We consider the following Polynomial Approximation Problem. Given distinct nodes xo,..., Xn €R and (not necessarily distinct) values yo,.…., yn € R, find a polynomial p(x) € Pn such that p(x;) = yi-
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![Just part (b) please
3. (Polynomial (Over-)Fitting).
(a) Let xo,...,X, €R be any n+1 points, which we shall call nodes.
1 Xo
1 x1
...
...
V:=
ER(n+1)x(d+1)
1 Xn
is called the Vandermonde matrix of the points xo,., Xxn. In first year linear algebra, you
probably proved the qualitative result that V is invertible if and only if n = d and all the x;
are distinct. Now we study its invertibility more quantitatively:
• Write a Matlab helper function [V] = vandermonde (x,d) which assembles the above
matrix, where d = size(x)-1, resulting in a square matrix.
• Generate equidistant points in [-1,1), and numerically calculate the condition num-
ber x(V) of V for several representative values d = ns 100 in some matrix norm.
(b) Denote by P, the space of polynomials of degree < n, equipped with the norm
lp|-1,11:= _max Ip(x)].
xEl-1,1]
We consider the following Polynomial Approximation Problem. Given distinct nodes Xo,...
Xn €Rand (not necessarily distinct) values yo..., yn € R, find a polynomial p(x) € Pn such
that p(x¡) = yi-
Show that p(x) = E"-0C;xi solves this problem if and only if the coefficients c; satisfy
Co
yo
yi
V
II](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4eca1263-44d2-4336-a09d-765b3e98edf6%2F96a396ea-2457-4295-b94f-1bc4b84a1448%2Fc59641e_processed.png&w=3840&q=75)
Step by step
Solved in 2 steps
