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Given the coordinates of the two points (Φ 1, L 1) and (Φ 2, L 2), the inverse problem finds the azimuths α 1, α 2 and the ellipsoidal distance s. Calculate U 1, U 2 and L, and set initial value of λ = L. Then iteratively evaluate the following equations until λ converges:
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
Sometimes, this multivalued inverse is called the full inverse of f, and the portions (such as √ x and − √ x) are called branches. The most important branch of a multivalued function (e.g. the positive square root) is called the principal branch , and its value at y is called the principal value of f −1 ( y ) .
In numerical analysis, inverse quadratic interpolation is a root-finding algorithm, meaning that it is an algorithm for solving equations of the form f(x) = 0. The idea is to use quadratic interpolation to approximate the inverse of f. This algorithm is rarely used on its own, but it is important because it forms part of the popular Brent's method.
A modular multiplicative inverse of an integer a with respect to the modulus m is a solution of the linear congruence a x ≡ 1 ( mod m ) . {\displaystyle ax\equiv 1{\pmod {m}}.} The previous result says that a solution exists if and only if gcd( a , m ) = 1 , that is, a and m must be relatively prime (i.e. coprime).
Condition numbers can also be defined for nonlinear functions, and can be computed using calculus.The condition number varies with the point; in some cases one can use the maximum (or supremum) condition number over the domain of the function or domain of the question as an overall condition number, while in other cases the condition number at a particular point is of more interest.
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Given a complex number z, there is not a unique complex number w satisfying erf w = z, so a true inverse function would be multivalued. However, for −1 < x < 1 , there is a unique real number denoted erf −1 x satisfying erf ( erf − 1 x ) = x . {\displaystyle \operatorname {erf} \left(\operatorname {erf} ^{-1}x\right)=x.}