equations 5.0.0

An equation-solving library that also works with complex numbers and fractions.

An equation-solving library written purely in Dart.

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The equations package is used to solve numerical analysis problems with ease. It's purely written in Dart, meaning it has no platform-specific dependencies and doesn't require Flutter to work. You can use equations, for example, in a Dart CLI project or a Flutter cross-platform application. Here's a summary of what you can do with this package:

• solve polynomial equations with Algebraic types;
• solve nonlinear equations with Nonlinear types;
• solve linear systems of equations with SystemSolver types;
• evaluate integrals with NumericalIntegration types;
• interpolate data points with Interpolation types.

In addition, you can also find utilities to work with:

• Real and complex matrices, using the Matrix<T> types;
• Complex number, using the Complex type;
• Fractions, using the Fraction and MixedFraction types.

This package has a type-safe API and requires Dart 3.0 (or higher versions). There is a demo project created with Flutter that shows an example of how this library could be used to develop a numerical analysis application 🚀

Equation Solver - Flutter web demo

The source code of the Flutter application can be found at example/flutter_example/. Visit the official pub.dev documentation for details about methods signatures and inline documentation.

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Algebraic (Polynomial equations)

Use one of the following classes to find the roots of a polynomial equation (also known as "algebraic equation"). 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
DurandKerner	Any polynomial P(xi) where xi are coefficients	xi ∈ C

There's a formula for polynomials up to the fourth degree, as explained by the Galois theory. Roots of polynomials whose degree is 5 or higher must be found using DurandKerner's method (or any other root-finding algorithm). For this reason, we suggest 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 DurandKerner to find the roots of a polynomial whose degree is 5 or higher.

Since DurandKerner works with any polynomial, you could also use it (for example) to solve a cubic equation. However, DurandKerner internally uses loops, derivatives, and other mechanics to approximate the actual roots. When the degree is 4 or lower, prefer working with Quartic, Cubic, Quadratic or Linear because they use direct formulas to find the roots (and thus they're more precise). Here's an example of how to find the roots of 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:
 *
 *  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 DurandKerner to solve the same equation:

// f(x) = (2-3i)x^3 + 6/5ix^2 - (-5+i)x - (9+6i)
final equation = DurandKerner(
  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);
}

As we've already pointed out, both ways are equivalent. However, DurandKerner is computationally slower than Cubic and doesn't always guarantee to converge to the correct roots. Use DurandKerner only when the degree of your polynomial is greater or equal to 5.

final quadratic = Algebraic.from(const [
  Complex(2, -3),
  Complex.i(),
  Complex(1, 6)
]);

final quartic = Algebraic.fromReal(const [
  1, -2, 3, -4, 5
]);

The factory constructor Algebraic.from() automatically returns the best type of Algebraic according to the number of parameters you've passed.

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
RegulaFalsi	a, b ∈ R

Expressions are parsed using petitparser: a fast, stable and well-tested grammar parser. Here's a simple example of how you can find the roots of an equation using Newton's method:

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 = 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;

Certain algorithms don't always guarantee to converge to the correct root so carefully read the documentation before choosing the method.

Systems of equations

Use one of the following classes to solve systems of linear equations. Only real coefficients are allowed (so double is ok, but Complex isn't) and you must define N equations in N variables (so square matrices only are allowed). This package supports the following algorithms:

Solver name	Iterative method
CholeskySolver	❌
GaussianElimination	❌
GaussSeidelSolver	✔️
JacobiSolver	✔️
LUSolver	❌
SORSolver	✔️

These solvers are used to find the x in the Ax = b equation. Methods require, at least, the system matrix A and the known values vector b. Iterative methods may require additional parameters such as an initial guess or a particular configuration value. For example:

// Solve a system using LU decomposition
final luSolver = LUSolver(
  equations: const [
    [7, -2, 1],
    [14, -7, -3],
    [-7, 11, 18]
  ],
  constants: const [12, 17, 5]
);

final solutions = luSolver.solve(); // [-1, 4, 3]
final determinant = luSolver.determinant(); // -84.0

If you just want to work with matrices (for operations, decompositions, eigenvalues, etc...) you can consider using either RealMatrix (to work with double) or ComplexMatrix (to work with Complex). Both are subclasses of Matrix<T> so they have the same public API:

final matrixA = RealMatrix.fromData(
  columns: 2,
  rows: 2,
  data: const [
    [2, 6],
    [-5, 0]
  ]
);

final matrixB = RealMatrix.fromData(
  columns: 2,
  rows: 2,
  data: const [
    [-4, 1],
    [7, -3],
  ]
);

final sum = matrixA + matrixB;
final sub = matrixA - matrixB;
final mul = matrixA * matrixB;
final div = matrixA / matrixB;

final lu = matrixA.luDecomposition();
final cholesky = matrixA.choleskyDecomposition();
final cholesky = matrixA.choleskyDecomposition();
final qr = matrixA.qrDecomposition();
final svd = matrixA.singleValueDecomposition();

final det = matrixA.determinant();
final rank = matrixA.rank();

final eigenvalues = matrixA.eigenvalues();
final characPolynomial = matrixA.characteristicPolynomial();

You can use toString() to print the matrix contents. The toStringAugmented() method prints the augmented matrix (the matrix + one extra column with the known values vector). For example:

final lu = LUSolver(
  equations: const [
    [7, -2, 1],
    [14, -7, -3],
    [-7, 11, 18]
  ],
  constants: const [12, 17, 5]
);

/*
 * Output with 'toString':
 *
 * [7.0, -2.0, 1.0]
 * [14.0, -7.0, -3.0]
 * [-7.0, 11.0, 18.0]
*/
print("$lu");

/*
 * Output with 'toStringAugmented':
 *
 * [7.0, -2.0, 1.0 | 12.0]
 * [14.0, -7.0, -3.0 | 17.0]
 * [-7.0, 11.0, 18.0 | 5.0]
*/
print("${lu.toStringAugmented()}");

The ComplexMatrix has the same API and the same usage as RealMatrix with the only difference that it works with complex numbers rather then real numbers.

Numerical integration

The "numerical integration" term refers to a group of algorithms for calculating the numerical value of a definite integral (on a given interval). The function must be continuous within the integration bounds. This package currently supports the following algorithms:

• MidpointRule
• SimpsonRule
• TrapezoidalRule

Other than the integration bounds (called lowerBound and lowerBound), the classes also have an optional parameter called intervals. It already has a good default value but of course you can change it!

const simpson = SimpsonRule(
  function: 'sin(x)*e^x',
  lowerBound: 2,
  upperBound: 4,
);

// Calculating the value of...
//
//   ∫ sin(x) * e^x dx
//
// ... between 2 and 4.
final results = simpson.integrate();

// Prints '-7.713'
print('${results.result.toStringAsFixed(3)}');

// Prints '32'
print('${results.guesses.length}');

The integrate() function returns an IntegralResults which simply is a wrapper for 2 values:

1. result: the value of the definite integral evaluated within lowerBound and lowerBound,
2. guesses: the various intermediate values that brought to the final result.

Interpolation

This package can also perform linear, polynomial or spline interpolations using the Interpolation types. You just need to give a few points in the constructor and then use compute(double x) to interpolate the value. The package currently supports the following algorithms:

• LinearInterpolation
• PolynomialInterpolation
• NewtonInterpolation
• SplineInterpolation

You'll always find the compute(double x) method in any Interpolation type, but some classes may have additional methods that others haven't. For example:

const newton = NewtonInterpolation(
  nodes: [
    InterpolationNode(x: 45, y: 0.7071),
    InterpolationNode(x: 50, y: 0.7660),
    InterpolationNode(x: 55, y: 0.8192),
    InterpolationNode(x: 60, y: 0.8660),
  ],
);

// Prints 0.788
final y = newton.compute(52);
print(y.toStringAsFixed(3));

// Prints the following:
//
// [0.7071, 0.05890000000000006, -0.005700000000000038, -0.0007000000000000339]
// [0.766, 0.053200000000000025, -0.006400000000000072, 0.0]
// [0.8192, 0.04679999999999995, 0.0, 0.0]
// [0.866, 0.0, 0.0, 0.0]
print('\n${newton.forwardDifferenceTable()}');

Since the Newton interpolation algorithm internally builds the "divided differences table", the API exposes two methods (forwardDifferenceTable() and backwardDifferenceTable()) to print those tables. Of course, you won't find forwardDifferenceTable() in other interpolation types because they just don't use it. By default, NewtonInterpolation uses the forward difference method but if you want the backward one, just pass forwardDifference: false in the constructor.

const polynomial = PolynomialInterpolation(
  nodes: [
    InterpolationNode(x: 0, y: -1),
    InterpolationNode(x: 1, y: 1),
    InterpolationNode(x: 4, y: 1),
  ],
);

// Prints -4.54
final y = polynomial.compute(-1.15);
print(y.toStringAsFixed(2));

// Prints -0.5x^2 + 2.5x + -1
print('\n${polynomial.buildPolynomial()}');

This is another example with a different interpolation strategy. The buildPolynomial() method returns the interpolation polynomial (as an Algebraic type) internally used to interpolate x.

Expressions parsing

You can use the ExpressionParser type to either parse numerical expressions or evaluate mathematical functions with the x variable:

const parser = ExpressionParser();

print(parser.evaluate('5*3-4')); // 11
print(parser.evaluate('sqrt(49)+10')); // 17
print(parser.evaluate('pi')); // 3.1415926535

print(parser.evaluateOn('6*x + 4', 3)); // 22
print(parser.evaluateOn('sqrt(x) - 3', 81)); // 6

This type is internally used by the library to evaluate nonlinear functions.