## Root Bracketing Algorithms

The root bracketing algorithms described in this section require an initial interval which is guaranteed to contain a root -- if @math{a} and @math{b} are the endpoints of the interval then @math{f(a)} must differ in sign from @math{f(b)}. This ensures that the function crosses zero at least once in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.

Note that a bracketing algorithm cannot find roots of even degree, since these do not cross the @math{x}-axis.

Solver: gsl_root_fsolver_bisection

The bisection algorithm is the simplest method of bracketing the roots of a function. It is the slowest algorithm provided by the library, with linear convergence.

On each iteration, the interval is bisected and the value of the function at the midpoint is calculated. The sign of this value is used to determine which half of the interval does not contain a root. That half is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small.

At any time the current estimate of the root is taken as the midpoint of the interval.

@image{roots-bisection,4in}

Four iterations of bisection, where @math{a_n} is @math{n}th position of the beginning of the interval and @math{b_n} is the @math{n}th position of the end. The midpoint of each interval is also indicated.

Solver: gsl_root_fsolver_falsepos

The false position algorithm is a method of finding roots based on linear interpolation. Its convergence is linear, but it is usually faster than bisection.

On each iteration a line is drawn between the endpoints @math{(a,f(a))} and @math{(b,f(b))} and the point where this line crosses the @math{x}-axis taken as a "midpoint". The value of the function at this point is calculated and its sign is used to determine which side of the interval does not contain a root. That side is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small.

The best estimate of the root is taken from the linear interpolation of the interval on the current iteration.

Solver: gsl_root_fsolver_brent

The Brent-Dekker method (referred to here as Brent's method) combines an interpolation strategy with the bisection algorithm. This produces a fast algorithm which is still robust.

On each iteration Brent's method approximates the function using an interpolating curve. On the first iteration this is a linear interpolation of the two endpoints. For subsequent iterations the algorithm uses an inverse quadratic fit to the last three points, for higher accuracy. The intercept of the interpolating curve with the @math{x}-axis is taken as a guess for the root. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary bisection step.

The best estimate of the root is taken from the most recent interpolation or bisection.
The text passage describes various root bracketing algorithms used to find the roots of a function, including the bisection method, false position method, and Brent's method. It explains the principles and steps involved in each algorithm but does not contain any specific mathematical questions or answers.
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## Root Bracketing Algorithms

The root bracketing algorithms described in this section require an initial interval which is guaranteed to contain a root -- if @math{a} and @math{b} are the endpoints of the interval then @math{f(a)} must differ in sign from @math{f(b)}. This ensures that the function crosses zero at least once in the interval. If a valid initial interval is used then these algorithm cannot fail, provided the function is well-behaved.

Note that a bracketing algorithm cannot find roots of even degree, since these do not cross the @math{x}-axis.

Solver: gsl_root_fsolver_bisection

The bisection algorithm is the simplest method of bracketing the roots of a function. It is the slowest algorithm provided by the library, with linear convergence.

On each iteration, the interval is bisected and the value of the function at the midpoint is calculated. The sign of this value is used to determine which half of the interval does not contain a root. That half is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small.

At any time the current estimate of the root is taken as the midpoint of the interval.

@image{roots-bisection,4in}

Four iterations of bisection, where @math{a_n} is @math{n}th position of the beginning of the interval and @math{b_n} is the @math{n}th position of the end. The midpoint of each interval is also indicated.

Solver: gsl_root_fsolver_falsepos

The false position algorithm is a method of finding roots based on linear interpolation. Its convergence is linear, but it is usually faster than bisection.

On each iteration a line is drawn between the endpoints @math{(a,f(a))} and @math{(b,f(b))} and the point where this line crosses the @math{x}-axis taken as a "midpoint". The value of the function at this point is calculated and its sign is used to determine which side of the interval does not contain a root. That side is discarded to give a new, smaller interval containing the root. This procedure can be continued indefinitely until the interval is sufficiently small.

The best estimate of the root is taken from the linear interpolation of the interval on the current iteration.

Solver: gsl_root_fsolver_brent

The Brent-Dekker method (referred to here as Brent's method) combines an interpolation strategy with the bisection algorithm. This produces a fast algorithm which is still robust.

On each iteration Brent's method approximates the function using an interpolating curve. On the first iteration this is a linear interpolation of the two endpoints. For subsequent iterations the algorithm uses an inverse quadratic fit to the last three points, for higher accuracy. The intercept of the interpolating curve with the @math{x}-axis is taken as a guess for the root. If it lies within the bounds of the current interval then the interpolating point is accepted, and used to generate a smaller interval. If the interpolating point is not accepted then the algorithm falls back to an ordinary bisection step.

The best estimate of the root is taken from the most recent interpolation or bisection.
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The text passage describes various root bracketing algorithms used to find the roots of a function, including the bisection method, false position method, and Brent's method. It explains the principles and steps involved in each algorithm but does not contain any specific mathematical questions or answers.

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