Use the information you found to write some software which runs Newton’s method
ID: 3766477 • Letter: U
Question
Use the information you found to write some software which runs Newton’s method to find the solutions to f(x) = 0. This will be much easier with a programminglanguage with built in linear algebra routines such as MATLAB, Python with ScyPy & NumPy, or Julia.
Consider an n × n syrnmetric matrix A. We seek an eigenvalue E R and eigenvector v E Rn pair which satisfy Av = Av. This is only n nonlinear equations for n + 1 unknowns ( and all n components of v). This fact is evidenced by the result that eigenvectors are only unique up to scaling. So, in order to add another equation and to make the eigenvector (almost) unique, we wil add the equation 2* In order to convert this into the form of a root finding problem, define a new variable x which is an n + 1 dimensional vector whose first n entries are v and whose last entry is . Then rewrite the above equations in the form f(x) = 0 by moving all terms to the left.Explanation / Answer
Geometrically this means that Av is in the same direction as v, since multiplying a vector by a number changes its length, but not its direction.
Matlab has a built-in routine for finding eigenvalues and eigenvectors:
> A = pascal(4)
> [V E] = eig(A)
The results are a matrix V that contains eigenvectors as columns and a diagonal matrix E that contains eigenvalues on the diagonal. We can check this by:
> v1 = V(:,1)
> A*v1
> E(1,1)*v1
It would be nice to factor out the v from the right-hand side of this equation, but we can’t because A is a matrix and is a number. However, since Iv = v, we can do the following:
Av v = Av Iv
= (A I)v
= 0
If v is nonzero, then by Theorem 3 the matrix (A I) must be singular. By the same theorem, we must have:
det(A I) = 0.
This is called the characteristic equation.
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