Univariate polynomial equations:

Polynomial equations with inexact coefficients:

Polynomial equations with multiple roots:

Find five roots of a polynomial of a high degree:

Algebraic equations:

Transcendental equations:

Find all solutions:

Specify the number of solutions returned:

Univariate elementary function equations over bounded regions:

Univariate holomorphic function equations over bounded regions:

Equation with a purely imaginary period over a vertical stripe in the complex plane:

Find all solutions:

Unrestricted transcendental function equations:

Systems of linear equations:

Linear equations with inexact coefficients:

Underdetermined systems of linear equations:

Linear equations with no solutions:

Systems of polynomial equations:

Find five out of a trillion roots of a polynomial system:

Underdetermined systems of polynomial equations:

Algebraic equations:

Transcendental equations:

Specify the number of solutions returned:

Polynomial equations:

Polynomial equations with multiple roots:

Algebraic equations:

Piecewise equations:

Transcendental equations, solvable using inverse functions:

Transcendental equations, solvable using special function zeros:

Transcendental inequalities, solvable using special function zeros:

Exp-log equations:

High-degree sparse polynomial equations:

Algebraic equations involving high-degree radicals:

Equations involving irrational real powers:

Tame elementary function equations:

Elementary function equations in bounded intervals:

Holomorphic function equations in bounded intervals:

Linear systems:

Polynomial systems:

Quantified polynomial systems:

Algebraic systems:

Piecewise systems:

Transcendental systems solvable using inverse functions:

Systems exp-log in the first variable and polynomial in the other variables:

Quantified system:

Systems elementary and bounded in the first variable and polynomial in the other variables:

Quantified system:

Systems holomorphic and bounded in the first variable and polynomial in the other variables:

Quantified system:

Mixed real and complex variables:

Solve over special regions in 2D:

Plot it:

Solve over special regions in 3D:

Plot it:

An implicitly defined region:

A parametrically defined region:

Derived regions:

Plot it:

Regions dependent on parameters:

Plot it:

Find values of parameters , , and for which the circles contain the given points:

Plot it:

Use to specify that is a vector in :

In this case is a vector in :

Find out of roots of a polynomial:

Find out of roots of a polynomial system:

Find out of infinitely many solutions of a transcendental system:

With the default Automatic setting NSolve may not give all solutions:

With MaxRootsInfinity, NSolve attempts to find all solutions:

Solve a square polynomial system using the automatically chosen method:

Use the "EndomorphismMatrix" method:

Use the "Homotopy" method:

Use the "Monodromy" method:

Use the "Symbolic" method:

This system has roots, which is strictly less than the bound of provided by the Bernstein-Khovanskii-Kushnirenko theorem:

The "Homotopy" method, used by default, returns multiple copies of some of the roots:

Remove the multiple copies:

The "Monodromy" method runs faster here and does not produce multiple copies of roots:

The "Monodromy" method returns finitely many solutions of transcendental systems:

Use the "Symbolic" method to obtain all solutions:

The Method option may also be used to locally set system options from the "NSolveOptions" group:

By default NSolve introduces slicing hyperplanes for underdetermined complex systems:

With Method->{"UseSlicingHyperplanes"->False}, NSolve gives parametric solutions:

NSolve verifies solutions obtained using non-equivalent transformations:

With VerifySolutions->False, NSolve does not verify the solutions:

Some of the solutions returned with VerifySolutions->False are not correct:

This uses a fast numeric test in an attempt to select correct solutions:

In this case the simple numeric verification gives the correct solution set:

By default, NSolve finds solutions of exact equations using machine-precision computations:

This computes the solutions using 50-digit precision:

Find intersection points of a circle and a parabola:

Find the intersection of InfiniteLine[{0,0},{1,1}] and InfiniteLine[{{0,1},{1,0}}]:

Plot it:

Find the intersections of InfiniteLine[{0,0},{1,1}] and Circle[{0,0},1]:

Plot it:

Find all pairwise intersections between five random lines:

Use BooleanCountingFunction to express that exactly two conditions are true:

Plot it:

Find the pairwise intersections of the circles Circle[{1/3 Cos[k 2π/5],1/3 Sin[k 2π/5]}] for k=0,…,4:

Plot it:

Find the intersection of InfiniteLine[{{-1,1,1},{1,1,1}}] and InfinitePlane[{{2,0,0},{0,2,0},{0,0,2}}]:

Plot it:

Find the intersections of InfiniteLine[{{-1,1,1},{1,1,1}}] and Sphere[{0,0,0},3]:

Plot it:

Find the intersections of InfiniteLine[{{-1,1/3,1/2},{1,1/3,1/2}}] and the boundary of Tetrahedron[{{0,0,0},{1,0,0},{0,1,0},{0,0,1}}]:

Plot it:

Find the intersection for three random planes:

Plot it:

Find the intersections of the spheres Sphere[{1/3 Cos[k 2π/3],1/3 Sin[k 2π/3],0}] for k=0,1,2:

Plot it:

Find all the intersections of exactly three planes among 10 random planes:

Use BooleanCountingFunction to find the condition of exactly three things being true:

Plot it:

A polynomial system for the equilibria of a certain chemical reaction network:

Since the variables represent chemical species quantities, we are interested in real-valued solutions with all components nonnegative::

Find the solutions:

The apparent multiplicity is an artifact of the Automatic method, which tends to be fast but can sometimes overstate multiplicity:

Direct kinematics of a Gough-Stewart parallel 6-degree of freedom platform:

This solution set claims multiple solutions; remove the multiple copies:

A different method does not produce multiple copies of solutions:

Set up an overdetermined system of six equations in four variables that arises from a camera pose estimation procedure:

Use the first four polynomials so the subsystem will be exactly determined:

For each solution the first four residuals from plugging in a solution are small whereas the last two are non-negligible:

Now polish each solution by using it as a starting point for minimizing the sum of squares of residuals:

A system of equations for static equilibria of a weight suspended by two massless cables in a plane:

Each real-valued solution corresponds to a distinct equilibrium position, and there are six such:

A reduced 8-dimensional system arising in economics; we use a non-default method to get better accuracy:

Find the real-valued solutions:

A nonlinear system of polynomial difference equations:

RSolve cannot find an analytic form for the solution:

Set up a polynomial system to find the asymptotic values:

Solve it and give asymptotic possible real asymptotic values (they depend on initial conditions):