diff --git a/README.md b/README.md index 8ba54e2..9c9171c 100644 --- a/README.md +++ b/README.md @@ -1,4 +1,4 @@ -[![Python 3](http://img.shields.io/badge/python-3-blue.svg)](https://docs.python.org/3/) +[![This project supports Python 3.10+](https://img.shields.io/badge/Python-3.10+-blue.svg)](https://python.org/downloads) [![pre-commit](https://img.shields.io/badge/pre--commit-enabled-brightgreen?logo=pre-commit&logoColor=white)](https://github.com/theochem/matrix-permanent/actions/workflows/pull_request.yml) [![GNU GPLv3](https://img.shields.io/badge/license-%20%20GNU%20GPLv3%20-green?style=plastic)](https://www.gnu.org/licenses/gpl-3.0.en.html) @@ -37,9 +37,10 @@ Compute the permanent of a matrix using the best algorithm for the shape of the Compute the permanent of a matrix combinatorically. **Formula:** -```math + +$$ \text{per}(A) = \sum_{\sigma \in P(N,M)}{\prod_{i=1}^M{a_{i,{\sigma(i)}}}} -``` +$$ **Parameters:** @@ -55,13 +56,13 @@ Compute the permanent of a matrix combinatorically. **Formula:** -```math +$$ \text{per}(A) = \frac{1}{2^{N-1}} \cdot \sum_{\delta \in \left[\delta_1 = 1,~ \delta_2 \dots \delta_N=\pm1\right]}{ \left(\sum_{k=1}^N{\delta_k}\right){\prod_{j=1}^N{\sum_{i=1}^N{\delta_i a_{i,j}}}}} -``` +$$ **Additional Information:** -The original formula has been generalized here to work with $M$-by-$N$ rectangular permanents with +The original formula has been generalized here to work with $M$ by $N$ rectangular permanents with $M \leq N$ by use of the following identity (shown here for $M \geq N$): ```math @@ -70,14 +71,14 @@ $M \leq N$ by use of the following identity (shown here for $M \geq N$): This can be neatly fit into the original formula by extending the inner sums over $\delta$ from $[1,M]$ to $[1,N]$: -```math +$$ \text{per}(A) = \frac{1}{2^{N-1}} \cdot \frac{1}{(N - M + 1)!}\cdot \sum_{\delta \in \left[\delta_1 = 1,~ \delta_2 \dots \delta_N=\pm1\right]}{ \left(\sum_{k=1}^N{\delta_k}\right) \prod_{j=1}^N{\left( \sum_{i=1}^M{\delta_i a_{i,j}} + \sum_{i=M+1}^N{\delta_i} \right)} } -``` +$$ **Parameters:** @@ -93,7 +94,7 @@ This can be neatly fit into the original formula by extending the inner sums ove **Formula:** -```math +$$ \text{per}(A) = \sum_{k=0}^{M-1}{ {(-1)}^k \binom{N - M + k}{k} @@ -103,7 +104,7 @@ This can be neatly fit into the original formula by extending the inner sums ove } } } -``` +$$ **Parameters:** @@ -181,4 +182,4 @@ is with pip. # License This code is distributed under the GNU General Public License version 3 (GPLv3). -See for more information. \ No newline at end of file +See for more information.