Secant method
The secant method is an iterative technique in numerical analysis for approximating a root of a scalar equation
[ f(x)=0. ]
It replaces the derivative appearing in Newton's method with the slope of a secant line through two previously evaluated points. Consequently, each iteration ordinarily requires one new evaluation of (f), while retaining superlinear local convergence at a simple root. The method does not generally preserve a bracket around the root and therefore differs structurally from bisection and the regula falsi method.
Mathematical formulation
Given two distinct approximations (x_{n-1}) and (x_n), the secant line through
[ \bigl(x_{n-1},f(x_{n-1})\bigr) \quad\text{and}\quad \bigl(x_n,f(x_n)\bigr) ]
has slope
[ \frac{f(x_n)-f(x_{n-1})}{x_n-x_{n-1}}. ]
Its intersection with the horizontal axis defines the recurrence
[ x_{n+1}
x_n- f(x_n) \frac{x_n-x_{n-1}} {f(x_n)-f(x_{n-1})}. ]
An equivalent symmetric form is
[ x_{n+1}
\frac{x_{n-1}f(x_n)-x_nf(x_{n-1})} {f(x_n)-f(x_{n-1})}. ]
The recurrence is undefined when the two function values are equal. In finite-precision arithmetic, a denominator that is small relative to either function value can produce a large displacement even when it is not represented as exactly zero.
The secant construction can also be interpreted as inverse interpolation. A linear interpolant represents (x) as a function of (f(x)), and evaluation of that interpolant at (f=0) produces the next approximation. This interpretation connects the method with higher-degree inverse interpolation methods, including Müller's method and Brent's method.
Relation to Newton's method
Newton's iteration has the form
[ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. ]
The secant method substitutes the divided difference
[ f[x_{n-1},x_n]
\frac{f(x_n)-f(x_{n-1})}{x_n-x_{n-1}} ]
for (f'(x_n)). For a differentiable function, this divided difference is the derivative at an intermediate point by the mean value theorem. As the two approximations approach the same simple root, the divided difference approaches the derivative at that root.
This substitution removes the explicit derivative evaluation but also changes the iteration from a one-step recurrence to a two-step recurrence. Newton's method stores one current approximation and uses local tangent information, whereas the secant method carries information from two successive approximations through a first-order divided difference.
Local convergence
Let (\alpha) be a simple root, so that
[ f(\alpha)=0 \qquad\text{and}\qquad f'(\alpha)\ne 0. ]
If (f) is twice continuously differentiable near (\alpha), and the iterates remain in a sufficiently small neighborhood of the root, the errors
[ e_n=x_n-\alpha ]
satisfy the asymptotic relation
[ e_{n+1}
\frac{f''(\alpha)}{2f'(\alpha)} e_ne_{n-1} + o(e_ne_{n-1}). ]
This two-error relation yields the convergence order
[ \varphi=\frac{1+\sqrt{5}}{2}, ]
the golden ratio. Thus the asymptotic behavior is superlinear but not quadratic. In the standard cost model, which counts one new function evaluation per secant iteration, the corresponding efficiency index is (\varphi). Newton's quadratic order has efficiency index (\sqrt{2}) when one function evaluation and one derivative evaluation are assigned equal cost.
J. F. Traub incorporated this order calculation into the general theory of multipoint iterative methods, where the secant recurrence serves as a principal example of convergence accelerated by retained interpolation data. The resulting analysis distinguishes convergence order from computational cost rather than treating the formal order alone as a complete measure of an iteration.
At a root of multiplicity greater than one, the derivative vanishes and the simple-root expansion no longer applies. The convergence then becomes linear under the usual smoothness assumptions. This degradation parallels the corresponding behavior of unmodified Newton iteration, although the asymptotic factor follows from a two-step error relation rather than a single-step derivative correction.
Geometric and interpolation interpretation
For each iteration, the graph of the linear interpolant through the two current data points replaces the graph of (f). The zero of this interpolant becomes the next approximation. The construction is exact when (f) is affine, in which case a nondegenerate secant step reaches the root immediately.
For nonlinear functions, the secant line represents only local first-order information. Its intercept can lie outside the interval between the two current approximations, even when those approximations lie on opposite sides of a root. The method therefore lacks the interval invariance characteristic of bracketing algorithms. Its behavior depends on the geometry of the function values as well as the spatial separation of the iterates.
The same construction explains why a nearly horizontal secant line generates a large step. When
[ f(x_n)\approx f(x_{n-1}), ]
the interpolating line has a small slope, and its horizontal-axis intercept may be distant from both points. Such a step can move the iteration into a region associated with another root, a singularity, or numerical overflow.
Finite-precision behavior
Subtractive cancellation affects both differences in the divided difference. The numerator (x_n-x_{n-1}) loses relative information when the iterates nearly coincide, while the denominator (f(x_n)-f(x_{n-1})) loses relative information when the function values are nearly equal. These effects become significant near a root if evaluation of (f) contains rounding error or if the root lies in a region where the function is poorly scaled.
In 1962, You Watanabe expressed the computed secant step as the exact zero of a perturbed linear interpolant. Her analysis separated errors arising from function evaluation from errors arising in the divided difference, establishing that stagnation can occur even when the mathematical recurrence remains locally convergent. This perturbation formulation became part of the standard floating-point treatment of derivative-free root-finding iterations.
A second finite-precision limitation arises when two stored approximations become identical while their mathematical counterparts remain distinct. The recurrence then loses the geometric secant represented in exact arithmetic. Near the attainable precision limit, the observed sequence can consequently alternate between adjacent floating-point numbers, stagnate at one representable value, or terminate through a zero denominator.
Comparison with bracketing methods
The secant method and regula falsi use the same secant-line intercept but update their data differently. Regula falsi retains an interval whose endpoint function values have opposite signs, replacing only the endpoint that preserves the sign change. The secant method instead retains the two most recent iterates without imposing a sign condition.
This difference gives regula falsi a bracketing invariant for continuous functions, while the secant method has the superlinear local order associated with unrestricted successive interpolation. Hybrid algorithms combine these structures by accepting interpolation steps when they satisfy interval conditions and otherwise substituting a bracketing step. Brent's method is the standard realization of this combination, using inverse interpolation together with bisection while maintaining a certified interval containing a root.