This is a visual demonstration of finding the root of an equation \(f(x) = 0\) on an interval using the Bisection Method.
Suppose we want to solve the equation \(f(x) = 0\). Given two points a and
b such that \(f(a)\) and \(f(b)\) have opposite signs, we know by the
intermediate value theorem that \(f\) must have at least one root in the
interval \([a, b]\) as long as \(f\) is continuous on this interval. The
bisection method divides the interval in two by computing \(c = (a + b) / 2\). There are now two possibilities: either \(f(a)\) and \(f(c)\) have
opposite signs, or \(f(c)\) and \(f(b)\) have opposite signs. The
bisection algorithm is then applied recursively to the sub-interval where the
sign change occurs.
During the process of searching, the mid-point of subintervals are annotated in the graph by both texts and blue straight lines, and the end-points are denoted in dashed red lines. The root of each iteration is also plotted in the right margin of the graph.
library(animation)
ani.options(nmax = 30)
## default example
xx = bisection.method()
xx$root # solution
## [1] 2
## a cubic curve
f = function(x) x^3 - 7 * x - 10
xx = bisection.method(f, c(-3, 5))
## interaction: use your mouse to select the two end-points
if (interactive()) bisection.method(f, c(-3, 5), interact = TRUE)
Donate
As a freelancer (currently working as a contractor) and a dad of three kids, I truly appreciate your donation to support my writing and open-source software development! Your contribution helps me cope with financial uncertainty better, so I can spend more time on producing high-quality content and software. You can make a donation through methods below.
-
Venmo:
@yihui_xie, or Zelle:xie@yihui.name -
Paypal
-
If you have a Paypal account, you can follow the link https://paypal.me/YihuiXie or find me on Paypal via my email
xie@yihui.name. Please choose the payment type as “Family and Friends” (instead of “Goods and Services”) to avoid extra fees. -
If you don’t have Paypal, you may donate through this link via your debit or credit card. Paypal will charge a fee on my side.
-
-
Other ways:
WeChat Pay (微信支付：谢益辉) Alipay (支付宝：谢益辉) 
When sending money, please be sure to add a note “gift” or “donation” if possible, so it won’t be treated as my taxable income but a genuine gift. Needless to say, donation is completely voluntary and I appreciate any amount you can give.
Please feel free to email me if you prefer a different way to give. Thank you very much!
I’ll give back a significant portion of the donations to the open-source community and charities. For the record, I received about $30,000 in total (before tax) in 2024-25, and gave back about $15,000 (after tax).