nonlinear_secant.pdf
版权申诉
161 浏览量
2023-06-18
13:14:59
上传
评论
收藏 281KB PDF 举报
Nonlinear equations:
The secant method
MATH2070: Numerical Methods in Scientific Computing I
Location: http://people.sc.fsu.edu/∼jburkardt/classes/math2070 2019/nonlinear secant/nonlinear secant.pdf
Use two samples to guess where a function crosses the x axis
Rootfinding (Secant version)
Given a function f (x) and two estimates for the root, use a linear model to predict a better root.
The bisection and regula falsi methods start and stay within a change-of-sign interval. This has the advantage
of a guaranteed bracketing of the solution, but requires the user to supply such an interval at the beginning,
and costs a certain amount of internal bookkeeping from step to step.
The secant method drops the change-of-sign requirement, resulting in simpler code, and faster convergence
when the initial estimates are close to a root.
1 The secant method is a linear model for f(x)
For a given function f(x) whose root we are seeking, suppose we have two sample data points, (a,f(a))
and (b,f(b)). Then the linear function that matches this data has the form:
y(x) =
f(a) (b − x) + f (b) (x − a)
b − a
If this linear function is a good model for f (x) locally, then we can estimate the root of f (x) by computing
the root of y(x):
c =
f(a) b − f (b) a
f(a) − f (b)
If our new root estimate is not good enough, we can update our data by a ← b and b ← c and apply a
new secant prediction. As the root estimates get better, the linear model approximates f(x) better, and
convergence may be expected to accelerate.
1
资源评论
