In
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, the secant method is a
root-finding algorithm
In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to the complex numbers ...
that uses a succession of
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of
secant lines to better approximate a root of a
function ''f''. The secant method can be thought of as a
finite-difference approximation of
Newton's method
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real- ...
. However, the secant method predates Newton's method by over 3000 years.
The method
For finding a zero of a function , the secant method is defined by the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
.
:
As can be seen from this formula, two initial values and are required. Ideally, they should be chosen close to the desired zero.
Derivation of the method
Starting with initial values and , we construct a line through the points and , as shown in the picture above. In slope–intercept form, the equation of this line is
:
The root of this linear function, that is the value of such that is
:
We then use this new value of as and repeat the process, using and instead of and . We continue this process, solving for , , etc., until we reach a sufficiently high level of precision (a sufficiently small difference between and ):
:
Convergence
The iterates
of the secant method converge to a root of
is,if the initial values
and
are sufficiently close to the root. The
order of convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of co ...
is "φ", where
:
is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
. In particular, the convergence is super linear, but not quite
quadratic.
This result only holds under some technical conditions, namely that
be twice continuously differentiable and the root in question be simple (i.e., with multiplicity 1).
If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval