InverseGammaRegularized

The inverse of the regularized incomplete gamma $Q(a, z)$ in $z$ : solves $s = Q(a, z)$ . Solved numerically by a safeguarded Newton's method against compute-engine's own GammaRegularized, using the Gamma density (in log space) as the derivative.

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

InverseGammaRegularized(a, s)solves

s = Q(a, z)

for

z

.

Typed signatures

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

Domain: Special functions

Details

$Q(a, 0) = 1$ and $Q(a, \infty) = 0$ hold for any $a$ , even a symbolic one, so InverseGammaRegularized(a, 1) = 0 and InverseGammaRegularized(a, 0) = \infty are exact regardless.
$Q(1, z) = e^{-z}$ inverts exactly to $-\ln s$ , symbolic $s$ included.
Otherwise: a safeguarded Newton's method (falls back to bisection whenever a step would leave the bracket) against the forward function GammaRegularized(a, z), whose own accuracy was checked directly against wolframscript to 20 digits.
N(x, d) past a double's digits polishes the double answer with Newton's method on the bignum series for $P(a, z)$ , for real $a > 0$ and $0 < s < 1$ .

Examples

2 more kept as data, checked against the oracles

Basic #

$Q(1, z) = e^{-z}$ , so the inverse at $1/2$ is $\ln 2$

In

\mathrm{InverseGammaRegularized}(1, 0.5)

Out

0.693\,147\,180\,559\,945\,3

The median of the Gamma(2, 1) distribution

In

\mathrm{InverseGammaRegularized}(2, 0.5)

Out

1.678\,346\,990\,016\,660\,8

Known value · Wolfram 15.0.0 InverseGammaRegularized

In

\mathrm{InverseGammaRegularized}(2.5, 0.3)

Out

3.032\,214\,992\,077\,453

Known value · Wolfram 15.0.0 InverseGammaRegularized

In

\mathrm{FullSimplify}(\operatorname{Normal}(\operatorname{Series}(\mathrm{InverseGammaRegularized}(2, x), x, -1, 1)))

Out

(x+1)\mathrm{InverseGammaRegularized}^{(0, 1)}(2, -1)+\mathrm{InverseGammaRegularized}(2, -1)

Scope #

Matrix function expressed through logarithms of the eigenvalues

In

\mathrm{FullSimplify}(\mathrm{MatrixFunction}(()↦\mathrm{InverseGammaRegularized}(1, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1/2, 1/5\bigr\rbrack, \bigl\lbrack1/3, 1/2\bigr\rbrack\bigr\rbrack))

Out

\bigl\lbrack\bigl\lbrack-\frac{1}{2}(\ln((1-(2×15^{1/2})/15)/2))-\frac{1}{2}(\ln((1+(2×15^{1/2})/15)/2)), \frac{1}{10}(\sqrt{15}\ln((1-(2×15^{1/2})/15)/2))-\frac{1}{10}(\sqrt{15}\ln((1+(2×15^{1/2})/15)/2))\bigr\rbrack, \bigl\lbrack\frac{1}{6}(\sqrt{15}\ln((1-(2×15^{1/2})/15)/2))-\frac{1}{6}(\sqrt{15}\ln((1+(2×15^{1/2})/15)/2)), -\frac{1}{2}(\ln((1-(2×15^{1/2})/15)/2))-\frac{1}{2}(\ln((1+(2×15^{1/2})/15)/2))\bigr\rbrack\bigr\rbrack

In

N(\mathrm{InverseGammaRegularized}(2, \frac{33}{100}), 50)

Out

2.303\,708\,141\,330\,049\,053\,229\,408\,137\,466\,818\,316\,144\,848\,837\,258\,7

In

\operatorname{Series}(\mathrm{InverseGammaRegularized}(a, s), s, \mathrm{s0}, 1)

Out

\mathrm{InverseGammaRegularized}(a, \mathrm{s0})+(s-\mathrm{s0})\frac{∂}{∂\mathrm{s0}} \mathrm{InverseGammaRegularized}(a, \mathrm{s0})+O\left((s-\mathrm{s0})^2\right)

Properties #

$Q(a, 0) = 1$

In

\mathrm{InverseGammaRegularized}(a, 1)

Out

0

$Q(a, \infty) = 0$

In

\mathrm{InverseGammaRegularized}(a, 0)

Out

∞

Order 1 inverts $e^{-z}$

In

\mathrm{InverseGammaRegularized}(1, s)

Out

-\ln(s)

At $a=1$ , $Q(1,z)=e^{-z}$ , so the value 0 gives $z=\infty$

In

\mathrm{InverseGammaRegularized}(1, 0)

Out

∞

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

The same pole at $a=2$

In

\mathrm{InverseGammaRegularized}(2, 0)

Out

∞

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

Probability 1 gives $z=0$ at $a=1$

In

\mathrm{InverseGammaRegularized}(1, 1)

Out

0

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

Probability 1 gives $z=0$ at $a=2$

In

\mathrm{InverseGammaRegularized}(2, 1)

Out

0

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

About

Examples	16
Checked against	Wolfram Language 13 agree2 differ1 unevaluated
Crosswalk	Wolfram Language

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 GammaRegularized.

See also: GammaRegularized, Gamma, InverseBetaRegularized