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FindRoot
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FindRoot

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FindRoot[f,{x,x0}]

searches for a numerical root of f, starting from the point x=x0.

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FindRoot[lhs==rhs,{x,x0}]

searches for a numerical solution to the equation lhs==rhs.

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FindRoot[{f1,f2,…},{{x,x0},{y,y0},…}]

searches for a simultaneous numerical root of all the fi.

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FindRoot[{eqn1,eqn2,…},{{x,x0},{y,y0},…}]

searches for a numerical solution to the simultaneous equations eqni.

Details and Options

  • If the starting point for a variable is given as a list, the values of the variable are taken to be lists with the same dimensions.
  • FindRoot returns a list of replacements for x, y, … , in the same form as obtained from Solve.
  • FindRoot first localizes the values of all variables, then evaluates f with the variables being symbolic, and then repeatedly evaluates the result numerically.
  • FindRoot has attribute HoldAll, and effectively uses Block to localize variables.
  • FindRoot[lhs==rhs,{x,x0,x1}] searches for a solution using x0 and x1 as the first two values of x, avoiding the use of derivatives.
  • FindRoot[lhs==rhs,{x,xstart,xmin,xmax}] searches for a solution, stopping the search if x ever gets outside the range xmin to xmax.
  • If you specify only one starting value of x, FindRoot searches for a solution using Newton methods. If you specify two starting values, FindRoot uses a variant of the secant method.
  • If all equations and starting values are real, then FindRoot will search only for real roots. If any are complex, it will also search for complex roots.
  • You can always tell FindRoot to search for complex roots by adding 0.I to the starting value.
  • The following options can be given:
  • AccuracyGoalAutomaticthe accuracy sought
    EvaluationMonitor Noneexpression to evaluate whenever equations are evaluated
    Jacobian Automaticthe Jacobian of the system
    MaxIterations 100maximum number of iterations to use
    TemplateBox[{Method, paclet:ref/Method}, RefLink, BaseStyle -> {3ColumnTableMod}] Automaticmethod to be used
    PrecisionGoalAutomaticthe precision sought
    StepMonitor Noneexpression to evaluate whenever a step is taken
    WorkingPrecision MachinePrecisionthe precision to use in internal computations
  • The default settings for AccuracyGoal and PrecisionGoal are WorkingPrecision/2.
  • The setting for AccuracyGoal specifies the number of digits of accuracy to seek in both the value of the position of the root, and the value of the function at the root.
  • The setting for PrecisionGoal specifies the number of digits of precision to seek in the value of the position of the root.
  • FindRoot continues until either of the goals specified by AccuracyGoal or PrecisionGoal is achieved.
  • If FindRoot does not succeed in finding a solution to the accuracy you specify within MaxIterations steps, it returns the most recent approximation to a solution that it found. You can then apply FindRoot again, with this approximation as a starting point.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Find a root of near :

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-e0n
Out[1]=1

Find a solution to near :

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-q7d
Out[1]=1

Solve a nonlinear system of equations:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-gj6nv
Out[1]=1

Scope  (4)Survey of the scope of standard use cases

Find the solution of a system of two nonlinear equations:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-cu4e6l
Out[1]=1

Find a root for a three-component function of three variables:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-bozf80
Out[1]=1

You can cause the search to use complex values by giving a complex starting value:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-bwqef8
Out[1]=1

When the function is complex for real input, a real starting value may give a complex result:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-j1hh6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cqyznmq-cnq18
Out[2]=2

Generalizations & Extensions  (1)Generalized and extended use cases

A variable may be considered as vector valued if specified in the starting value:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-uai3d
In[2]:=2
✖
https://wolfram.com/xid/0cqyznmq-fphe7i
Out[2]=2

Options  (9)Common values & functionality for each option

AccuracyGoal and PrecisionGoal  (1)

Change tolerances for error estimates:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-jtov3i
Out[1]=1

Relax error tolerances for stopping:

In[2]:=2
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https://wolfram.com/xid/0cqyznmq-b1fqiy
Out[2]=2

Make estimated relative distance to the root the main criterion for stopping:

In[3]:=3
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https://wolfram.com/xid/0cqyznmq-clbvh8
Out[3]=3

DampingFactor  (1)

DampingFactor can be used to help speed convergence to higher-order roots:

In[1]:=1
✖
https://wolfram.com/xid/0cqyznmq-h6azqo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cqyznmq-fu1y9v
Out[2]=2

EvaluationMonitor  (1)

EvaluationMonitor can be used to keep track of function evaluations used:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-k9lg6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0cqyznmq-lg1msc
Out[2]=2

Jacobian  (1)

Specify the Jacobian for a "black-box" function:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-bzb824
In[2]:=2
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https://wolfram.com/xid/0cqyznmq-bewro8
In[3]:=3
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https://wolfram.com/xid/0cqyznmq-h0emjy
Out[3]=3

Without a specified Jacobian, extra evaluations are used to compute finite differences:

In[4]:=4
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https://wolfram.com/xid/0cqyznmq-dg97nw
Out[4]=4

If you just know the sparse form, specifying the sparse pattern template saves evaluations:

In[5]:=5
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https://wolfram.com/xid/0cqyznmq-cujvd8
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0cqyznmq-dn54r
Out[6]=6

Inspect the number of Jacobian evaluations needed by different methods:

In[7]:=7
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https://wolfram.com/xid/0cqyznmq-oh5glq
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0cqyznmq-epfhp1
Out[8]=8

MaxIterations  (1)

Limit or increase the number of steps taken:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-fsnzkw
Out[1]=1

The default number of iterations is 100:

In[2]:=2
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https://wolfram.com/xid/0cqyznmq-c44dwc
Out[2]=2

Eventually the algorithm stalls out since this mollifier function has all derivatives 0 at :

In[3]:=3
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https://wolfram.com/xid/0cqyznmq-bz8fio
Out[3]=3

Method  (2)

Method options are also explained in Unconstrained Optimization.

Find a root for using different methods:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-cbnivr
Out[1]=1

Define a function that monitors the steps and evaluations used by FindRoot:

In[2]:=2
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https://wolfram.com/xid/0cqyznmq-dp8mcj

The default (Newton's) method:

In[3]:=3
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https://wolfram.com/xid/0cqyznmq-m0oq43
Out[3]=3

Brent's root-bracketing method requiring two initial conditions bracketing the root:

In[4]:=4
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https://wolfram.com/xid/0cqyznmq-rvy67
Out[4]=4

Secant method, starting with two initial conditions:

In[5]:=5
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https://wolfram.com/xid/0cqyznmq-bx1ah
Out[5]=5

Select the affine covariant Newton method:

In[6]:=6
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https://wolfram.com/xid/0cqyznmq-bpiom4
Out[6]=6

StepMonitor  (1)

Monitor when iterative steps have been taken:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-eb9hep
In[2]:=2
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https://wolfram.com/xid/0cqyznmq-jp9vv1
Out[2]=2

Show the steps on a contour plot of :

In[3]:=3
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https://wolfram.com/xid/0cqyznmq-dd1sd7
Out[3]=3

Show steps (red) and evaluations (green). A step may require several evaluations:

In[4]:=4
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https://wolfram.com/xid/0cqyznmq-c68nn1
In[5]:=5
✖
https://wolfram.com/xid/0cqyznmq-gcg7f2
Out[5]=5

WorkingPrecision  (1)

Find a root using 100-digit precision arithmetic:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-gtsklo
Out[1]=1

Find the root starting with machine precision and adaptively working up to precision 100:

In[2]:=2
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https://wolfram.com/xid/0cqyznmq-xj0z
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Computing Inverse Functions  (1)

For an isomorphism , the inverse is the root of :

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-d7534v

An approximate inverse for the exponential function:

In[2]:=2
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https://wolfram.com/xid/0cqyznmq-csmeod
Out[2]=2

It is very close to the built-in Log function:

In[3]:=3
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https://wolfram.com/xid/0cqyznmq-tktrx
Out[3]=3

A "black-box" function giving the period of an oscillation:

In[4]:=4
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https://wolfram.com/xid/0cqyznmq-c4ky56
In[5]:=5
✖
https://wolfram.com/xid/0cqyznmq-bcp5a0
Out[5]=5

Plot its inverse:

In[6]:=6
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https://wolfram.com/xid/0cqyznmq-odifmw
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0cqyznmq-bbsmdk
Out[7]=7

Solving Boundary Value Problems  (2)

Solve a boundary value problem , using a shooting method:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-ihlyq8
In[2]:=2
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https://wolfram.com/xid/0cqyznmq-b3h5j6
Out[2]=2

Use points on either side of the root to give bracketing starting values:

In[3]:=3
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https://wolfram.com/xid/0cqyznmq-bj4jrq
Out[3]=3

Plot the solution:

In[4]:=4
✖
https://wolfram.com/xid/0cqyznmq-d1bz28
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0cqyznmq-kf0i1
Out[5]=5

Solve the boundary-value problem , with n collocation points:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-nrmujs

Consider as a first-order system :

In[2]:=2
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https://wolfram.com/xid/0cqyznmq-k7k7fv

Equations for collocation using the trapezoidal rule:

In[3]:=3
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https://wolfram.com/xid/0cqyznmq-hab678

Use 0 as a starting value:

In[4]:=4
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https://wolfram.com/xid/0cqyznmq-lsn4de

Find a solution for a particular value of ϵ:

In[5]:=5
✖
https://wolfram.com/xid/0cqyznmq-biyvby
Out[5]=5

Properties & Relations  (2)Properties of the function, and connections to other functions

For a polynomial system of equations, NSolve finds all solutions and FindRoot finds one:

In[1]:=1
✖
https://wolfram.com/xid/0cqyznmq-ctwrsp
Out[1]=1

FindRoot will find a single solution using an iterative method:

In[2]:=2
✖
https://wolfram.com/xid/0cqyznmq-dmofgf
Out[2]=2

NSolve will find all solutions using a direct method:

In[3]:=3
✖
https://wolfram.com/xid/0cqyznmq-k17ab7
Out[3]=3

For equations involving parameters or exact solutions use Solve, Reduce, or FindInstance:

In[1]:=1
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https://wolfram.com/xid/0cqyznmq-bxft1x

Solve will return some solutions:

In[2]:=2
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https://wolfram.com/xid/0cqyznmq-emd72l
Out[2]=2

Reduce will enumerate all solutions:

In[3]:=3
✖
https://wolfram.com/xid/0cqyznmq-gm0hf3
Out[3]=3

FindInstance will find particular instances:

In[4]:=4
✖
https://wolfram.com/xid/0cqyznmq-biaafk
Out[4]=4

Possible Issues  (2)Common pitfalls and unexpected behavior

If a function is complex, variables are allowed to have complex values:

In[1]:=1
✖
https://wolfram.com/xid/0cqyznmq-hetx8k
Out[1]=1

If the function is kept real, variables are also taken to be real:

In[2]:=2
✖
https://wolfram.com/xid/0cqyznmq-om8qq
Out[2]=2

It can be time-consuming to compute functions symbolically:

In[1]:=1
✖
https://wolfram.com/xid/0cqyznmq-1e272o
In[2]:=2
✖
https://wolfram.com/xid/0cqyznmq-jetqzm
Out[2]=2

Restricting the function definition avoids symbolic evaluation:

In[3]:=3
✖
https://wolfram.com/xid/0cqyznmq-oukqck
In[4]:=4
✖
https://wolfram.com/xid/0cqyznmq-7zmo2z
Out[4]=4
Wolfram Research (1988), FindRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/FindRoot.html (updated 2003).
✖
Wolfram Research (1988), FindRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/FindRoot.html (updated 2003).

Text

Wolfram Research (1988), FindRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/FindRoot.html (updated 2003).

✖
Wolfram Research (1988), FindRoot, Wolfram Language function, https://reference.wolfram.com/language/ref/FindRoot.html (updated 2003).

CMS

Wolfram Language. 1988. "FindRoot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/FindRoot.html.

✖
Wolfram Language. 1988. "FindRoot." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2003. https://reference.wolfram.com/language/ref/FindRoot.html.

APA

Wolfram Language. (1988). FindRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindRoot.html

✖
Wolfram Language. (1988). FindRoot. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindRoot.html

BibTeX

@misc{reference.wolfram_2025_findroot, author="Wolfram Research", title="{FindRoot}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/FindRoot.html}", note=[Accessed: 02-April-2025 ]}

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@misc{reference.wolfram_2025_findroot, author="Wolfram Research", title="{FindRoot}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/FindRoot.html}", note=[Accessed: 02-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_findroot, organization={Wolfram Research}, title={FindRoot}, year={2003}, url={https://reference.wolfram.com/language/ref/FindRoot.html}, note=[Accessed: 02-April-2025 ]}

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@online{reference.wolfram_2025_findroot, organization={Wolfram Research}, title={FindRoot}, year={2003}, url={https://reference.wolfram.com/language/ref/FindRoot.html}, note=[Accessed: 02-April-2025 ]}