equations 2.0.0-nullsafety.1

An equation solving library written in Dart. It also works with complex numbers and fractions.

An equation solving library written purely in Dart

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Thanks to equations you're able to solve polynomial and nonlinear equations with ease. It's been written in "pure" Dart, meaning that it has no
dependency on any framework. It can be used with Flutter for web, desktop and mobile. Here's a summary of the contents of the package:

• Algebraic and all of its subtypes, which can be used to solve algebraic equations (also known as polynomial equations);
• Nonlinear and all of its subtypes, which can be used to solve nonlinear equations;
• Complex, which is used to easily handle complex numbers;
• Fraction, from the fraction package which helps you working with fractions.

This package is meant to be used with Dart 2.12 or higher because the code is entirely null safe!

Algebraic equations

Use one of the following classes to find the roots of a polynomial. You can use both complex numbers and fractions as coefficients.

Solver name	Equation	Params field
Constant	f(x) = a	a ∈ C
Linear	f(x) = ax + b	a, b ∈ C
Quadratic	f(x) = ax2 + bx + c	a, b, c ∈ C
Cubic	f(x) = ax3 + bx2 + cx + d	a, b, c, d ∈ C
Quartic	f(x) = ax4 + bx3 + cx2 + dx + e	a, b, c, d, e ∈ C
Laguerre	Any polynomial P(xi) where xi are coefficients	xi ∈ C

There's a formula for polynomials up to the fourth degree, as explained by Galois Theory. Roots of polynomials whose degree is 5 or higher, must be seeked using Laguerre's method or any other root-finding algorithm. For this reason, we suggest to go for the following approach:

• Use Linear to find the roots of a polynomial whose degree is 1.
• Use Quadratic to find the roots of a polynomial whose degree is 2.
• Use Cubic to find the roots of a polynomial whose degree is 3.
• Use Quartic to find the roots of a polynomial whose degree is 4.
• Use Laguerre to find the roots of a polynomial whose degree is 5 or higher.

Note that Laguerre can be used with any polynomials, so you could use it (for example) to solve a cubic equation as well. Laguerre internally uses loops, derivatives and other mechanics that are much slower than Quartic, Cubic, Quadratic and Linear so use it only onle when really needed. Here's how you can solve a cubic:

// f(x) = (2-3i)x^3 + 6/5ix^2 - (-5+i)x - (9+6i)
final equation = Cubic(
  a: Complex(2, -3),
  b: Complex.fromImaginaryFraction(Fraction(6, 5)),
  c: Complex(5, -1),
  d: Complex(-9, -6)
);

final degree = equation.degree; // 3
final isReal = equation.isRealEquation; // false
final discr = equation.discriminant(); // -31299.688 + 27460.192i

// f(x) = (2 - 3i)x^3 + 1.2ix^2 + (5 - 1i)x + (-9 - 6i)
print("$equation");
// f(x) = (2 - 3i)x^3 + 6/5ix^2 + (5 - 1i)x + (-9 - 6i)
print(equation.toStringWithFractions());

/*
 * Prints the roots of the equation:
 *
 *  x1 = 0.348906207844 - 1.734303423032i
 *  x2 = -1.083892638909 + 0.961044482775
 *  x3 = 1.011909507988 + 0.588643555642
 * */
for (final root in equation.solutions()) {
  print(root);
}

Alternatively, you could have used Laguerre to solve the same equation:

// f(x) = (2-3i)x^3 + 6/5ix^2 - (-5+i)x - (9+6i)
final equation = Laguerre(
  coefficients: [
    Complex(2, -3),
    Complex.fromImaginaryFraction(Fraction(6, 5)),
    Complex(5, -1),
    Complex(-9, -6),
  ]
);

/*
 * Prints the roots of the equation:
 *
 *  x1 = 1.0119095 + 0.5886435
 *  x2 = 0.3489062 - 1.7343034i
 *  x3 = -1.0838926 + 0.9610444
 * */ 
for (final root in equation.solutions()) {
  print(root);
}

Nonlinear equations

Use one of the following classes, representing a root-finding algorithm, to find a root of an equation. Only real numbers are allowed. This package
supports the following root finding methods:

Solver name	Params field
Bisection	a, b ∈ R
Chords	a, b ∈ R
Netwon	x0 ∈ R
Secant	a, b ∈ R
Steffensen	x0 ∈ R
Brent	a, b ∈ R

Expressions are parsed using petitparser, a fasts, stable and well tested grammar parser. These algorithms only
work with real numbers. Here's a simple example of how you can find the roots of an equation:

final newton = Newton("2*x+cos(x)", -1, maxSteps: 5);

final steps = newton.maxSteps; // 5
final tol = newton.tolerance; // 1.0e-10
final fx = newton.function; // 2*x+cos(x)
final guess = newton.x0; // -1

final solutions = await newton.solve();

final convergence = solutions.convergence.round(); // 2
final solutions = solutions.efficiency.round(); // 1

/*
 * The getter `solutions.guesses` returns the list of values computed by the algorithm
 *
 * -0.4862880170389824
 * -0.45041860473199363
 * -0.45018362150211116
 * -0.4501836112948736
 * -0.45018361129487355
 */
final List<double> guesses = solutions.guesses;

Note that certain algorithms don't guarantee the convergence to a root so read the documentation carefully before choosing the method.