1 Solving equations Flashcards
(18 cards)
What is the basic goal of numerical root-finding methods?
To find values of
𝑥 such that 𝑓(𝑥)=0
f(x)=0, using systematic, computer-based approximations
What is the key difference between root-finding and optimization problems?
Root location is where 𝑓(𝑥)=0 f
Optimization finds where the derivative 𝑓′(𝑥)=0 involves searching for the function’s extreme points (min. or max)
What are bracketing methods?
Methods that start with two initial guesses that contain (bracket) the root, then systematically reduce the width of the bracket
What are open methods for finding roots?
Methods that use one or more initial guesses but do not require them to bracket the root .
What are two key advantages of open methods compared to bracketing methods?
1) Faster convergence; 2) More computationally efficient .
What is the main disadvantage of open methods?
They are not guaranteed to work (may fail to converge) .
What is an example of a hybrid root-finding method?
Brent’s method — combines the reliability of bracketing with the speed of open methods .
Which MATLAB function is based on Brent’s method?
fzero .
How does fzero work with a single initial guess?
It searches outward from the guess to find a sign change, then applies a combination of secant, inverse quadratic interpolation, and bisection methods .
How can you use fzero with bracketing?
Provide a vector of two guesses [x0 x1] that bracket a root (i.e., f(x0)f(x0) and f(x1)f(x1) have opposite signs) .
What options can be controlled with MATLAB’s optimset for fzero?
Display of iterations (‘Display’,’iter’) and termination tolerance on xx (‘TolX’) .
What is an error estimate in root-finding?
A measure of how close the current approximation is to the true root, helping determine when to stop iterating .
How do bracketing and open methods compare in terms of accuracy?
Bracketing methods are guaranteed to converge but slowly; open methods are faster but may fail or give less reliable results .
Why is fzero often preferred in engineering applications?
It combines robustness (via bracketing and bisection) with speed (via open methods), making it highly reliable and efficient .
What are some bracketing methods?
- Incremental search (forarbejdet / en forudsætning)
- Bisection
- False position
What are some open methods?
- Fixed point iteration
- Newton-Raphson
- Secant metoden (udvidet Newton-Raphson)
Bisection
Bracket method:
intervallet deles i halve –> hvis funktionen skifter fortegn i et interval, vil funktionsværdien ved midt-punktet evalueres
How does the fzero function work?
x = fzero (function, x0)
[x,fx]= fzero (function, x0)
x0, Initial guess
x, location of the root
fx, function being evaluated at the root
Initial brackets:
x = fzero(function, [x0,x1]
