bisectionfixed-point-theorems
I need to answer a multiple choice question that goes like this –
Does fixed point iteration method always faster than bisection method?"
Now, I know that fixed point is faster, I have seen the math behind the convergence
But I wonder if there's like a specific case where I would pick bisection over fixed point ?
How would you answer that question cause I feel like I didn't get enough information or the question isn't well phrased.
What definition of trig functions (also called "circular functions") are you using?
The definitions in terms of right angles cannot be used in general as that requires the angle to be between 0 and 90 degrees but, in order to use them as functions we want them defined for all real numbers.
Of the "circle" definition is used: Construct a unit circle (center at (0, 0), radius 1) in an xy-coordinate system. Given non-negative number, t, measure, counter clockwise, around the circumference a distance t (for negative t, measure clockwise). The endpoint has coordinates $(\cos(t), \sin(t))$. That is, whatever the end point is, the x coordinate is, by definition, $\cos(t)$, the y coordinate is $\sin(t)$.
Notice that "$t$" is NOT in degrees because it is not an angle at all, it is a distance around the circumference. We can think of that as an angle by identifying the angle with the distance subtended on a unit circle which is simply the definition of "radian".
$$ \exp(x)-1-x=\frac12x^2\left(1+\frac13x+\frac1{12}x^2+...\right) $$ has a minimum in $x=0$ and behaves locally like $\frac12 x^2$.
Note that you pretend to not know the root. Starting the one of the methods at the point $x=0$ has thus to be considered extremely unlikely.
Best Answer
Fixed point iteration is not always faster than bisection. Both methods generally observe linear convergence. The rates of convergence are $|f'(x)|$ for fixed-point iteration and $1/2$ for bisection, assuming continuously differentiable functions in one dimension.
It's easy to construct examples where fixed-point iteration will converge much slower than bisection (sublinear convergence). For example, the iterations $x_{n+1}=\sin(x_n)$ are well-known to converge very slowly, much slower than using bisection on $f(x)=x-\sin(x)$.
The main advantage of bisection is the fact that it is guaranteed to give linear convergence. The main drawback is also that it is guaranteed to only give linear convergence, whereas other methods can converge faster in some situations.
Bisection is also only applicable in some scenarios (the function must be defined over an ordered domain (e.g. not $\mathbb R^2$) and known to attain two different signs (e.g. not $x^2$)). It also does not give an analytic estimate of the solution, so it's less useful from a theoretical standpoint.