marketing head takes leave to undergo breast cancer treatment
  • Kaspersky Exposes Labubu Doll Scams on Fraudulent Sites
  • IRCAI & Zindi, in Partnership with AWS, Announce AI for Equity Challenge Winners
  • Questions about Gemini, Claude, and ChatGPT? Prompt engineering is the answer
  • How is Technology Modernizing Recruitment in Temporary Employment Services

Categories

  • Industry News
  • Programming
  • RSS Fetched Articles
  • Uncategorized

Archives

  • June 2025
  • May 2025
  • April 2025
  • February 2025
  • January 2025
  • December 2024
  • November 2024
  • October 2024
  • September 2024
  • August 2024
  • July 2024
  • June 2024
  • May 2024
  • April 2024
  • March 2024
  • February 2024
  • January 2024
  • December 2023
  • November 2023
  • October 2023
  • September 2023
  • August 2023
  • July 2023
  • June 2023
  • May 2023
  • April 2023

Tap into the power of Microservices, MVC Architecture, Cloud, Containers, UML, and Scrum methodologies to bolster your project planning, execution, and application development processes.

Solutions

  • IT Consultation
  • Agile Transformation
  • Software Development
  • DevOps & CI/CD

Regions Covered

  • Montreal
  • New York
  • Paris
  • Mauritius
  • Abidjan
  • Dakar

Subscribe to Newsletter

Join our monthly newsletter subscribers to get the latest news and insights.

© Copyright 2023. All Rights Reserved by Soatdev IT Consulting Inc.