From 6bf4b90c90f15f4ab60833bddf5b5756d1a6b1f6 Mon Sep 17 00:00:00 2001 From: Elizabeth Alexander Hunt Date: Thu, 2 Jul 2026 11:55:17 -0700 Subject: Init --- Homework/math4610/notes/Oct-18.org | 18 ++++++++++++++++++ 1 file changed, 18 insertions(+) create mode 100644 Homework/math4610/notes/Oct-18.org (limited to 'Homework/math4610/notes/Oct-18.org') diff --git a/Homework/math4610/notes/Oct-18.org b/Homework/math4610/notes/Oct-18.org new file mode 100644 index 0000000..0104164 --- /dev/null +++ b/Homework/math4610/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 v1.3