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Questions tagged [bisection]

Use this tag for questions related to the bisection method, which is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing.

The bisection method is a root-finding method that bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a simple and robust method, but it is also slow, hence its use is often restricted to obtaining quickly a starting point for more rapidly-converging methods. The bisection method is also called the interval halving method, binary search method, or dichotomy method.

125 questions
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What is a proof that a diameter bisects a circle?

One of the major contributions Thales is said to have given is the proof that a diameter of a circle bisects the circle, yet Euclid doesn't even bat an eye. Then again, Euclid skipped over other things like needing to assume that the plane was…
Aaron Goldsmith
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Simple bisection geometry

Let $\triangle ABC$ have incenter $D$ and let the incircle intersect sides $BC,AB,AC$ at $E,F,G$ respectively. Extend $AB$ and $AC$ to meet the circumcircle of $\triangle ADE$ at $K$ and $I$ respectively. Prove that $FG$ bisects $KI$. I have solved…
user1164127
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Consider the triangle ABC in which $AC(AB+ AC)={BC}^{2}$ Show that angles $BAC = 2\cdot ABC$.

Consider the triangle ABC in which $AC(AB+ AC)={BC}^{2}$ Show that angles $BAC = 2\cdot ABC$. MY IDEAS MY DRAWING So I processed the equality that was given $AC(AB+ AC)={BC}^{2}$ $AC=\frac{{BC}^{2}}{AB+AC}$ As you can see, I put a point AA'…
user1104319
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Bisection method for $f(x)=x^4-x-2$

Finding a root for the polynomial $$f(x)=x^4-x-2$$ I just to double-check if I've done this correctly, as it's my first time doing so. My work so far In above function, $a=1$ and $b=2$ works, as $f(a)$ and $f(b)$ have opposite…
Jessie
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When to use Newtons's, bisection, fixed-point iteration and the secant methods?

I've been introduced more or less to these methods of finding a root of a function (a point where it intersects the $x$ axis), but I'm not sure when they should be used and what are the advantages of one method over the other. I think that it would…
user168764
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What about this method to solve equation systems?

Assume you want to solve a system of non-linear equations $$f_1(x,y)=0\\ f_2(x,y)=0$$ $x,y \in [0,1]$ and $f_i : [0,1]^2 \rightarrow \mathbb{R}$, but derivatives and finite differences are not available or too costly. The literature for this case is…
darksun
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Relationship of the Egg Riddle solution to bisection and $\log(N)$ function?

I stumbled upon the video "Can you solve the egg drop riddle? - Yossi Elran" on the TED-Ed YouTube channel. What I found interesting is that in the solution, given that the triangle numbers grow quadratically, it means that in this kind of search,…
trojj
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Location of roots using Darboux property

I was studying the Bisection method for root finding. It states that for a continuous function we can find out a root if it exists by the Intermediate value theorem.My question is why do we need Continuity of a function? Can't I just work with a…
Upstart
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Why is the formula for relative approximation error with respect to the current approximation?

I have to approximate the root of the equation $x^3 -4x - 9 = 0$ by using the Bisection method from my numerical methods class. The question also states that I have to iterate the process until my relative approximation error gets below a specified…
Makeda
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prove analyticaly that f converges to a root

Let $f,g \subset C[a,b] $ where $f$ satisfies $f(a)f(b)<0$. Consider the bisection method for the numerical approximation of the root in $fg^2$ has atleast one root in $[a,b]$, and the sequence $y_1, y_2$ the method produces. Explain why $fg^2$ has…
user919693
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Bisection method with geometric mean

The bisection method is a well-known method for root-finding. Given a continuous function $f$ and an interval $[a,b]$ where $f(a)$ and $f(b)$ have opposite signs, a root can be guaranteed to be in $(a,b)$. The bisection method computes…
Simply Beautiful Art
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Roots of the function $f(x) = \frac{x}{2} - \sin x + \frac{\pi}{6} - \frac{\sqrt{3}}{2}=0$ using bisection method.

Suppose we wish to compute the zeros of the function $f(x) = \frac{x}{2} - \sin x + \frac{\pi}{6} - \frac{\sqrt{3}}{2}=0$, which has exactly two roots in the interval $[-\pi,\pi]$. Is it possible to apply the bisection method to compute both roots?…
user839136
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Euler method and bisection method

I'd like to solve the equation $$ \phi''(x) = \lambda \sin (\phi(x)) $$ where $x \in (0,L)$, $\phi'(0) = 0$, $\phi'(L) = 0$. Let $ \psi = \phi'$ and $$ \phi'(x) - \psi(x) = 0$$ $$ \psi'(x) - \lambda \sin (\phi(x)) = 0$$ for $x \in (0,L)$ and…
DMan
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Reasoning of optimality of the Two-Egg Drop Problem solution

I come across the interesting egg drop problems and this two-egg variation. Briefly, the problem is: Given a building with n floors and two eggs, we want to find the critical floor f, where the egg breaks if dropped from floor $≥$ f, while stays…
PkDrew
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Can the order of convergence be defined for a bisection method?

$\sqrt{2}$ can be approximated by bisection method to $f(x)=x^2-2$ on $[0, 1]$. I expected this to be linear convergence, but thought it might not be. The definition of the order of convergence of $\lim\limits_{n\to\infty} p_n=p$ is the positive…
Dinosaur
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