Built-in Mathematica Symbol

NSolve

NSolve[lhs

rhs, var]
gives a list of numerical approximations to the roots of a polynomial equation.

NSolve[{eqn₁, eqn₂, ...}, {var₁, var₂, ...}]
solves a system of polynomial equations.

MORE INFORMATION

NSolve[eqns, vars, n] gives results to n-digit precision.

NSolve[eqns, vars] gives the same final result as N[Solve[eqns, vars]], apart from issues of numerical precision.

EXAMPLES

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Basic Examples (1)

Approximate solutions to a polynomial equation:

Approximate solutions to a system of polynomial equations:

Approximate solutions to a polynomial equation:

In[1]:=

Out[1]=

Approximate solutions to a system of polynomial equations:

In[2]:=

Out[2]=

Scope (5)

A univariate polynomial equation:

A system of polynomial equations with a finite number of solutions:

A system of polynomial equations with infinitely many solutions:

Polynomial equations with inexact coefficients:

Equations involving radicals:

Generalizations & Extensions (3)

Eliminate a variable and solve the resulting system:

Polynomial equations with parameters:

Transcendental equations:

Options (2)

By default NSolve computes a Gröbner basis in DegreeReverseLexicographic order:

Here using the Lexicographic order is faster, since polys already form a Gröbner basis:

By default the solutions are machine-precision numbers:

The values of polys at some of the machine precision-solutions are relatively large:

This computes solutions using 50 digits of precision:

The solutions computed with 50 digits of precision satisfy the equations much more accurately:

Applications (2)

Solve a polynomial equation with inexact coefficients:

Find intersection points of a circle and a parabola:

Properties & Relations (6)

NSolve gives approximate solutions to the equations:

Solutions are given as replacement rules and can be directly used for substitution:

NSolve uses {} to represent the empty or no solution:

NSolve uses {{}} to represent the universal solution or that all points satisfy the equations:

NSolve will produce solutions according to their multiplicity:

NSolve computes numeric approximations of solutions:

Use Solve to get exact values of solutions:

NSolve finds solutions in complex numbers:

Use Reduce to find solutions over specified domains:

Use N and ToRules to get approximations of the solutions given as replacement rules:

Use FindRoot to find an approximate solution instance starting the search at x=1 and y=-1:

Use FindInstance to find exact solution instances:

Use N to get a numeric approximation of the solution:

Use NDSolve to solve differential equations numerically:

Possible Issues (2)

If the solutions set is infinite, NSolve gives its intersection with random hyperplanes:

Solutions obtained with machine-precision numeric computations may not be accurate: