From 9e7166a52e94d8e15bf2dbfe00026f21f76630a9 Mon Sep 17 00:00:00 2001 From: Elizabeth Hunt Date: Wed, 18 Oct 2023 12:49:39 -0600 Subject: oct 16, 18 notes. add unit tests with utest, and bisection root finding methods --- notes/Oct-18.org | 18 ++++++++++++++++++ 1 file changed, 18 insertions(+) create mode 100644 notes/Oct-18.org (limited to 'notes/Oct-18.org') diff --git a/notes/Oct-18.org b/notes/Oct-18.org new file mode 100644 index 0000000..0104164 --- /dev/null +++ b/notes/Oct-18.org @@ -0,0 +1,18 @@ +Error Analysis Of Bisection Root Finding: + +e_0 \le b - a = b_0 - a_0 +e_1 \le b_1 - a_1 = 1/2(b_0 - a_0) +e_2 \le b_2 - a_2 = 1/2(b_1 - a_1) = (1/2)^2(b_0 - a_0) +e_k \le b_k - a_k = 1/2(b_{k-1} - a_{k-1}) = \cdots = (1/2)^k (b_0 - a_0) + + +e_k \le (1/2)^k (b_0 - a_0) = tolerance +\Rightarrow log(1/2^k) + log(b_0 - a_0) = log(tolerance) +\Rightarrow k log(1/2) + log(tolerance) - log(b_0 - a_0) +\Rightarrow k log(1/2) = log(tolerance / (b_0 - a_0)) +\Rightarrow k \ge log(tolerance / (b_0 - a_0)) / log(1/2) + +The Bisection Method applied to an interval [a, b] for a continous function will reduce the error +each time through by at least one half. + +| x_{k+1} - x_k | \le 1/2|x_k - x^* | -- cgit v1.2.3-70-g09d2