This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# NSolve

 NSolveattempts to find numerical approximations to the solutions of the system expr of equations or inequalities for the variables vars. NSolvefinds solutions over the domain of real numbers.
• The system expr can be any logical combination of:
 lhs==rhs equations lhs!=rhs inequations or inequalities exprdom domain specifications ForAll[x,cond,expr] universal quantifiers Exists[x,cond,expr] existential quantifiers
• A single variable or a list of variables can be specified.
• NSolve gives solutions in terms of rules of the form .
• When there are several variables, the solution is given in terms of lists of rules: .
• When there are several solutions, NSolve gives a list of them.
• When a single variable is specified and a particular root of an equation has multiplicity greater than one, NSolve gives several copies of the corresponding solution.
• NSolve assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
• In NSolve all variables, parameters, constants, and function values are restricted to be real.
• NSolve[expr&&varsReals, vars, Complexes] solves for real values of variables, but function values are allowed to be complex.
• NSolve deals primarily with linear and polynomial equations.
• The following options can be given:
 Method Automatic what method should be used WorkingPrecision Automatic precision to be used in computations
• NSolve gives if there are no solutions to the equations.
• NSolve gives if the set of solutions is full dimensional.
Approximate solutions to a polynomial equation:
Approximate real solutions to a polynomial equation:
Approximate solutions to a system of polynomial equations:
Approximate real solutions to a system of polynomial equations:
Approximate solutions to a polynomial equation:
 Out[1]=
Approximate real solutions to a polynomial equation:
 Out[2]=
Approximate solutions to a system of polynomial equations:
 Out[3]=
Approximate real solutions to a system of polynomial equations:
 Out[4]=
 Scope   (38)
Univariate polynomial equations:
Polynomial equations with inexact coefficients:
Polynomial equations with multiple roots:
Algebraic equations:
Transcendental equations:
Univariate elementary function equations over bounded regions:
Univariate holomorphic function equations over bounded regions:
Here NSolve finds some solutions but is not able to prove there are no other solutions:
Equation with a purely imaginary period over a vertical stripe in the complex plane:
Systems of linear equations:
Linear equations with inexact coefficients:
Underdetermined systems of linear equations:
Linear equations with no solutions:
Systems of polynomial equations:
Underdetermined systems of polynomial equations:
Algebraic equations:
Transcendental equations:
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:
Equations involving irrational real powers:
Equation with a double root:
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:
All variables are solved for:
Working precision can be given as the last argument:
 Options   (2)
By default NSolve introduces slicing hyperplanes for underdetermined complex systems:
With Method->{"UseSlicingHyperplanes"->False}, NSolve gives parametric solutions:
By default, NSolve finds solutions of exact equations using machine-precision computations:
This computes the solutions using 50-digit precision:
 Applications   (2)
Approximate solutions of a polynomial equation:
Find intersection points of a circle and a parabola:
Solutions approximately satisfy 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:
For univariate equations, NSolve repeats solutions according to their multiplicity:
Find solutions over specified domains:
NSolve is a global equation solver:
FindRoot is a local equation solver:
NSolve gives approximate results:
Use Solve to get exact solutions:
Use FindInstance to get exact solution instances:
Use NDSolve to solve differential equations numerically:
Solutions obtained with machine-precision numeric computations may not be accurate:
With higher WorkingPrecision more accurate results are produced:
Approximate solutions may not satisfy the equations due to numeric errors:
The equations are satisfied up to a certain tolerance:
Using higher WorkingPrecision will give solutions with smaller tolerance:
If the solutions set is infinite, NSolve gives its intersection with random hyperplanes:
Use ContourPlot and ContourPlot3D to view the real part of solutions:
Solve the equation :