Bisection method

This article is about searching zeros of continuous functions. For searching a finite sorted array, see binary search algorithm. For the method of determining what software change caused a change in behavior, see Bisection (software engineering).

A few steps of the bisection method applied over the starting range [a₁;b₁]. The bigger red dot is the root of the function.

In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods. The method is also called the interval halving method, the binary search method, or the dichotomy method.

For polynomials, more elaborate methods exist for testing the existence of a root in an interval (Descartes' rule of signs, Sturm's theorem, Budan's theorem). They allow extending the bisection method into efficient algorithms for finding all real roots of a polynomial; see Real-root isolation.

The method

The method is applicable for numerically solving the equation $f(x)=0$ for the real variable $x$, where $f$ is a continuous function defined on an interval $[a,b]$ and where $f(a)$ and $f(b)$ have opposite signs. In this case $a$ and $b$ are said to bracket a root since, by the intermediate value theorem, the continuous function $f$ must have at least one root in the interval $(a,b)$.

At each step the method divides the interval in two parts/halves by computing the midpoint $c=(a+b)/2$ of the interval and the value of the function $f(c)$ at that point. If $c$ itself is a root then the process has succeeded and stops. Otherwise, there are now only two possibilities: either $f(a)$ and $f(c)$ have opposite signs and bracket a root, or $f(c)$ and $f(b)$ have opposite signs and bracket a root. The method selects the subinterval that is guaranteed to be a bracket as the new interval to be used in the next step. In this way an interval that contains a zero of $f$ is reduced in width by 50% at each step. The process is continued until the interval is sufficiently small.

Explicitly, if $f(c)=0$ then $c$ may be taken as the solution and the process stops. Otherwise, if $f(a)$ and $f(c)$ have opposite signs, then the method sets $c$ as the new value for $b$, and if $f(b)$ and $f(c)$ have opposite signs then the method sets $c$ as the new $a$. In both cases, the new $f(a)$ and $f(b)$ have opposite signs, so the method is applicable to this smaller interval.

Generalization to higher dimensions

The bisection method has been generalized to multi-dimensional functions. Such methods are called generalized bisection methods.

Methods based on degree computation

Some of these methods are based on computing the topological degree, which for a bounded region $\Omega \subseteq \mathbb {R} ^{n}$ and a differentiable function $f:\mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}$ is defined as a sum over its roots:

$\deg(f,\Omega ):=\sum _{y\in f^{-1}(\mathbf {0} )}\operatorname {sgn} \det(Df(y))$,

where $Df(y)$ is the Jacobian matrix, $\mathbf {0} =(0,0,...,0)^{T}$, and

$\operatorname {sgn}(x)={\begin{cases}1,&x>0\\0,&x=0\\-1,&x<0\\\end{cases}}$

is the sign function. In order for a root to exist, it is sufficient that $\deg(f,\Omega )\neq 0$, and this can be verified using a surface integral over the boundary of $\Omega$.

Characteristic bisection method

The characteristic bisection method uses only the signs of a function in different points. Lef f be a function from R^d to R^d, for some integer d ≥ 2. A characteristic polyhedron (also called an admissible polygon) of f is a polytope in R^d, having 2^d vertices, such that in each vertex v, the combination of signs of f(v) is unique and the topological degree of f on its interior is not zero (a necessary criterion to ensure the existence of a root). For example, for d=2, a characteristic polyhedron of f is a quadrilateral with vertices (say) A,B,C,D, such that:

• ⁠$\operatorname {sgn} f(A)=(-,-)$⁠, that is, f₁(A)<0, f₂(A)<0.
• ⁠$\operatorname {sgn} f(B)=(-,+)$⁠, that is, f₁(B)<0, f₂(B)>0.
• ⁠$\operatorname {sgn} f(C)=(+,-)$⁠, that is, f₁(C)>0, f₂(C)<0.
• ⁠$\operatorname {sgn} f(D)=(+,+)$⁠, that is, f₁(D)>0, f₂(D)>0.

A proper edge of a characteristic polygon is a edge between a pair of vertices, such that the sign vector differs by only a single sign. In the above example, the proper edges of the characteristic quadrilateral are AB, AC, BD and CD. A diagonal is a pair of vertices, such that the sign vector differs by all d signs. In the above example, the diagonals are AD and BC.

At each iteration, the algorithm picks a proper edge of the polyhedron (say, A—B), and computes the signs of f in its mid-point (say, M). Then it proceeds as follows:

• If ⁠$\operatorname {sgn} f(M)=\operatorname {sgn}(A)$⁠, then A is replaced by M, and we get a smaller characteristic polyhedron.
• If ⁠$\operatorname {sgn} f(M)=\operatorname {sgn}(B)$⁠, then B is replaced by M, and we get a smaller characteristic polyhedron.
• Else, we pick a new proper edge and try again.

Suppose the diameter (= length of longest proper edge) of the original characteristic polyhedron is D. Then, at least $\log _{2}(D/\varepsilon )$ bisections of edges are required so that the diameter of the remaining polygon will be at most ε. If the topological degree of the initial polyhedron is not zero, then there is a procedure that can choose an edge such that the next polyhedron also has nonzero degree.

See also

• Binary search algorithm
• Lehmer–Schur algorithm, generalization of the bisection method in the complex plane
• Nested intervals

References

• Burden, Richard L.; Faires, J. Douglas (2014). "2.1 The Bisection Algorithm". Numerical Analysis (10th ed.). Cengage Learning. ISBN 978-0-87150-857-7.

Further reading

• Corliss, George (1977). "Which root does the bisection algorithm find?". SIAM Review. 19 (2): 325–327. doi:10.1137/1019044. ISSN 1095-7200.
• Kaw, Autar; Kalu, Egwu (2008). Numerical Methods with Applications (1st ed.). Archived from the original on 2009-04-13.

External links

• Weisstein, Eric W. "Bisection". MathWorld.
• Bisection Method Notes, PPT, Mathcad, Maple, Matlab, Mathematica from Holistic Numerical Methods Institute