Recently in Inversion Category

Found In Space

Some time ago we saw how Newton's method used the derivative of a univariate scalar valued function to guide the search for an argument at which it took a specific value. A related problem is finding a vector at which a multivariate vector valued function takes one, or at least comes as close as possible to it. In particular, we should often like to fit an arbitrary parametrically defined scalar valued functional form to a set of points with possibly noisy values, much as we did using linear regression to find the best fitting weighted sum of a given set of functions, and in this post we shall see how we can generalise Newton's method to solve such problems.

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Every Secant Counts

In recent posts we have concerned ourselves with numerically approximating inverses of functions, seeking a value x for which f(x) equals some desired value y.
We began with the bisection method which, whilst terrifically stable, was unfortunately rather slow, so we went on to discuss Newton's method which used the derivative of the function to make a potentially far more accurate estimate of the location of the inverse, albeit at the risk of making a catastrophically worse one; a risk we eliminated by switching back to bisection in the event that it did so.
Unfortunately, it's not always easy to calculate the derivative of a function, especially if we're numerically approximating it, so ideally we should seek an algorithm that is not quite so blind to the behaviour of the function as is the bisection method but that does not require the explicit caclulation of derivatives.

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On The Shoulders Of Gradients

In the last post we implemented the bisection method for the numerical approximation of the inverse of a function. Now this is an extremely stable algorithm, but annoyingly not a particularly efficient one.
Fortunately, the great Isaac Newton is coming to our rescue with a significantly more efficient approach...

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Rooting Around For Answers

A common task in numerical computing is to approximate the inverse of a function. Specifically, given a function f such that

  y = f(x)

we shall often want to find a function f-1 such that

  x = f-1(y)

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