General
What is the primary purpose of numerical analysis?
The numerical approximation of problems when symbolic manipulation is difficult/impossible
For the Bisection method to work, what conditions must the initial guesses satisfy?
a and b must be on opposite sides of the root (i.e opposite signs)
what are the two things you need for this method
the function and its derivative
Why is the secant method considered a variation of the NR method
Instead of using the derivative you use the secant line
What is the purpose of Lagrange polynomials
They are used for polynomial interpretation
Main purpose of the algorithms we learned today
Finding the root/zero of the equation
Main advantage of the Bisection Method
If there is a root between the two guesses it will eventually converge
Main advantage of NR method
Calculation time
What is a secant
A line that intersects a curve at 2 points
If you have a set of N+1 data points, what degree is your polynomial
N degree
What is the core mathematical theorem that the Bisection method relies on?
IVT
Each step of the NR method finds the root of what
the tangent line of the function at the given point
Advantage of Secant over NR method
the derivative is not required
In each iteration of the Bisection method, how are the new points a,b calculated?
Diff signs: b = x
Same signs: a = x
Under what condition (in relation to the derivative) will the NR method fail
If the tangent is parallel or nearly parallel to the x axis, or if there is a stationary point nearby
What is the iterative formula of this method?
x_n = x_(n-2)-f(x_n-2)*((x_n-1)-(x_n-2))/(f(x_(n-1)-f(x_n-2))
Using your code (or logic) if the user inputs the values of a=1 and b = 5 to the function (x-3.4)(x-8) and the tolerance is 1e-6, what are the number of iterations that must be done?
22
What is the iterative formula of this method
x_n+1 = x_n -f(x_n)/f'(x_n)
Potential issue of secant method in a concave graph
One of the endpoints can become fixed, slowing the process down or even making it fail