Question

2 Sentences each

1 advantage of Newton interpolation over Vandermonde matrix method:

1 advantage of Langrange interpolation over Newton interpolation:

Is Bisection search a fixed point method?

Answer 1

1 advantage of Newton interpolation over Vandermonde matrix method:

Newton's form is more efficient when you have to interpolate data incrementally. The evaluation can be done quickly and straightforwardly using Horner's method, and adding an additional node just amounts to adding a single additional term.

1 advantage of Langrange interpolation over Newton interpolation:

Lagrange's form is more efficient when you have to interpolate several data sets on the same data points. The Lagrange approach amounts to diagonalizing the problem of finding the coefficients, so it takes only linear time to find the coefficients. This is good if you need to use the same set of points repeatedly.

Is Bisection search a fixed point method?

NO, the bisection section search not a fixed point method beacuse the bisection method is a root-finding method that applies to any continuous functions for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and then selecting the subinterval in which the function changes sign, and therefore must contain a root.

As you can see in the figure a few steps of the bisection method applied over the starting range [a₁;b₁]. The bigger red dot is the root of the function.