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1 Solving equations Flashcards

(18 cards)

1
Q

What is the basic goal of numerical root-finding methods?

A

To find values of
𝑥 such that 𝑓(𝑥)=0
f(x)=0, using systematic, computer-based approximations

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2
Q

What is the key difference between root-finding and optimization problems?

A

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)

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3
Q

What are bracketing methods?

A

Methods that start with two initial guesses that contain (bracket) the root, then systematically reduce the width of the bracket

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4
Q

What are open methods for finding roots?

A

Methods that use one or more initial guesses but do not require them to bracket the root .

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5
Q

What are two key advantages of open methods compared to bracketing methods?

A

1) Faster convergence; 2) More computationally efficient .

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6
Q

What is the main disadvantage of open methods?

A

They are not guaranteed to work (may fail to converge) .

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7
Q

What is an example of a hybrid root-finding method?

A

Brent’s method — combines the reliability of bracketing with the speed of open methods .

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8
Q

Which MATLAB function is based on Brent’s method?

A

fzero .

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9
Q

How does fzero work with a single initial guess?

A

It searches outward from the guess to find a sign change, then applies a combination of secant, inverse quadratic interpolation, and bisection methods .

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10
Q

How can you use fzero with bracketing?

A

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) .

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11
Q

What options can be controlled with MATLAB’s optimset for fzero?

A

Display of iterations (‘Display’,’iter’) and termination tolerance on xx (‘TolX’) .

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12
Q

What is an error estimate in root-finding?

A

A measure of how close the current approximation is to the true root, helping determine when to stop iterating .

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13
Q

How do bracketing and open methods compare in terms of accuracy?

A

Bracketing methods are guaranteed to converge but slowly; open methods are faster but may fail or give less reliable results .

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14
Q

Why is fzero often preferred in engineering applications?

A

It combines robustness (via bracketing and bisection) with speed (via open methods), making it highly reliable and efficient .

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15
Q

What are some bracketing methods?

A
  • Incremental search (forarbejdet / en forudsætning)
  • Bisection
  • False position
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16
Q

What are some open methods?

A
  • Fixed point iteration
  • Newton-Raphson
  • Secant metoden (udvidet Newton-Raphson)
17
Q

Bisection

A

Bracket method:

intervallet deles i halve –> hvis funktionen skifter fortegn i et interval, vil funktionsværdien ved midt-punktet evalueres

18
Q

How does the fzero function work?

A

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]