2.7.6 Conjugate Gradient Method for a Sparse Sy stem Shi & Bo What is sparse system A system of linear equations is called sparse if only a relatively small number of its matrix elements are n
onzero. It is wasteful to use general methods of line ar algebra on such problems, because most of the O () arithmetic operations devoted to solving the set of equations or inverting the matrix involve zero opera nds. Furthermore, you might wish to work problems so large as to tax your available memory space, and it is wasteful to reserve storage for unfruitful zero el ements. Note that there are two distinct (and not al ways compatible) goals for any sparse matrix metho
Where A is n*n symmetric positive definit e matrix, b is known n-d vector. Solving Ax=b is equivalent to minimizing f=Ax-b steepest descent method Steepest Descent
Start at a initial guess point adjust until close en ough to the exact solution: Where i is number of iterations is step size, is A djustment Direction. How to choose direction and step size?
Choose direction Choose the direction which f decreases mos t quickly. Move from point to the point by minimizing along the line from in the direc tion opposite to . Hense, .
Choose step size Step size should minimize f, along the dire ction of ,which means When to stop should give a stopping criterion because We
there may have many errors and noises. (1) only for exact arithmetic, not in practice (2) in practice unstable alg for general A stopping criterion is or with an given small . Symmetric but non-positive definite A With the choice instead of . In this case a
nd for all k. This algorithm is equivalent t o the ordinary conjugate gradient algorith m, but with all dot products replaced by . It is called the minimum residual algorith m, because it corresponds to successive m inimizations of the function any nonsingular matrix A, is symmetric For
and positive-definite. But we cant use . Because the condition number of the matri x is the square of the condition number of A. Thanks Reference Numerical Recipes
Steepest Decent and Conjugate Gradients, w3.pppl.gov/m3d/reference/SteepestDecent andCG.ppt Reference Numerical Recipes Steepest Decent and Conjugate Gradients
A redox reaction is therefore a reaction in which changes in oxidation numbers occur. Rules for assigning oxidation numbers: An atom in its elemental state, 0. An atom in a simple monoatomic ion, charge on the ion. The sum of...
The Role of Venture Capitalists New Start-Ups Udacity Udemy University Now Coursebook Coursekit Courseload CourseRank 8. The Models of College University of the People (UoPeople): Western Governors University (WGU) now also WGU Indiana, WGU Washington (state), WGU Texas, WGU Tennessee,...
DNA looks like a twisted ladder. This arrangement is called a . double helix. Alternating sugar and phosphates make up the backbone (the "rails" of the ladder) Bases make up the "rungs" and are held together by weak hydrogen bonds....
3-phase levels $120 (B-dressage Tests) 2-phase levels $50 (A-dressage Test) DRESSAGE ONLY $30 Extra Dressage Round $20 Starter - 2019 Intro Test B. Jump 12" (only 2-phase) 2019 Intro Test B Pre-Elementary- 2019 Intro Test B. Jump 18" (Only Vista...
Ready to download the document? Go ahead and hit continue!