linear-algebra

An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
The Conjugate Gradient Method is the most prominent iterative method for solving sparse systems of linear equations.
Unfortunately, many textbook treatments of the topic are written with neither illustrations nor intuition, and their
victims can be found to this day babbling senselessly in the corners of dusty libraries. For this reason, a deep,
geometric understanding of the method has been reserved for the elite brilliant few who have painstakingly decoded
the mumblings of their forebears. Nevertheless,the Conjugate Gradient Method is a composite ofsimple, elegantideas
that almost anyone can understand. Of course, a reader as intelligent as yourself will learn them almost effortlessly.
The idea of quadratic forms is introduced and used to derive the methods of Steepest Descent, Conjugate Directions,
and Conjugate Gradients. Eigenvectors are explained and used to examine the convergence of the Jacobi Method,
Method. I have taken pains to make this article easy to read. Sixty-six illustrations are provided. Dense prose is
avoided. Concepts are explained in several different ways. Most equations are coupled with an intuitive interpretation.
linear-algebra  academia  papers  cmu  computer-science  matrix-algebra  math
19 days ago by amsone
Making Sense of Bivector Addition, viXra.org e-Print archive, viXra:1807.0234
As a demonstration of the coherence of Geometric Algebra's (GA's) geometric and algebraic concepts of bivectors, we add three geometric bivectors according to the procedure described by Hestenes and Macdonald, then use bivector identities to determine, from the result, two bivectors whose outer product is equal to the initial sum. In this way, we show that the procedure that GA's inventors dened for adding geometric bivectors is precisely that which is needed to give results that coincide with those obtained by calculating outer products of vectors that are expressed in terms of a 3D basis. We explain that that accomplishment is no coincidence: it is a consequence of the attributes that GA's designers assigned (or didn't) to bivectors.
linear-algebra  algebra  define-your-terms  rather-interesting  to-write-about  nudge-targets  consider:representation  Grassmannian  wedge-product
27 days ago by Vaguery