The impetus to write this material down was a question from my student Michael Campbell. May it go some way to relieving the reader’s tedium, as it did mine.
To help with the numerous numpy arrays that needed to be typeset as matrices in latex, I wrote this small python package: np2latex.
#QR ALGORITHM SYMBOLIC CALCULATOR CODE#
In this post, I will be providing python code implementing the various algorithms hopefully this will be more accessible to many readers. I first learned about the -algorithm when writing this C library, which implements many of the standard numerical linear algebraic algorithms. In this essay we will hopefully de-mystify this process, and give the reader some insight into this important algorithm. This process eventually leads us (in the limit) to an upper triangular matrix (though it is not at all obvious why this would be the case), and the diagonal entries on this limit are the eigenvalues of.
The algorithm then iterates this factor-and-reverse process There is no immediate geometric or intuitive interpretation of multiplying the two matrices in reverse order. The next step is the mystery, we multiply the factors in the reverse order The first step of the -algorithm is to factor into the product of an orthogonal and an upper triangular matrix (this is the -factorization mentioned above) Suppose we are interested in computing the eigenvalues of a matrix. The details of the -algorithm are mysterious.
As the reader can surely guess, this involves the -factorization of the matrix in question (as a quick reminder, the -factorization encodes the Gram–Schmidt process for orthonormalizing a basis). The standard algorithm for computing eigenvalues is called the -algorithm. In fact, popular methods for polynomial root finding are closely related to the eigenvalue finding algorithms discussed below. Even though techniques like Newton’s method exist, the sets of initial points that converge to each individual root are extremely complex, so choosing a collection initial points so that all the roots are found is very difficult. Computing the roots to a polynomial equation is also a difficult problem.Computing the characteristic equation of a matrix involves symbolically computing a determinant, which is expensive, and should be avoided if possible.While this is the textbook method, it is a poor method for numerical computation, because The reader familiar with a standard linear algebra curriculum may wonder why this question is even interesting, all that must be done is to compute the characteristic equation of the matrix, then compute its roots. We’ve already discussed a method for solving linear equations in A Deep Dive Into How R Fits a Linear Model, so for this post I thought we should complete the circle with a discussion of methods which compute eigenvalues and eigenvectors. The two most practically important problems in computational mathematics are solving systems of linear equations, and computing the eigenvalues and eigenvectors of a matrix.
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