Math.NET Symbolics

Math.NET Symbolics is a basic open source computer algebra library for .NET, Silverlight and Mono written entirely in F#.

This project does not aim to become a full computer algebra system. If you need such a system, have a look at Axiom or Maxima instead, or for proprietary commercial solutions Maple, Mathematica or Wolfram Alpha.

You'll find a large set of expression and algebraic operator examples in the Unit Tests (yes, they're actually very readable). A few examples:

• (3Q + 2)*4/6 → 10/3.
• (a/b/(c*a))*(c*d/a)/d → 1/(a*b)
• (a+b)/(b+a)**2 → 1/(a + b)
• Algebraic.expand ((a+b)**3) → a^3 + 3*a^2*b + 3*a*b^2 + b^3
• Exponential.expand (exp(2*x+y)) → exp(x)^2*exp(y)
• Exponential.contract (exp(x)*(exp(x) + exp(y))) → exp(2*x) + exp(x + y)
• Exponential.simplify (1/(exp(x)*(exp(y)+exp(-x))) - (exp(x+y)-1)/((exp(x+y))**2-1)) → 0
• Trigonometric.expand (sin(2*x)) → 2*sin(x)*cos(x)
• Trigonometric.contract (sin(x)**2*cos(x)**2) → 1/8 - (1/8)*cos(4*x)
• Trigonometric.simplify ((cos(x)+sin(x))**4 + (cos(x)-sin(x))**4 + cos(4*x) - 3) → 0
• Polynomial.polynomialDivision x (x**3 - 2*x**2 - 4) (x-3) → (3 + x + x^2, 5)
• Polynomial.polynomialExpansion x y (x**5 + 11*x**4 + 51*x**3 + 124*x**2 + 159*x + 86) (x**2 + 4*x + 5) → 1 + x + (2 + x)*y + (3 + x)*y^2
• Polynomial.gcd x (x**7 - 4*x**5 - x**2 + 4) (x**5 - 4*x**3 - x**2 + 4) → 4 - 4*x - x^2 + x^3
• Rational.rationalize (1+1/(1+1/x)) → (1 + 2*x)/(1 + x)
• Rational.simplify x ((x**2-1)/(x+1)) → -1 + x

let taylor (k:int) symbol x a =
    let rec impl n nf acc dxn =
        if n = k then acc else
        impl (n+1) (nf*(n+1)) (acc + (dxn |> Structure.substitute symbol a)/nf*(symbol-a)**n) (Calculus.differentiate symbol dxn)
    impl 0 1 zero x |> Algebraic.expand

taylor 3 x (1/(1-x)) 0Q       → 1 + x + x^2
taylor 3 x (1/x) 1Q           → 3 - 3*x + x^2
taylor 3 x (ln(x)) 1Q         → -3/2 + 2*x - (1/2)*x^2
taylor 4 x (ln(x)) 1Q         → -11/6 + 3*x - (3/2)*x^2 + (1/3)*x^3
taylor 4 x (sin(x)+cos(x)) 0Q → 1 + x - (1/2)*x^2 - (1/6)*x^3

Literature

• Computer Algebra and Symbolic Computation - Elementary Algorithms, Joel. S. Cohen
• Computer Algebra and Symbolic Computation - Mathematical Methods, Joel. S. Cohen
• Modern Computer Algebra, Second Edition, Joachim von zur Gathen, Jürgen Gerhard
• Symbolic Integration I - Transcendental Functions, Second Edition, Manuel Bronstein
• Concrete Mathematics, Second Edition, Graham, Knuth, Patashnik
• ... and of course the fundamental theory by Euclid, Newton, Gauss, Fermat and Hilbert.

Project

Windows (.NET):
Linux (Mono):

Maintained by Christoph Rüegg and part of the Math.NET initiative (see also Math.NET Numerics). It is covered under the terms of the MIT/X11 open source license. See also the license file in the root folder. We accept contributions!

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