Illinois Algorithm
While it is a misunderstanding to think that the method of false position is a good method, it is equally a mistake to think that it is unsalvageable. The failure mode is easy to detect (the same end-point is retained twice in a row) and easily remedied by next picking a modified false position, such as
or
down-weighting one of the endpoint values to force the next ck to occur on that side of the function. The factor of 2 above looks like a hack, but it guarantees superlinear convergence (asymptotically, the algorithm will perform two regular steps after any modified step). There are other ways to pick the rescaling which give even better superlinear convergence rates.
The above adjustment to regula falsi is sometimes called the Illinois algorithm. Ford (1995) summarizes and analyzes this and other similar superlinear variants of the method of false position. Judging from the bibliography, modified regula falsi methods were well known in the 1970s and have been subsequently forgotten or misremembered in current textbooks.
Read more about this topic: False Position Method
Famous quotes containing the word illinois:
“An Illinois woman has invented a portable house which can be carried about in a cart or expressed to the seashore. It has also folding furniture and a complete camping outfit.”
—Lydia Hoyt Farmer (1842–1903)