diff --git a/CHANGELOG.md b/CHANGELOG.md --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -1,3 +1,8 @@ 1.0.0.0 ------- * initial release + +1.1.0.0 +------- +* upgrade version bounds of the 'containers' dependency +* fixed LaTeX code in README diff --git a/README.md b/README.md --- a/README.md +++ b/README.md @@ -2,39 +2,39 @@ ## Evaluation of the hypergeometric function of a matrix argument (Koev & Edelman's algorithm) -Let $(a\_1, \ldots, a\_p)$ and $(b\_1, \ldots, b\_q)$ be two vectors of real or +Let $(a_1, \ldots, a_p)$ and $(b_1, \ldots, b_q)$ be two vectors of real or complex numbers, possibly empty, $\alpha > 0$ and $X$ a real symmetric or a complex Hermitian matrix. The corresponding *hypergeometric function of a matrix argument* is defined by -$${}\_pF\_q^{(\alpha)} \left(\begin{matrix} a\_1, \ldots, a\_p \\\\ b\_1, \ldots, b\_q\end{matrix}; X\right) = \sum\_{k=0}^{\infty}\sum\_{\kappa \vdash k} \frac{{(a\_1)}\_{\kappa}^{(\alpha)} \cdots {(a\_p)}\_{\kappa}^{(\alpha)}} {{(b\_1)}\_{\kappa}^{(\alpha)} \cdots {(b\_q)}\_{\kappa}^{(\alpha)}} \frac{C\_{\kappa}^{(\alpha)}(X)}{k!}.$$ +$${}_pF_q^{(\alpha)} \left(\begin{matrix} a_1, \ldots, a_p \\\\ b_1, \ldots, b_q\end{matrix}; X\right) = \sum_{k=0}^{\infty}\sum_{\kappa \vdash k} \frac{{(a_1)}_{\kappa}^{(\alpha)} \cdots {(a_p)}_{\kappa}^{(\alpha)}} {{(b_1)}_{\kappa}^{(\alpha)} \cdots {(b_q)}_{\kappa}^{(\alpha)}} \frac{C_{\kappa}^{(\alpha)}(X)}{k!}.$$ The inner sum is over the integer partitions $\kappa$ of $k$ (which we also -denote by $|\kappa| = k$). The symbol ${(\cdot)}\_{\kappa}^{(\alpha)}$ is the +denote by $|\kappa| = k$). The symbol ${(\cdot)}_{\kappa}^{(\alpha)}$ is the *generalized Pochhammer symbol*, defined by -$${(c)}^{(\alpha)}\_{\kappa} = \prod\_{i=1}^{\ell}\prod\_{j=1}^{\kappa\_i} \left(c - \frac{i-1}{\alpha} + j-1\right)$$ +$${(c)}^{(\alpha)}_{\kappa} = \prod_{i=1}^{\ell}\prod_{j=1}^{\kappa_i} \left(c - \frac{i-1}{\alpha} + j-1\right)$$ -when $\kappa = (\kappa\_1, \ldots, \kappa\_\ell)$. -Finally, $C\_{\kappa}^{(\alpha)}$ is a *Jack function*. +when $\kappa = (\kappa_1, \ldots, \kappa_\ell)$. +Finally, $C_{\kappa}^{(\alpha)}$ is a *Jack function*. Given an integer partition $\kappa$ and $\alpha > 0$, and a real symmetric or complex Hermitian matrix $X$ of order $n$, the Jack function -$$C\_{\kappa}^{(\alpha)}(X) = C\_{\kappa}^{(\alpha)}(x\_1, \ldots, x\_n)$$ +$$C_{\kappa}^{(\alpha)}(X) = C_{\kappa}^{(\alpha)}(x_1, \ldots, x_n)$$ is a symmetric homogeneous polynomial of degree $|\kappa|$ in the -eigen values $x\_1$, $\ldots$, $x\_n$ of $X$. +eigen values $x_1$, $\ldots$, $x_n$ of $X$. The series defining the hypergeometric function does not always converge. See the references for a discussion about the convergence. The inner sum in the definition of the hypergeometric function is over all partitions $\kappa \vdash k$ but actually -$C\_{\kappa}^{(\alpha)}(X) = 0$ when $\ell(\kappa)$, the number of non-zero +$C_{\kappa}^{(\alpha)}(X) = 0$ when $\ell(\kappa)$, the number of non-zero entries of $\kappa$, is strictly greater than $n$. -For $\alpha=1$, $C\_{\kappa}^{(\alpha)}$ is a *Schur polynomial* and it is +For $\alpha=1$, $C_{\kappa}^{(\alpha)}$ is a *Schur polynomial* and it is a *zonal polynomial* for $\alpha = 2$. In random matrix theory, the hypergeometric function appears for $\alpha=2$ and $\alpha$ is omitted from the notation, implicitely assumed to be $2$. @@ -42,19 +42,19 @@ Koev and Edelman (2006) provided an efficient algorithm for the evaluation of the truncated series -$$\sideset{\_p^m}{\_q^{(\alpha)}}F \left(\begin{matrix} a\_1, \ldots, a\_p \\\\ b\_1, \ldots, b\_q\end{matrix}; X\right) = \sum\_{k=0}^{m}\sum\_{\kappa \vdash k} \frac{{(a\_1)}\_{\kappa}^{(\alpha)} \cdots {(a\_p)}\_{\kappa}^{(\alpha)}} {{(b\_1)}\_{\kappa}^{(\alpha)} \cdots {(b\_q)}\_{\kappa}^{(\alpha)}} -\frac{C\_{\kappa}^{(\alpha)}(X)}{k!}.$$ +$$\sideset{_p^m}{_q^{(\alpha)}}F \left(\begin{matrix} a_1, \ldots, a_p \\\\ b_1, \ldots, b_q\end{matrix}; X\right) = \sum_{k=0}^{m}\sum_{\kappa \vdash k} \frac{{(a_1)}_{\kappa}^{(\alpha)} \cdots {(a_p)}_{\kappa}^{(\alpha)}} {{(b_1)}_{\kappa}^{(\alpha)} \cdots {(b_q)}_{\kappa}^{(\alpha)}} +\frac{C_{\kappa}^{(\alpha)}(X)}{k!}.$$ Hereafter, $m$ is called the *truncation weight of the summation* (because $|\kappa|$ is called the weight of $\kappa$), the vector -$(a\_1, \ldots, a\_p)$ is called the vector of *upper parameters* while -the vector $(b\_1, \ldots, b\_q)$ is called the vector of *lower parameters*. -The user has to supply the vector $(x\_1, \ldots, x\_n)$ of the eigenvalues +$(a_1, \ldots, a_p)$ is called the vector of *upper parameters* while +the vector $(b_1, \ldots, b_q)$ is called the vector of *lower parameters*. +The user has to supply the vector $(x_1, \ldots, x_n)$ of the eigenvalues of $X$. For example, to compute -$$\sideset{\_2^{15}}{\_3^{(2)}}F \left(\begin{matrix} 3, 4 \\\\ 5, 6, 7\end{matrix}; 0.1, 0.4\right)$$ +$$\sideset{_2^{15}}{_3^{(2)}}F \left(\begin{matrix} 3, 4 \\\\ 5, 6, 7\end{matrix}; 0.1, 0.4\right)$$ you have to enter @@ -91,10 +91,10 @@ For $n = 1$, the hypergeometric function of a matrix argument is known as the [generalized hypergeometric function](https://mathworld.wolfram.com/HypergeometricFunction.html). -It does not depend on $\alpha$. The case of $\sideset{\_{2\thinspace}^{}}{\_1^{}}F$ is the most known, +It does not depend on $\alpha$. The case of $\sideset{_{2\thinspace}^{}}{_1^{}}F$ is the most known, this is the Gauss hypergeometric function. Let's check a value. It is known that -$$\sideset{\_{2\thinspace}^{}}{\_1^{}}F \left(\begin{matrix} 1/4, 1/2 \\\\ 3/4\end{matrix}; 80/81\right) = 1.8.$$ +$$\sideset{_{2\thinspace}^{}}{_1^{}}F \left(\begin{matrix} 1/4, 1/2 \\\\ 3/4\end{matrix}; 80/81\right) = 1.8.$$ Since $80/81$ is close to $1$, the convergence is slow. We compute the truncated series below for $m = 300$. diff --git a/hypergeomatrix.cabal b/hypergeomatrix.cabal --- a/hypergeomatrix.cabal +++ b/hypergeomatrix.cabal @@ -1,49 +1,55 @@ -cabal-version: 2.2 -name: hypergeomatrix -version: 1.0.0.0 -synopsis: Hypergeometric function of a matrix argument -description: Evaluation of hypergeometric functions of a matrix argument, - following Koev & Edelman's algorithm. -homepage: https://github.com/stla/hypergeomatrix#readme -license: BSD-3-Clause -license-file: LICENSE -author: Stéphane Laurent -maintainer: laurent_step@outlook.fr -copyright: 2022 Stéphane Laurent -category: Math, Numeric -build-type: Simple -extra-source-files: README.md - CHANGELOG.md +cabal-version: 2.2 +name: hypergeomatrix +version: 1.1.0.0 +license: BSD-3-Clause +license-file: LICENSE +copyright: 2022 Stéphane Laurent +maintainer: laurent_step@outlook.fr +author: Stéphane Laurent +homepage: https://github.com/stla/hypergeomatrix#readme +synopsis: Hypergeometric function of a matrix argument +description: + Evaluation of hypergeometric functions of a matrix argument, + following Koev & Edelman's algorithm. +category: Math, Numeric +build-type: Simple +extra-source-files: + README.md + CHANGELOG.md + +source-repository head + type: git + location: https://github.com/stla/hypergeomatrix + library - hs-source-dirs: src - exposed-modules: Math.HypergeoMatrix - other-modules: Math.HypergeoMatrix.HypergeoMatrix - , Math.HypergeoMatrix.Internal - , Math.HypergeoMatrix.Gaussian - build-depends: base >= 4.7 && < 5 - , array >= 0.5.4.0 && < 0.6 - , containers >= 0.6.4.1 && < 0.7 - , cyclotomic >= 1.1.1 && < 1.2 - other-extensions: BangPatterns - , DefaultSignatures - , ScopedTypeVariables - , TypeFamilies - , TypeSynonymInstances - default-language: Haskell2010 - ghc-options: -Wall + exposed-modules: Math.HypergeoMatrix + hs-source-dirs: src + other-modules: + Math.HypergeoMatrix.HypergeoMatrix + Math.HypergeoMatrix.Internal + Math.HypergeoMatrix.Gaussian -test-suite unit-tests - type: exitcode-stdio-1.0 - main-is: Main.hs - hs-source-dirs: tests/ - other-modules: Approx - Build-Depends: base >= 4.7 && < 5 - , tasty - , tasty-hunit - , hypergeomatrix - Default-Language: Haskell2010 + default-language: Haskell2010 + other-extensions: + BangPatterns DefaultSignatures ScopedTypeVariables TypeFamilies + TypeSynonymInstances -source-repository head - type: git - location: https://github.com/stla/hypergeomatrix + ghc-options: -Wall + build-depends: + base >=4.7 && <5, + array >=0.5.4.0 && <0.6, + containers >=0.6.5.1 && <0.7, + cyclotomic >=1.1.1 && <1.2 + +test-suite unit-tests + type: exitcode-stdio-1.0 + main-is: Main.hs + hs-source-dirs: tests/ + other-modules: Approx + default-language: Haskell2010 + build-depends: + base >=4.7 && <5, + tasty >=1.4.2.3 && <1.5, + tasty-hunit >=0.10.0.3 && <0.11, + hypergeomatrix -any