# scipy.optimize.brenth#

scipy.optimize.brenth(f, a, b, args=(), xtol=2e-12, rtol=8.881784197001252e-16, maxiter=100, full_output=False, disp=True)[source]#

Find a root of a function in a bracketing interval using Brent’s method with hyperbolic extrapolation.

A variation on the classic Brent routine to find a zero of the function f between the arguments a and b that uses hyperbolic extrapolation instead of inverse quadratic extrapolation. Bus & Dekker (1975) guarantee convergence for this method, claiming that the upper bound of function evaluations here is 4 or 5 times lesser than that for bisection. f(a) and f(b) cannot have the same signs. Generally, on a par with the brent routine, but not as heavily tested. It is a safe version of the secant method that uses hyperbolic extrapolation. The version here is by Chuck Harris, and implements Algorithm M of [BusAndDekker1975], where further details (convergence properties, additional remarks and such) can be found

Parameters:
ffunction

Python function returning a number. f must be continuous, and f(a) and f(b) must have opposite signs.

ascalar

One end of the bracketing interval [a,b].

bscalar

The other end of the bracketing interval [a,b].

xtolnumber, optional

The computed root `x0` will satisfy ```np.allclose(x, x0, atol=xtol, rtol=rtol)```, where `x` is the exact root. The parameter must be nonnegative. As with `brentq`, for nice functions the method will often satisfy the above condition with `xtol/2` and `rtol/2`.

rtolnumber, optional

The computed root `x0` will satisfy ```np.allclose(x, x0, atol=xtol, rtol=rtol)```, where `x` is the exact root. The parameter cannot be smaller than its default value of `4*np.finfo(float).eps`. As with `brentq`, for nice functions the method will often satisfy the above condition with `xtol/2` and `rtol/2`.

maxiterint, optional

If convergence is not achieved in maxiter iterations, an error is raised. Must be >= 0.

argstuple, optional

Containing extra arguments for the function f. f is called by `apply(f, (x)+args)`.

full_outputbool, optional

If full_output is False, the root is returned. If full_output is True, the return value is `(x, r)`, where x is the root, and r is a `RootResults` object.

dispbool, optional

If True, raise RuntimeError if the algorithm didn’t converge. Otherwise, the convergence status is recorded in any `RootResults` return object.

Returns:
x0float

Zero of f between a and b.

r`RootResults` (present if `full_output = True`)

Object containing information about the convergence. In particular, `r.converged` is True if the routine converged.

References

Bus, J. C. P., Dekker, T. J., “Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero of a Function”, ACM Transactions on Mathematical Software, Vol. 1, Issue 4, Dec. 1975, pp. 330-345. Section 3: “Algorithm M”. DOI:10.1145/355656.355659

Examples

```>>> def f(x):
...     return (x**2 - 1)
```
```>>> from scipy import optimize
```
```>>> root = optimize.brenth(f, -2, 0)
>>> root
-1.0
```
```>>> root = optimize.brenth(f, 0, 2)
>>> root
1.0
```