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Secant method

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In <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a>, the secant method is a <a href="Root-finding_algorithm" title="Root-finding algorithm">root-finding algorithm</a> that uses a succession of <a href="Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> of <a href="Secant_line" title="Secant line">secant lines</a> to better approximate a root of a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> f. The secant method can be thought of as a <a href="Finite-difference" class="mw-redirect" title="Finite-difference">finite-difference</a> approximation of <a href="Newton's_method" title="Newton's method">Newton's method</a>, so it is considered a <a href="Quasi-Newton_method" title="Quasi-Newton method">quasi-Newton method</a>. Historically, it is as an evolution of the <a href="Regula_falsi" title="Regula falsi">method of false position</a>, which predates Newton's method by over 3000 years.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="The_method">The method</h2></div>

The secant method is an iterative numerical method for finding a zero of a function f. Given two initial values x0 and x1, the method proceeds according to the <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>

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<annotation encoding="application/x-tex">{\displaystyle x_{n}=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}={\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}.}</annotation>

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</math><img src="./93f178fd896cad8b93b2fa4e22880fb8fe29a28d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:72.214ex; height:6.509ex;" alt="{\displaystyle x_{n}=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}={\frac {x_{n-2}f(x_{n-1})-x_{n-1}f(x_{n-2})}{f(x_{n-1})-f(x_{n-2})}}.}" loading="lazy"></dd></dl>

This is a nonlinear second-order recurrence that is well-defined given f and the two initial values x0 and x1. Ideally, the initial values should be chosen close to the desired zero.

<div class="mw-heading mw-heading2"><h2 id="Derivation_of_the_method">Derivation of the method</h2></div>

Starting with initial values x0 and x1, we construct a line through the points (x0, f(x0)) and (x1, f(x1)), as shown in the picture above. In point–point form,<a href="#cite_note-2">[2]</a> the equation of this line is

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<annotation encoding="application/x-tex">{\displaystyle y={\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{1})+f(x_{1}).}</annotation>

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</math><img src="./208363b89cd54601ad808416c455017b377a672a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.196ex; height:6.009ex;" alt="{\displaystyle y={\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{1})+f(x_{1}).}" loading="lazy"></dd></dl>

The root of this linear function, that is the value of x such that y = 0 is

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<annotation encoding="application/x-tex">{\displaystyle x=x_{1}-f(x_{1}){\frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.}</annotation>

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</math><img src="./ce2bae249f0a7bd7316259506001e3ede137a878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.391ex; height:5.843ex;" alt="{\displaystyle x=x_{1}-f(x_{1}){\frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.}" loading="lazy"></dd></dl>

We then use this new value of x as x2 and repeat the process, using x1 and x2 instead of x0 and x1. We continue this process, solving for x3, x4, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between xn and xn−1):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x_{2}&=x_{1}-f(x_{1}){\frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}},\\[6pt]x_{3}&=x_{2}-f(x_{2}){\frac {x_{2}-x_{1}}{f(x_{2})-f(x_{1})}},\\[6pt]&\,\,\,\vdots \\[6pt]x_{n}&=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}.\end{aligned}}}</annotation>

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</math><img src="./b876ff00b7fcfd76e25e4e2efa3a28bc5bb283f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.318ex; margin-bottom: -0.186ex; width:41.42ex; height:26.176ex;" alt="{\displaystyle {\begin{aligned}x_{2}&=x_{1}-f(x_{1}){\frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}},\\[6pt]x_{3}&=x_{2}-f(x_{2}){\frac {x_{2}-x_{1}}{f(x_{2})-f(x_{1})}},\\[6pt]&\,\,\,\vdots \\[6pt]x_{n}&=x_{n-1}-f(x_{n-1}){\frac {x_{n-1}-x_{n-2}}{f(x_{n-1})-f(x_{n-2})}}.\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Convergence">Convergence</h2></div>

The iterates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

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</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> of the secant method converge to a root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> if the initial values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}">

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</math><img src="./a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy"> are sufficiently close to the root and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is well-behaved. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is twice continuously differentiable and the root in question is a simple root, i.e., it has multiplicity 1, the <a href="Order_of_convergence" class="mw-redirect" title="Order of convergence">order of convergence</a> is the <a href="Golden_ratio" title="Golden ratio">golden ratio</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =(1+{\sqrt {5}})/2\approx 1.618.}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi =(1+{\sqrt {5}})/2\approx 1.618.}</annotation>

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</math><img src="./97ae7078d10e2ba2e5e7e1009eb3ab1055b1a639.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.896ex; height:3.009ex;" alt="{\displaystyle \varphi =(1+{\sqrt {5}})/2\approx 1.618.}" loading="lazy"><a href="#cite_note-:0-3">[3]</a> This convergence is superlinear but subquadratic.

If the initial values are not close enough to the root or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is not well-behaved, then there is no guarantee that the secant method converges at all. There is no general definition of "close enough", but the criterion for convergence has to do with how "wiggly" the function is on the interval between the initial values. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is differentiable on that interval and there is a point where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'=0}">

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</math><img src="./b4f1fa9ac45077caae50849fcc3556be013af2e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.266ex; height:2.843ex;" alt="{\displaystyle f'=0}" loading="lazy"> on the interval, then the algorithm may not converge.

<div class="mw-heading mw-heading2"><h2 id="Comparison_with_other_root-finding_methods">Comparison with other root-finding methods</h2></div>

The secant method does not require or guarantee that the root remains bracketed by sequential iterates, like the <a href="Bisection_method" title="Bisection method">bisection method</a> does, and hence it does not always converge. The <a href="False_position_method" class="mw-redirect" title="False position method">false position method</a> (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n-1}}">

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</math><img src="./56ef8a744e0bb10d512f2e77625811bff9771d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.649ex; height:2.009ex;" alt="{\displaystyle x_{n-1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n-2}}">

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</math><img src="./737e524ecd78befd521ba26382cc6fc18f0cd8c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.649ex; height:2.009ex;" alt="{\displaystyle x_{n-2}}" loading="lazy">, like the secant method, but on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n-1}}">

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</math><img src="./56ef8a744e0bb10d512f2e77625811bff9771d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.649ex; height:2.009ex;" alt="{\displaystyle x_{n-1}}" loading="lazy"> and on the last iterate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

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</math><img src="./20d432970858c21563481c08f9c2bf50dffbeb06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.506ex; height:2.843ex;" alt="{\displaystyle f(x_{k})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{n-1})}">

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</math><img src="./5a4c19b6ae9574db80733f29275ab95986767687.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.737ex; height:2.843ex;" alt="{\displaystyle f(x_{n-1})}" loading="lazy"> have a different sign. This means that the <a href="False_position_method" class="mw-redirect" title="False position method">false position method</a> always converges; however, only with a linear order of convergence. Bracketing with a super-linear order of convergence as the secant method can be attained with improvements to the false position method (see <a href="Regula_falsi#Improvements_in_''regula_falsi''" title="Regula falsi">Regula falsi § Improvements in regula falsi</a>) such as the <a href="ITP_Method" class="mw-redirect" title="ITP Method">ITP method</a> or the <a href="Illinois_Method" class="mw-redirect" title="Illinois Method">Illinois method</a>.

The recurrence formula of the secant method can be derived from the formula for <a href="Newton's_method" title="Newton's method">Newton's method</a>

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<annotation encoding="application/x-tex">{\displaystyle x_{n}=x_{n-1}-{\frac {f(x_{n-1})}{f'(x_{n-1})}}}</annotation>

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by using the <a href="Finite-difference" class="mw-redirect" title="Finite-difference">finite-difference</a> approximation, for a small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon =x_{n-1}-x_{n-2}}">

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<annotation encoding="application/x-tex">{\displaystyle \epsilon =x_{n-1}-x_{n-2}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle f'(x_{n-1})=\lim _{\epsilon \rightarrow 0}{\frac {f(x_{n-1})-f(x_{n-1}-\epsilon )}{\epsilon }}\approx {\frac {f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}}}</annotation>

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</math><img src="./05bd2d7a00a7e2ce188adf4214808e2a5df15511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:60.361ex; height:6.176ex;" alt="{\displaystyle f'(x_{n-1})=\lim _{\epsilon \rightarrow 0}{\frac {f(x_{n-1})-f(x_{n-1}-\epsilon )}{\epsilon }}\approx {\frac {f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}}}" loading="lazy">

The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a <a href="Quasi-Newton_method" title="Quasi-Newton method">quasi-Newton method</a>.

If we compare Newton's method with the secant method, we see that Newton's method converges faster (order 2 against order the <a href="Golden_ratio" title="Golden ratio">golden ratio</a> φ ≈ 1.6).<a href="#cite_note-:0-3">[3]</a> However, Newton's method requires the evaluation of both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and its derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f'}</annotation>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Therefore, the secant method may sometimes be faster in practice. For instance, if we assume that evaluating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> takes as much time as evaluating its derivative and we neglect all other costs, we can do two steps of the secant method (decreasing the logarithm of the error by a factor φ2 ≈ 2.6) for the same cost as one step of Newton's method (decreasing the logarithm of the error by a factor of 2), so the secant method is faster. In higher dimensions, the full set of <a href="Partial_derivative" title="Partial derivative">partial derivatives</a> required for Newton's method, that is, the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix</a>, may become much more expensive to calculate than the function itself. If, however, we consider parallel processing for the evaluation of the derivative or derivatives, Newton's method can be faster in clock time though still costing more computational operations overall.

<div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2></div>

<a href="Broyden's_method" title="Broyden's method">Broyden's method</a> is a generalization of the secant method to more than one dimension.

The following graph shows the function f in red and the last secant line in bold blue. In the graph, the x intercept of the secant line seems to be a good approximation of the root of f.

<div class="mw-heading mw-heading2"><h2 id="Computational_example">Computational example</h2></div>

Below, the secant method is implemented in the <a href="Python_(programming_language)" title="Python (programming language)">Python</a> programming language.

It is then applied to find a root of the function f(x) = x2 − 612 with initial points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=10}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}=10}</annotation>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{1}=30}</annotation>

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</math><img src="./81d920ebde88f718fe6ad606a20a7e1668295952.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.807ex; height:2.509ex;" alt="{\displaystyle x_{1}=30}" loading="lazy">

<div class="mw-highlight mw-highlight-lang-python3 mw-content-ltr" dir="ltr"><pre>def secant_method(f, x0: int, x1: int, iterations: int) -> float:

"""Return the root calculated using the secant method."""

for i in range(iterations):

x2 = x1 - f(x1) * (x1 - x0) / float(f(x1) - f(x0))

x0, x1 = x1, x2

# Apply a stopping criterion here (see below)

return x2

def f_example(x):

return x ** 2 - 612

root = secant_method(f_example, 10, 30, 5)

print(f"Root: {root}") # Root: 24.738633748750722

</pre></div>

It is very important to have a good stopping criterion above, otherwise, due to limited numerical precision of floating point numbers, the algorithm can return inaccurate results if running for too many iterations. For example, the loop above can stop when one of these is reached first: <style data-mw-deduplicate="TemplateStyles:r886049734">

/* start https://en.wikipedia.org/ */

.mw-parser-output .monospaced{font-family:monospace,monospace}

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</style>abs(x0 - x1) < tol, or abs(x0/x1-1) < tol, or abs(f(x1)) < tol. <a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFPapakonstantinouTapia2013" class="citation journal cs1">Papakonstantinou, Joanna; Tapia, Richard (2013). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/10.4169/amer.math.monthly.120.06.500">"Origin and evolution of the secant method in one dimension"</a>. American Mathematical Monthly. 120 (6): 500–518. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2Famer.math.monthly.120.06.500">10.4169/amer.math.monthly.120.06.500</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/10.4169/amer.math.monthly.120.06.500">10.4169/amer.math.monthly.120.06.500</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17645996">17645996</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFMarsden1985" class="citation book cs1">Marsden, Jerrold (1985). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-1-4612-5024-1">Calculus I</a>. Springer-Verlag New York Inc. p. 31. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-5024-1</bdi>.</cite>

</li>

<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <cite id="CITEREFChanson2024" class="citation web cs1">Chanson, Jeffrey R. (October 3, 2024). <a rel="nofollow" class="external text" href="https://math.libretexts.org/Bookshelves/Applied_Mathematics/Numerical_Methods_(Chasnov)/02%3A_Root_Finding/2.04%3A_Order_of_Convergence">"Order of Convergence"</a>. LibreTexts Mathematics. Retrieved October 3, 2024.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cfm.brown.edu/people/dobrush/am33/Matlab/ch3/secant.html">"MATLAB Tutorial for the First Course. Part 1.3: Secant Methods"</a>.</cite>

</li>

</ol></div></div>

<ul><li><a href="False_position_method" class="mw-redirect" title="False position method">False position method</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFAvriel1976" class="citation book cs1">Avriel, Mordecai (1976). Nonlinear Programming: Analysis and Methods. Prentice Hall. pp. 220–221. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-623603-0</bdi>.</cite></li>

<li><cite id="CITEREFAllenIsaacson1998" class="citation book cs1">Allen, Myron B.; Isaacson, Eli L. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PpB9cjOxQAQC">Numerical analysis for applied science</a>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. 188–195. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-55266-6</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://numericalmethods.eng.usf.edu/topics/secant_method.html">Secant Method</a> Notes, PPT, Mathcad, Maple, Mathematica, Matlab at <a rel="nofollow" class="external text" href="http://numericalmethods.eng.usf.edu">Holistic Numerical Methods Institute</a></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SecantMethod.html">"Secant Method"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul>

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</style><div id="Root-finding_algorithms311" style="font-size:114%;margin:0 4em"><a href="Root-finding_algorithm" title="Root-finding algorithm">Root-finding algorithms</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Root-finding_algorithm#Bracketing_methods" title="Root-finding algorithm">Bracketing (no derivative)</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Bisection_method" title="Bisection method">Bisection method</a></li>

<li><a href="Regula_falsi" title="Regula falsi">Regula falsi</a></li>

<li><a href="ITP_method" title="ITP method">ITP method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Householder's_method" title="Householder's method">Householder</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Newton's_method" title="Newton's method">Newton's method</a></li>

<li><a href="Halley's_method" title="Halley's method">Halley's method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi-Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Broyden's_method" title="Broyden's method">Broyden's method</a></li>

<li><a href="Newton%E2%80%93Krylov_method" title="Newton–Krylov method">Newton–Krylov method</a></li>

<li><a href="Steffensen's_method" title="Steffensen's method">Steffensen's method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Root-finding_algorithm#Combinationations_of_methods" title="Root-finding algorithm">Hybrid methods</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Brent's_method" title="Brent's method">Brent's method</a></li>

<li><a href="Ridders'_method" title="Ridders' method">Ridders' method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polynomial_root-finding_algorithms" class="mw-redirect" title="Polynomial root-finding algorithms">Polynomial methods</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Aberth_method" title="Aberth method">Aberth method</a></li>

<li><a href="Bairstow's_method" title="Bairstow's method">Bairstow's method</a></li>

<li><a href="Bernoulli's_method" title="Bernoulli's method">Bernoulli's method</a></li>

<li><a href="Durand%E2%80%93Kerner_method" title="Durand–Kerner method">Durand–Kerner method</a></li>

<li><a href="Graeffe's_method" title="Graeffe's method">Graeffe's method</a></li>

<li><a href="Jenkins%E2%80%93Traub_algorithm" title="Jenkins–Traub algorithm">Jenkins–Traub algorithm</a></li>

<li><a href="Lehmer%E2%80%93Schur_algorithm" title="Lehmer–Schur algorithm">Lehmer–Schur algorithm</a></li>

<li><a href="Laguerre's_method" title="Laguerre's method">Laguerre's method</a></li>

<li><a href="Splitting_circle_method" title="Splitting circle method">Splitting circle method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other methods</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0;line-height:1.4em; padding:0.33em 0;"><div style="padding:0 0.25em">

<ul><li><a href="Fixed-point_iteration" title="Fixed-point iteration">Fixed-point iteration</a></li>

<li><a href="Inverse_quadratic_interpolation" title="Inverse quadratic interpolation">Inverse quadratic interpolation</a></li>

<li><a href="Muller's_method" title="Muller's method">Muller's method</a></li>

<li><a href="Sidi's_generalized_secant_method" title="Sidi's generalized secant method">Sidi's generalized secant method</a></li></ul>

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