InverseBetaRegularized

The inverse of the regularized incomplete beta $I_x(a, b)$ in $x$: the Beta-distribution quantile. Solved the same way as InverseGammaRegularized — a safeguarded Newton's method against native BetaRegularized.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageInverseBetaRegularized✓↗

InverseBetaRegularized(s, a, b)solves $s = I_x(a, b)$ for $x$.

mpmathenumeratio_inverse_beta_regularized(s, a, b)Rustinverse_beta_regularized(a, b, s)

Typed signatures

Type	Package	Applies
(number, number, number) -> number	analytic	always

Domain: Special functions

Details

• $I_x(1, 1) = x$ is its own inverse, and $I_x(a, 1) = x^a$ inverts to $s^{1/a}$ — both exact, symbolic $s$ (and $a$) included.
• Otherwise: a safeguarded Newton's method in $\ln x$ against BetaRegularized(x, a, b) (above $s = 1/2$, on the upper tail $I_y(b, a) = 1 - s$), using the Beta density $x^{a-1}(1-x)^{b-1}/B(a,b)$ (in log space) as the derivative; checked against wolframscript to 20 digits.

Plot

Examples

6 more kept as data, checked against the oracles

Basic #

The median of Beta(2, 3)

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In$\mathrm{InverseBetaRegularized}(0.5, 2, 3)$Out$0.385\,727\,568\,132\,389\,5$

Known value · Wolfram 15.0.0 InverseBetaRegularized

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In$\mathrm{InverseBetaRegularized}(0.3, 2.5, 1.5)$Out$0.509\,497\,412\,428\,341\,3$

Known value · Wolfram 15.0.0 InverseBetaRegularized

$I_x(2, 1) = x^2$, so the inverse at $1/4$ is $1/2$

live ↗

In$\mathrm{InverseBetaRegularized}(0.25, 2, 1)$Out$0.5$

Properties #

$I_x(1, 1) = x$ is its own inverse

live ↗

In$\mathrm{InverseBetaRegularized}(s, 1, 1)$Out$s$

$I_x(a, 1) = x^a$ inverts to $s^{1/a}$

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In$\mathrm{InverseBetaRegularized}(s, a, 1)$Out$\sqrt[a]{s}$

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In$\mathrm{InverseBetaRegularized}(0, 2, 3)$Out$0$

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In$\mathrm{InverseBetaRegularized}(1, 2, 3)$Out$1$

About

Examples	13
Checked against	Wolfram Language 6 agree2 unevaluatedmpmath 10 agree1 errorRust 7 agree3 differ1 error
Crosswalk	Wolfram LanguagempmathRust

Implementation

primitive · kernela kernel — an algorithm over mutable state that would not be readable as a tree

nativeenginetypescriptpackages/symbols/analysis/analytic/src/inverse-regularized.ts

Safeguarded Newton/bisection against native BetaRegularized.

mappedrustpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

See also: BetaRegularized, InverseGammaRegularized