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 $z$.

mpmathenumeratio_inverse_gamma_regularized(a, s)

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 in $\ln z$ (falls back to bisection whenever a step would leave the bracket) against the forward function — GammaRegularized(a, z), or the continued fraction for $Q$ where $z \ge a + 1$, which keeps its digits in the far tail — whose own accuracy was checked directly against wolframscript to 20 digits. A root the double solve cannot vouch for is left unevaluated.
• 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 $0 < s < 1$.

Plot

Examples

6 more kept as data, checked against the oracles

Basic #

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

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In$\mathrm{InverseGammaRegularized}(1, 0.5)$Out$0.693\,147\,180\,559\,945\,3$

The median of the Gamma(2, 1) distribution

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In$\mathrm{InverseGammaRegularized}(2, 0.5)$Out$1.678\,346\,990\,016\,660\,8$

Known value · Wolfram 15.0.0 InverseGammaRegularized

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In$\mathrm{InverseGammaRegularized}(2.5, 0.3)$Out$3.032\,214\,992\,077\,453$

Known value · Wolfram 15.0.0 InverseGammaRegularized

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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

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In$\mathrm{FullSimplify}(\mathrm{MatrixFunction}(\operatorname{\_1}↦\mathrm{InverseGammaRegularized}(1, \operatorname{\_1}), \bigl\lbrack\bigl\lbrack1/2, 1/5\bigr\rbrack, \bigl\lbrack1/3, 1/2\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack-\frac{1}{2}\ln((1-(2×15^{1/2})/15)/2)-\frac{1}{2}\ln((1+(2×15^{1/2})/15)/2), \frac{1}{10}\sqrt{15}\ln((1-(2×15^{1/2})/15)/2)-\frac{1}{10}\sqrt{15}\ln((1+(2×15^{1/2})/15)/2)\bigr\rbrack, \bigl\lbrack\frac{1}{6}\sqrt{15}\ln((1-(2×15^{1/2})/15)/2)-\frac{1}{6}\sqrt{15}\ln((1+(2×15^{1/2})/15)/2), -\frac{1}{2}\ln((1-(2×15^{1/2})/15)/2)-\frac{1}{2}\ln((1+(2×15^{1/2})/15)/2)\bigr\rbrack\bigr\rbrack$

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In$N(\mathrm{InverseGammaRegularized}(2, \frac{33}{100}), 50)$Out$2.303\,708\,141\,330\,049\,053\,229\,408\,137\,466\,818\,316\,144\,848\,837\,258\,7$

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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 #

$Q(a, 0) = 1$

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In$\mathrm{InverseGammaRegularized}(a, 1)$Out$0$

$Q(a, \infty) = 0$

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In$\mathrm{InverseGammaRegularized}(a, 0)$Out$∞$

Order 1 inverts $e^{-z}$

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In$\mathrm{InverseGammaRegularized}(1, s)$Out$-\ln(s)$

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

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In$\mathrm{InverseGammaRegularized}(1, 0)$Out$∞$

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

The same pole at $a=2$

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In$\mathrm{InverseGammaRegularized}(2, 0)$Out$∞$

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

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

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In$\mathrm{InverseGammaRegularized}(1, 1)$Out$0$

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

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

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In$\mathrm{InverseGammaRegularized}(2, 1)$Out$0$

Known value · Wolfram MathematicalFunctionData, InverseGammaRegularized

About

Examples	20
Checked against	Wolfram Language 13 agree2 differ1 unevaluatedmpmath 6 agree
Crosswalk	Wolfram Languagempmath

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.

mappedmpmathpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

SymPy has no inverse of the regularized incomplete gamma (no gammaincinv); lowergamma, uppergamma and gammainc exist but nothing solves for the argument.

See also: GammaRegularized, Gamma, InverseBetaRegularized