SciPy · Function

optimize.fsolve

Finds roots of scalar or vector nonlinear equations from an initial estimate. It can use a supplied Jacobian or estimate one numerically, with optional convergence diagnostics.

How it works

fsolve applies MINPACK’s hybrid algorithms to seek a vector x whose function values are zero. It starts from x0, estimates the Jacobian unless fprime is supplied, and can optionally return diagnostic information about convergence.

Syntax

scipy.optimize.fsolve(func, x0, args=(), fprime=None, full_output=False, col_deriv=False, xtol=1.49012e-08, maxfev=0, band=None, epsfcn=None, factor=100, diag=None)

Parameters

Name	Type	Description
`func`	`callable`	Callable f(x, *args) returning values with the same length as its argument.
`x0`	`ndarray`	Initial estimate for the roots.
`args`	`tuple, optional`	Extra arguments passed to func and fprime.
`fprime`	`callable, optional`	Jacobian callable; estimated numerically when omitted.
`full_output`	`bool, optional`	Whether to return optional diagnostic outputs.
`col_deriv`	`bool, optional`	Whether fprime computes derivatives down columns.
`xtol`	`float, optional`	Relative-error termination tolerance between consecutive iterates.
`maxfev`	`int, optional`	Maximum function calls; zero selects 100*(N+1).
`band`	`2-tuple, optional`	Jacobian band widths for banded problems.
`epsfcn`	`float, optional`	Step length for forward-difference Jacobian approximation.
`factor`	`float, optional`	Initial step-bound multiplier; must be in (0.1, 100).
`diag`	`sequence, optional`	Positive variable scale factors.

Returns

x ndarray: the solution, or the result of the last iteration for an unsuccessful call. With full_output=True, optional diagnostic outputs are also returned.

Key points

Use fsolve to locate zeros of a scalar or vector-valued function.
For systems, func must return as many values as x has elements.
Define func so its output length matches the input vector and accept any extra args passed through args.
Choose x0 as the initial guess; convergence can depend strongly on this estimate.
Set full_output=True when convergence diagnostics or intermediate information are needed.
Provide fprime to avoid numerical Jacobian estimation when an analytic Jacobian is available.

Common mistakes

The returned array can be only the final iterate when the solver does not converge; inspect the diagnostic output when using full_output=True.
If no fprime is supplied, SciPy approximates the Jacobian numerically, which can add function evaluations and sensitivity to finite-difference settings.
factor must remain between 0.1 and 100 because it controls the initial step bound.

Example: Solve a system of equations

import numpy as np
from scipy.optimize import fsolve

def equations(x):
    a, b = x
    return [a + b - 3, a - b - 1]

root = fsolve(equations, x0=[1.0, 1.0])
print(root)  # [2. 1.]

Solves two nonlinear equations simultaneously from an initial estimate.

When to use it

Nonlinear systems

Solve coupled nonlinear equations in several unknowns.

Scalar root finding

Find a scalar x where a numerical expression evaluates to zero.

Jacobian-aware solving

Use an analytic Jacobian to guide a computationally expensive solve.

Interview questions

QWhat problem does fsolve solve?

fsolve finds roots of a vector function by solving func(x)=0 from a supplied starting estimate x0. It is intended for scalar or multidimensional systems whose function returns a value array matching the input length.

Follow-up: How can the Jacobian be provided? Supply fprime; its derivatives are arranged across rows unless col_deriv indicates derivatives down columns.

QWhat is the role of fprime?

fprime is optional. Without it, the Jacobian is estimated numerically; with it, the callable computes the Jacobian of func.

Follow-up: When might a user adjust epsfcn? It specifies a suitable step length for the forward-difference Jacobian approximation when fprime is omitted.

QHow should convergence be assessed?

With full_output=True, fsolve returns optional diagnostic outputs in addition to x. The primary x result is the solution, or the last iteration if the call is unsuccessful.

Follow-up: What is the default maximum number of function calls? When maxfev is zero, it is 100*(N+1), where N is the number of variables.

scipy.optimize.root
scipy.optimize.leastsq
scipy.optimize.minimize

Source: fsolve. Checked 2026-10-15.