数学表达式计算-AngouriMath

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AngouriMath

asc-community/AngouriMath: New open-source cross-platform symbolic algebra library for C# and F#. Can be used for both production and research purposes. (github.com)

What is it about?

AngouriMath is an open source symbolic algebra library. That is, via AngouriMath, you can automatically solve equations, systems of equations, differentiate, parse from string, compile expressions, work with matrices, find limits, convert an expression to LaTeX, and many other things.

AngouriMath for research

As F#, great first-functional language, skyrocketing in the area of data analysis and interactive research, AngouriMath offers a few ways to conveniently work with symbolic expressions.

Notebooks

Notebooks provide amazing experience working with function visualization, for functions over one and two variables. With dotnet/interactive, it can be used in Visual Studio Code notebooks as well as Jupyter notebooks. To install the package, simply run this in the notebook:

#r "nuget:AngouriMath.Interactive,*-*"

Terminal

As both a demonstration sample and a convenient tool, this repository includes tool called AngouriMath.Terminal. It is a CLI-based program to interact with AngouriMath (as opposed to API-based interaction, that is, consuming it as a lib).

[ Download ]

Or build from sources:

git clone https://github.com/asc-community/AngouriMath
cd AngouriMath/Sources/Terminal/AngouriMath.Terminal
dotnet run -c release

See the online Jupyter notebook on how to use the F# API of AngouriMath. Note, that the C# API is still available via open AngouriMath command, then you can call the main library's methods.

See its source folder.

Read more about using AngouriMath for research on the website.

💻

Software Development [click 🖱️]

It is installed from nuget for both C# and F# and can be used by Web/Desktop/Mobile development.

Installing the library

Install AngouriMath from NuGet.
Write the following code:

using AngouriMath; using System;
Entity expr = "x + sin(y)";
Console.WriteLine(expr);

Run.

More detailed Quick Start.

If you are new to AM, we suggest you checking out some samples instead of reading boring documentation. If you want to contribute, we would be happy to welcome you in our community.

For any questions, feel free to contact us via Discord.

Official website: am.angouri.org.

Examples

Expand any section to see. Examples with live shell are on the website.

Computations

Use as a simple calculator:

Entity expr = "1 + 2 * log(3, 9)";
Console.WriteLine(expr.EvalNumerical());

$5$

Console.WriteLine("2 / 3 + sqrt(-16)".EvalNumerical());
>>> 2 / 3 + 4i

$\frac{2}{3} + 4i$

Console.WriteLine("(-2) ^ 3".EvalNumerical());

$-8$

Build expressions with variables and substitute them:

Entity expr = "2x + sin(x) / sin(2 ^ x)";
var subs = expr.Substitute("x", 0.3m);
Console.WriteLine(subs);

$2\times \frac{3}{10}+\frac{\sin\left(\frac{3}{10}\right)}{\sin\left(\sqrt[10]{2}^{3}\right)}$

Simplify complicated expressions:

Console.WriteLine("2x + x + 3 + (4 a * a^6) / a^3 / 5".Simplify());

$3+\frac{4}{5}\times {a}^{4}+3\times x$

var expr = "1/2 + sin(pi / 4) + (sin(3x)2 + cos(3x)2)";
Console.WriteLine(expr.Simplify());

$\frac{1}{2}\times \left(1+\sqrt{2}\right)+1$

Compiled functions work 15x+ faster

var x = MathS.Variable("x");
var expr = MathS.Sin(x) + MathS.Sqrt(x) / (MathS.Sqrt(x) + MathS.Cos(x)) + MathS.Pow(x, 3);
var func = expr.Compile(x);
Console.WriteLine(func.Substitute(3));

var expr = "sin(x) + sqrt(x) / (sqrt(x) + cos(x)) + x3";
var compiled = expr.Compile("x");
Console.WriteLine(compiled.Substitute(4));

Algebra

Start with boolean algebra:

Entity expr1 = "a and b or c";

// Those are the same
Entity expr3 = "a -> b";
Entity expr3 = "a implies b";

Entity expr = "a -> true";
Console.WriteLine(MathS.SolveBooleanTable(expr, "a"));

>>> Matrix[2 x 1]
>>> False
>>> True

Next, solve some equations:

Console.WriteLine("x^2 + x + a".SolveEquation("x"));

$\left\{\frac{-1-\sqrt{1-4\times a}}{2},\frac{-1+\sqrt{1-4\times a}}{2}\right\}$

Under developing now and forever (always available)

Entity expr = "(sin(x)^2 - sin(x) + a)(b - x)((-3) * x + 2 + 3 * x ^ 2 + (x + (-3)) * x ^ 3)";
Console.WriteLine(expr.SolveEquation("x").Latexise());

$\left\{-\left(-\arcsin\left(\frac{1-\sqrt{1-4\times a}}{2}\right)-2\times \pi\times n_{1}\right),-\left(-\pi--\arcsin\left(\frac{1-\sqrt{1-4\times a}}{2}\right)-2\times \pi\times n_{1}\right),-\left(-\arcsin\left(\frac{1+\sqrt{1-4\times a}}{2}\right)-2\times \pi\times n_{1}\right),-\left(-\pi--\arcsin\left(\frac{1+\sqrt{1-4\times a}}{2}\right)-2\times \pi\times n_{1}\right),\frac{-b}{-1},-i,i,1,2\right\}$

Try some inequalities:

Console.WriteLine("(x - 6)(x + 9) >= 0".Solve("x"));

$\left\{-9,6\right\}\cup\left(-\infty;-9\right)\cup\left(6;\infty\right)$

Systems of equations:

var system = MathS.Equations(
    "x^2 + y + a",
    "y - 0.1x + b"
);
Console.WriteLine(system);
var solutions = system.Solve("x", "y");
Console.WriteLine(solutions);

System:

$\begin{cases}{x}^{2}+y+a = 0\\y-\frac{1}{10}\times x+b = 0\\\end{cases}$

Result:

var system = MathS.Equations(
    "cos(x2 + 1)^2 + 3y",
    "y * (-1) + 4cos(x2 + 1)"
);
Console.WriteLine(system.Latexise());
var solutions = system.Solve("x", "y");
Console.WriteLine(solutions);

$\begin{cases}{\cos\left({x}^{2}+1\right)}^{2}+3\times y = 0\\y\times -1+4\times \cos\left({x}^{2}+1\right) = 0\\\end{cases}$ (solution matrix is too complicated to show)

Calculus

Find derivatives:

Entity func = "x^2 + ln(cos(x) + 3) + 4x";
Entity derivative = func.Differentiate("x");
Console.WriteLine(derivative.Simplify());

$4+\frac{\sin\left(x\right)}{{\ln\left(\cos\left(x\right)+3\right)}^{2}\times \left(\cos\left(x\right)+3\right)}+2\times x$

Find limits:

WriteLine("(a x^2 + b x) / (e x - h x^2 - 3)".Limit("x", "+oo").InnerSimplified);

$\frac{a}{-h}$

Find integrals:

WriteLine("x^2 + a x".Integrate("x").InnerSimplified);

$\frac{{x}^{3}}{3}+a\times \frac{{x}^{2}}{2}$

Sets

There are four types of sets:

WriteLine("{ 1, 2 }".Latexise());
WriteLine("[3; +oo)".Latexise());
WriteLine("RR".Latexise());
WriteLine("{ x : x^8 + a x < 0 }".Latexise());

$\left\{ 1, 2 \right\}$ $\left[3; \infty \right)$ $\mathbb{R}$ $\left\{ x : {x}^{8}+ a\times x < 0 \right\}$

And there operators:

WriteLine(@"A \/ B".Latexise());
WriteLine(@"A /\ B".Latexise());
WriteLine(@"A \ B".Latexise());

$A\cup B$ $A\cap B$ $A\setminus B$

Syntax

You can build LaTeX with AngouriMath:

var expr = "x ^ y + sqrt(x) + integral(sqrt(x) / a, x, 1) + derive(sqrt(x) / a, x, 1) + limit(sqrt(x) / a, x, +oo)";
Console.WriteLine(expr.Latexise());
>>> {x}^{y}+\sqrt{x}+\int \left[\frac{\sqrt{x}}{a}\right] dx+\frac{d\left[\frac{\sqrt{x}}{a}\right]}{dx}+\lim_{x\to \infty } \left[\frac{\sqrt{x}}{a}\right]

${x}^{y}+\sqrt{x}+\int\left[\frac{\sqrt{x}}{a}\right]dx+\frac{d\left[\frac{\sqrt{x}}{a}\right]}{dx}+\lim_{x\to\infty}\left[\frac{\sqrt{x}}{a}\right]$

You can parse Entity from string with

var expr = MathS.FromString("x + 2 + sqrt(x)");
Entity expr = "x + 2 + sqrt(x)";

A few convenient features: x2 => x^2, a x => a * x, (...)2 => (...)^2, 2(...) => 2 * (...)

Compilation

Now you can compile expressions with pritimives into native lambdas. They will be at least as fast as if you wrote them in line in code, or faster if you have same subexpressions in your expression.

Entity expr = "a and x > 3";
var func = expr.Compile<bool, double, bool>("a", "x");
WriteLine(func(true, 6));
WriteLine(func(false, 6));
WriteLine(func(true, 2));
WriteLine(func(false, 2));

Output:

True
False
False
False

Multithreading

You are guaranteed that all functions in AM run in one thread. It is also guaranteed that you can safely run multiple functions from AM in different threads, that is, all static variables and lazy properties are thread-safe.

There is also support of cancellation a task. However, to avoid injecting the cancellation token argument into all methods, we use AsyncLocal<T> instead. That is why instead of passing your token to all methods what you need is to pass it once to the MathS.Multithreading.SetLocalCancellationToken(CancellationToken) method.

There is a sample code demonstrating cancellation:

var cancellationTokenSource = new CancellationTokenSource();

// That goes instead of passing your token to methods
MathS.Multithreading.SetLocalCancellationToken(cancellationTokenSource.Token);

// Then you normally run your task
var currTask = Task.Run(() => InputText.Text.Solve("x"), cancellationTokenSource.Token);

try
{
    await currTask;
    LabelState.Text = currTask.Result.ToString();
}
catch (OperationCanceledException)
{
    LabelState.Text = "Operation canceled";
}

Download

Not everything is supported directly from F#, so if something missing, you will need to call the necessary methods from AngouriMath.

open Functions
open Operators
open Shortcuts

printfn "%O" (solutions "x" "x + 2 = 0")

printfn "%O" (simplified (solutions "x" "x^2 + 2 a x + a^2 = 0"))

printfn "%O" (``dy/dx`` "x^2 + a x")

printfn "%O" (integral "x" "x2 + e")

printfn "%O" (``lim x->0`` "sin(a x) / x")

printfn "%O" (latex "x / e + alpha + sqrt(x) + integral(y + 3, y, 1)")

C++ (Experimental)

At the moment, AngouriMath.CPP is in the experimental phase. See how to get AngouriMath for C++.

#include <AngouriMath.h>

int main()
{
    AngouriMath::Entity expr = "x y + 2";
    std::cout << expr.Differentiate("x");
}

Contribution

AngouriMath is a free open-source project, there is no big company backing us. That is why we warmly welcome any contributors to the project. Aside from volunteer donations, you can help developing the project: check the guide for developers.