Modified Secant Method Formula
2021年6月18日Download here: http://gg.gg/v1ozc
Modified Regula Falsi Method generates the approximations in the same manner as the Regula Falsi Method does. But for faster convergence some modifications are made. We first choose the initial approximations and with. The approximation is chosen as the x-intercept of the line joining. To decide which secant line to use to compute, we. In this study, a modification of the classical Secant method for solving nonlinear, univariate and unconstrained optimization problems based on the development of the cubic approximation is presented. The iteration formula including an approximation of the third derivative of f(x) by using the Taylor series expansion is derived. The performance of the new method is analyzed in terms of the. GET SECANT Value using SEC Function in Excel. The SEC function in Excel calculates and returns the secant of a given radians. The Secant is the reciprocal of Cosine. SEC (θ) = Hypotenuse / Adjacent = c/b. Secant of the angle θ, is the ratio of the hypotenuse side – c, to the adjacent side – b. By Newton’s method: b = af(a) f0(a) Some will say that Newton’s method is very fast and has a convergence factor of 2. However, for every one step of Newton’s method, two steps of the secant method can be done, because Newton’s method requires the taking of a derivative and then nding two function evaluations. The secant can be calculated using known trigonometric properties if the angle is known, but it also can be calculated through simply the equation of H/A where H is the length of the hypotenuse of a triangle and A is the length of the adjacent side. For this example we will assume the length of both of these sides is known.
*Python Function For Modified Secant Method Formula
*Secant Equation
*Newton Secant Method
*Secant Method ExampleThe first two iterations of the secant method. The red curve shows the function f, and the blue lines are the secants. For this particular case, the secant method will not converge to the visible root.
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a functionf. The secant method can be thought of as a finite-difference approximation of Newton’s method. However, the secant method predates Newton’s method by over 3000 years.[1]The method[edit]
The secant method is defined by the recurrence relationxn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2)=xn−2f(xn−1)−xn−1f(xn−2)f(xn−1)−f(xn−2).{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})}}.}
As can be seen from the recurrence relation, the secant method requires two initial values, x0 and x1, which should ideally be chosen to lie close to the root.Derivation of the method[edit]
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 slope–intercept form, the equation of this line isy=f(x1)−f(x0)x1−x0(x−x1)+f(x1).{displaystyle y={frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{1})+f(x_{1}).}
The root of this linear function, that is the value of x such that y = 0 isx=x1−f(x1)x1−x0f(x1)−f(x0).{displaystyle x=x_{1}-f(x_{1}){frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.}
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):x2=x1−f(x1)x1−x0f(x1)−f(x0),x3=x2−f(x2)x2−x1f(x2)−f(x1),⋮xn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2).{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}}}Convergence[edit]
The iterates xn{displaystyle x_{n}} of the secant method converge to a root of f{displaystyle f} if the initial values x0{displaystyle x_{0}} and x1{displaystyle x_{1}} are sufficiently close to the root. The order of convergence is φ, whereφ=1+52≈1.618{displaystyle varphi ={frac {1+{sqrt {5}}}{2}}approx 1.618}
is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic.
This result only holds under some technical conditions, namely that f{displaystyle f} 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 [x0,x1]{displaystyle [x_{0},x_{1}]}. For example, if f{displaystyle f} is differentiable on that interval and there is a point where f′=0{displaystyle f’=0} on the interval, then the algorithm may not converge.Comparison with other root-finding methods[edit]
The secant method does not require that the root remain bracketed, like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on xn−1{displaystyle x_{n-1}} and xn−2{displaystyle x_{n-2}}, like the secant method, but on xn−1{displaystyle x_{n-1}} and on the last iterate xk{displaystyle x_{k}} such that f(xk){displaystyle f(x_{k})} and f(xn−1){displaystyle f(x_{n-1})} have a different sign. This means that the false position method always converges.
The recurrence formula of the secant method can be derived from the formula for Newton’s methodxn=xn−1−f(xn−1)f′(xn−1){displaystyle x_{n}=x_{n-1}-{frac {f(x_{n-1})}{f’(x_{n-1})}}}
Ppsspp game download android mobile. by using the finite-difference approximationf′(xn−1)≈f(xn−1)−f(xn−2)xn−1−xn−2.{displaystyle f’(x_{n-1})approx {frac {f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}}.}
The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method.
If we compare Newton’s method with the secant method, we see that Newton’s method converges faster (order 2 against φ ≈ 1.6). However, Newton’s method requires the evaluation of both f{displaystyle f} and its derivative f′{displaystyle f’} at every step, while the secant method only requires the evaluation of f{displaystyle f}. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating f{displaystyle f} 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 2), so the secant method is faster. If, however, we consider parallel processing for the evaluation of the derivative, Newton’s method proves its worth, being faster in time, though still spending more steps.Python Function For Modified Secant Method FormulaGeneralizations[edit]
Broyden’s method 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.Computational example[edit]
Below, the secant method is implemented in the Python programming language.
It is then applied to find a root of the function f(x) = x2 − 612 with initial points x0=10{displaystyle x_{0}=10} and x1=30{displaystyle x_{1}=30}Notes[edit]
*^Papakonstantinou, J., The Historical Development of the Secant Method in 1-D, retrieved 2011-06-29See also[edit]References[edit]
*Avriel, Mordecai (1976). Nonlinear Programming: Analysis and Methods. Prentice Hall. pp. 220–221. ISBN0-13-623603-0.
*Allen, Myron B.; Isaacson, Eli L. (1998). Numerical analysis for applied science. John Wiley & Sons. pp. 188–195. ISBN978-0-471-55266-6.External links[edit]
*Secant Method Notes, PPT, Mathcad, Maple, Mathematica, Matlab at Holistic Numerical Methods Institute
*Weisstein, Eric W.’Secant Method’. MathWorld.Retrieved from ’https://en.wikipedia.org/w/index.php?title=Secant_method&oldid=988064369’The first two iterations of the secant method. The red curve shows the function f, and the blue lines are the secants. For this particular case, the secant method will not converge to the visible root.
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a functionf. The secant method can be thought of as a finite-difference approximation of Newton’s method. However, the secant method predates Newton’s method by over 3000 years.[1]The method[edit]
The secant method is defined by the recurrence relationxn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2)=xn−2f(xn−1)−xn−1f(xn−2)f(xn−1)−f(xn−2).{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})}}.}
As can be seen from the recurrence relation, the secant method requires two initial values, x0 and x1, which should ideally be chosen to lie close to the root.Derivation of the method[edit]
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 slope–intercept form, the equation of this line isSecant Equationy=f(x1)−f(x0)x1−x0(x−x1)+f(x1).{displaystyle y={frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{1})+f(x_{1}).}
The root of this linear function, that is the value of x such that y = 0 isx=x1−f(x1)x1−x0f(x1)−f(x0).{displaystyle x=x_{1}-f(x_{1}){frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.}
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):x2=x1−f(x1)x1−x0f(x1)−f(x0),x3=x2−f(x2)x2−x1f(x2)−f(x1),⋮xn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2).{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}}}Convergence[edit]
The iterates xn{displaystyle x_{n}} of the secant method converge to a root of f{displaystyle f} if the initial values x0{displaystyle x_{0}} and x1{displaystyle x_{1}} are sufficiently close to the root. The order of convergence is φ, whereφ=1+52≈1.618{displaystyle varphi ={frac {1+{sqrt {5}}}{2}}approx 1.618}
is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic.
This result only holds under some technical conditions, namely that f{displaystyle f} 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 [x0,x1]{displaystyle [x_{0},x_{1}]}. For example, if f{displaystyle f} is differentiable on that interval and there is a point where f′=0{displaystyle f’=0} on the interval, then the algorithm may not converge.Comparison with other root-finding methods[edit]Newton Secant Method
The secant method does not require that the root remain bracketed, like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on xn−1{displaystyle x_{n-1}} and xn−2{displaystyle x_{n-2}}, like the secant method, but on xn−1{displaystyle x_{n-1}} and on the last iterate xk{displaystyle x_{k}} such that f(xk){displaystyle f(x_{k})} and f(xn−1){displaystyle f(x_{n-1})} have a different sign. This means that the false position method always converges.Secant Method Example
The recurrence formula of the secant method can be derived from the formula for Newton’s methodxn=xn−1−f(xn−1)f′(xn−1){displaystyle x_{n}=x_{n-1}-{frac {f(x_{n-1})}{f’(x_{n-1})}}}
by using the finite-difference approximationf′(xn−1)≈f(xn−1)−f(xn−2)xn−1−xn−2.{displaystyle f’(x_{n-1})approx {frac {f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}}.}
The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method.
If we compare Newton’s method with the secant method, we see that Newton’s method converges faster (order 2 against φ ≈ 1.6). However, Newton’s method requires the evaluation of both f{displaystyle f} and its derivative f′{displaystyle f’} at every step, while the secant method only requires the evaluation of f{displaystyle f}. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating f{displaystyle f} 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 2), so the secant method is faster. If, however, we consider parallel processing for the evaluation of the derivative, Newton’s method proves its worth, being faster in time, though still spending more steps.Generalizations[edit]
Broyden’s method 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.Computational example[edit]
Below, the secant method is implemented in the Python programming language.
It is then applied to find a root of the function f(x) = x2 − 612 with initial points x0=10{displaystyle x_{0}=10} and x1=30{displaystyle x_{1}=30}Notes[edit]
*^Papakonstantinou, J., The Historical Development of the Secant Method in 1-D, retrieved 2011-06-29See also[edit]References[edit]
*Avriel, Mordecai (1976). Nonlinear Programming: Analysis and Methods. Prentice Hall. pp. 220–221. ISBN0-13-623603-0.
*Allen, Myron B.; Isaacson, Eli L. (1998). Numerical analysis for applied science. John Wiley & Sons. pp. 188–195. ISBN978-0-471-55266-6.External links[edit]
*Secant Method Notes, PPT, Mathcad, Maple, Mathematica, Matlab at Holistic Numerical Methods Institute
*Weisstein, Eric W.’Secant Method’. MathWorld.Retrieved from ’https://en.wikipedia.org/w/index.php?title=Secant_method&oldid=988064369’
Download here: http://gg.gg/v1ozc
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Modified Regula Falsi Method generates the approximations in the same manner as the Regula Falsi Method does. But for faster convergence some modifications are made. We first choose the initial approximations and with. The approximation is chosen as the x-intercept of the line joining. To decide which secant line to use to compute, we. In this study, a modification of the classical Secant method for solving nonlinear, univariate and unconstrained optimization problems based on the development of the cubic approximation is presented. The iteration formula including an approximation of the third derivative of f(x) by using the Taylor series expansion is derived. The performance of the new method is analyzed in terms of the. GET SECANT Value using SEC Function in Excel. The SEC function in Excel calculates and returns the secant of a given radians. The Secant is the reciprocal of Cosine. SEC (θ) = Hypotenuse / Adjacent = c/b. Secant of the angle θ, is the ratio of the hypotenuse side – c, to the adjacent side – b. By Newton’s method: b = af(a) f0(a) Some will say that Newton’s method is very fast and has a convergence factor of 2. However, for every one step of Newton’s method, two steps of the secant method can be done, because Newton’s method requires the taking of a derivative and then nding two function evaluations. The secant can be calculated using known trigonometric properties if the angle is known, but it also can be calculated through simply the equation of H/A where H is the length of the hypotenuse of a triangle and A is the length of the adjacent side. For this example we will assume the length of both of these sides is known.
*Python Function For Modified Secant Method Formula
*Secant Equation
*Newton Secant Method
*Secant Method ExampleThe first two iterations of the secant method. The red curve shows the function f, and the blue lines are the secants. For this particular case, the secant method will not converge to the visible root.
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a functionf. The secant method can be thought of as a finite-difference approximation of Newton’s method. However, the secant method predates Newton’s method by over 3000 years.[1]The method[edit]
The secant method is defined by the recurrence relationxn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2)=xn−2f(xn−1)−xn−1f(xn−2)f(xn−1)−f(xn−2).{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})}}.}
As can be seen from the recurrence relation, the secant method requires two initial values, x0 and x1, which should ideally be chosen to lie close to the root.Derivation of the method[edit]
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 slope–intercept form, the equation of this line isy=f(x1)−f(x0)x1−x0(x−x1)+f(x1).{displaystyle y={frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{1})+f(x_{1}).}
The root of this linear function, that is the value of x such that y = 0 isx=x1−f(x1)x1−x0f(x1)−f(x0).{displaystyle x=x_{1}-f(x_{1}){frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.}
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):x2=x1−f(x1)x1−x0f(x1)−f(x0),x3=x2−f(x2)x2−x1f(x2)−f(x1),⋮xn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2).{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}}}Convergence[edit]
The iterates xn{displaystyle x_{n}} of the secant method converge to a root of f{displaystyle f} if the initial values x0{displaystyle x_{0}} and x1{displaystyle x_{1}} are sufficiently close to the root. The order of convergence is φ, whereφ=1+52≈1.618{displaystyle varphi ={frac {1+{sqrt {5}}}{2}}approx 1.618}
is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic.
This result only holds under some technical conditions, namely that f{displaystyle f} 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 [x0,x1]{displaystyle [x_{0},x_{1}]}. For example, if f{displaystyle f} is differentiable on that interval and there is a point where f′=0{displaystyle f’=0} on the interval, then the algorithm may not converge.Comparison with other root-finding methods[edit]
The secant method does not require that the root remain bracketed, like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on xn−1{displaystyle x_{n-1}} and xn−2{displaystyle x_{n-2}}, like the secant method, but on xn−1{displaystyle x_{n-1}} and on the last iterate xk{displaystyle x_{k}} such that f(xk){displaystyle f(x_{k})} and f(xn−1){displaystyle f(x_{n-1})} have a different sign. This means that the false position method always converges.
The recurrence formula of the secant method can be derived from the formula for Newton’s methodxn=xn−1−f(xn−1)f′(xn−1){displaystyle x_{n}=x_{n-1}-{frac {f(x_{n-1})}{f’(x_{n-1})}}}
Ppsspp game download android mobile. by using the finite-difference approximationf′(xn−1)≈f(xn−1)−f(xn−2)xn−1−xn−2.{displaystyle f’(x_{n-1})approx {frac {f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}}.}
The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method.
If we compare Newton’s method with the secant method, we see that Newton’s method converges faster (order 2 against φ ≈ 1.6). However, Newton’s method requires the evaluation of both f{displaystyle f} and its derivative f′{displaystyle f’} at every step, while the secant method only requires the evaluation of f{displaystyle f}. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating f{displaystyle f} 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 2), so the secant method is faster. If, however, we consider parallel processing for the evaluation of the derivative, Newton’s method proves its worth, being faster in time, though still spending more steps.Python Function For Modified Secant Method FormulaGeneralizations[edit]
Broyden’s method 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.Computational example[edit]
Below, the secant method is implemented in the Python programming language.
It is then applied to find a root of the function f(x) = x2 − 612 with initial points x0=10{displaystyle x_{0}=10} and x1=30{displaystyle x_{1}=30}Notes[edit]
*^Papakonstantinou, J., The Historical Development of the Secant Method in 1-D, retrieved 2011-06-29See also[edit]References[edit]
*Avriel, Mordecai (1976). Nonlinear Programming: Analysis and Methods. Prentice Hall. pp. 220–221. ISBN0-13-623603-0.
*Allen, Myron B.; Isaacson, Eli L. (1998). Numerical analysis for applied science. John Wiley & Sons. pp. 188–195. ISBN978-0-471-55266-6.External links[edit]
*Secant Method Notes, PPT, Mathcad, Maple, Mathematica, Matlab at Holistic Numerical Methods Institute
*Weisstein, Eric W.’Secant Method’. MathWorld.Retrieved from ’https://en.wikipedia.org/w/index.php?title=Secant_method&oldid=988064369’The first two iterations of the secant method. The red curve shows the function f, and the blue lines are the secants. For this particular case, the secant method will not converge to the visible root.
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a functionf. The secant method can be thought of as a finite-difference approximation of Newton’s method. However, the secant method predates Newton’s method by over 3000 years.[1]The method[edit]
The secant method is defined by the recurrence relationxn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2)=xn−2f(xn−1)−xn−1f(xn−2)f(xn−1)−f(xn−2).{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})}}.}
As can be seen from the recurrence relation, the secant method requires two initial values, x0 and x1, which should ideally be chosen to lie close to the root.Derivation of the method[edit]
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 slope–intercept form, the equation of this line isSecant Equationy=f(x1)−f(x0)x1−x0(x−x1)+f(x1).{displaystyle y={frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}(x-x_{1})+f(x_{1}).}
The root of this linear function, that is the value of x such that y = 0 isx=x1−f(x1)x1−x0f(x1)−f(x0).{displaystyle x=x_{1}-f(x_{1}){frac {x_{1}-x_{0}}{f(x_{1})-f(x_{0})}}.}
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):x2=x1−f(x1)x1−x0f(x1)−f(x0),x3=x2−f(x2)x2−x1f(x2)−f(x1),⋮xn=xn−1−f(xn−1)xn−1−xn−2f(xn−1)−f(xn−2).{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}}}Convergence[edit]
The iterates xn{displaystyle x_{n}} of the secant method converge to a root of f{displaystyle f} if the initial values x0{displaystyle x_{0}} and x1{displaystyle x_{1}} are sufficiently close to the root. The order of convergence is φ, whereφ=1+52≈1.618{displaystyle varphi ={frac {1+{sqrt {5}}}{2}}approx 1.618}
is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic.
This result only holds under some technical conditions, namely that f{displaystyle f} 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 [x0,x1]{displaystyle [x_{0},x_{1}]}. For example, if f{displaystyle f} is differentiable on that interval and there is a point where f′=0{displaystyle f’=0} on the interval, then the algorithm may not converge.Comparison with other root-finding methods[edit]Newton Secant Method
The secant method does not require that the root remain bracketed, like the bisection method does, and hence it does not always converge. The false position method (or regula falsi) uses the same formula as the secant method. However, it does not apply the formula on xn−1{displaystyle x_{n-1}} and xn−2{displaystyle x_{n-2}}, like the secant method, but on xn−1{displaystyle x_{n-1}} and on the last iterate xk{displaystyle x_{k}} such that f(xk){displaystyle f(x_{k})} and f(xn−1){displaystyle f(x_{n-1})} have a different sign. This means that the false position method always converges.Secant Method Example
The recurrence formula of the secant method can be derived from the formula for Newton’s methodxn=xn−1−f(xn−1)f′(xn−1){displaystyle x_{n}=x_{n-1}-{frac {f(x_{n-1})}{f’(x_{n-1})}}}
by using the finite-difference approximationf′(xn−1)≈f(xn−1)−f(xn−2)xn−1−xn−2.{displaystyle f’(x_{n-1})approx {frac {f(x_{n-1})-f(x_{n-2})}{x_{n-1}-x_{n-2}}}.}
The secant method can be interpreted as a method in which the derivative is replaced by an approximation and is thus a quasi-Newton method.
If we compare Newton’s method with the secant method, we see that Newton’s method converges faster (order 2 against φ ≈ 1.6). However, Newton’s method requires the evaluation of both f{displaystyle f} and its derivative f′{displaystyle f’} at every step, while the secant method only requires the evaluation of f{displaystyle f}. Therefore, the secant method may occasionally be faster in practice. For instance, if we assume that evaluating f{displaystyle f} 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 2), so the secant method is faster. If, however, we consider parallel processing for the evaluation of the derivative, Newton’s method proves its worth, being faster in time, though still spending more steps.Generalizations[edit]
Broyden’s method 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.Computational example[edit]
Below, the secant method is implemented in the Python programming language.
It is then applied to find a root of the function f(x) = x2 − 612 with initial points x0=10{displaystyle x_{0}=10} and x1=30{displaystyle x_{1}=30}Notes[edit]
*^Papakonstantinou, J., The Historical Development of the Secant Method in 1-D, retrieved 2011-06-29See also[edit]References[edit]
*Avriel, Mordecai (1976). Nonlinear Programming: Analysis and Methods. Prentice Hall. pp. 220–221. ISBN0-13-623603-0.
*Allen, Myron B.; Isaacson, Eli L. (1998). Numerical analysis for applied science. John Wiley & Sons. pp. 188–195. ISBN978-0-471-55266-6.External links[edit]
*Secant Method Notes, PPT, Mathcad, Maple, Mathematica, Matlab at Holistic Numerical Methods Institute
*Weisstein, Eric W.’Secant Method’. MathWorld.Retrieved from ’https://en.wikipedia.org/w/index.php?title=Secant_method&oldid=988064369’
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