Finding Real Solutions of Polynomial Equations Calculator

dart_equations logo

An equation solving library written purely in Dart

Thanks to the equations package you will be able to solve numerical analysis problems with ease. It's been written purely in Dart, meaning that it has no platform-specific dependencies and it doens't require Flutter to work. You can use, for example, equations with Flutter for web, desktop and mobile. 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 is meant to be used with Dart 2.12 or higher because the code is entirely null safe. There is a demo, built with Flutter, that shows an example on how this library can be used (especially for numerical analysis apps) 🚀

Equation Solver logo

Equation Solver - Flutter web demo

The source code of the above website can be found at example/flutter_example. In the following lines, you'll find an overview of the various types in this package and their APIs; make sure to visit the pub.dev documentation for details about methods signatures and docstring comments.

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(x_i) where x_i are coefficients	x_i ∈ 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 DurandKerner'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 DurandKerner to find the roots of a polynomial whose degree is 5 or higher.

Note that DurandKerner can be used with any polynomials, so you could use it (for example) to solve a cubic equation as well. However, DurandKerner internally uses loops, derivatives, and other mechanics to approximate the actual roots. When trying to solve a polynomial equation whose degree is 4 or lower, prefer using Quartic, Cubic, Quadratic and Linear since they use direct formulas to find the roots. 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 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 but DurandKerner is computationally slower than Cubic and it doesn't guaranteed to always converge to the correct roots. Use DurandKerner only when the degree of your polynomial is greater or equal than 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 with the number of parameters you've passed.

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 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 always guarantee t converge to a correct root so read the documentation carefully before choosing the method.

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

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

In any case, solvers only work with square matrices so N equations in N variables. These solvers are used to find the x in the Ax = b equation. Methods require at least the equation system matrix A and the vector b containing the unknowns. Iterative methods may require additional parameters such as an initial guess or a particular configuration value.

          // 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, LU/Cholesky/QR/SVD decompositions, etc...) you can consider using either RealMatrix (to work with the double data type) or ComplexMatrix (to work with the Complex data type). Both classes are of type 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(); finak rank = matrixA.rank();

You can use toString() to print the content of the matrix but there's also the possibility to use toStringAugmented() which prints the augmented matrix (the matrix + one extra column with the known values vector).

          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.

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:

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 if you wanted to change the number of parts in which the interval will be split, just make sure to set 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 is a simple wrapper for 2 values:

result: the actual result, which is the value of the definite integral evaluated within lowerBound and lowerBound,
guesses: the various intermetiate values that brought to the final result.

If you want to perform linear, polynomial or spline interpolation, then you can use the Interpolation types provided by this package. You just need to provide a few points in the constructor and then use compute(double x) to interpolate the value. This package currently supports the following algorithms:

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 newtown interpolation algorithm internally builds the "divided differences table", the API exposes two methods (forwardDifferenceTable() and forwardDifferenceTable()) 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 backwards 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() returns the interpolation polynomial (as an Algebraic type) internally used to interpolate x.

Finding Real Solutions of Polynomial Equations Calculator

Source: https://pub.dev/packages/equations

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Finding Real Solutions of Polynomial Equations Calculator

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