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Numerical Analysis. EE, NCKU Tien-Hao Chang (Darby Chang). In the previous slide. Error estimation in system of equations vector/matrix norms LU decomposition split a matrix into the product of a lower and a upper triangular matrices
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Numerical Analysis EE, NCKU Tien-Hao Chang (Darby Chang)
In the previous slide • Error estimation in system of equations • vector/matrix norms • LU decomposition • split a matrix into the product of a lower and a upper triangular matrices • efficient in dealing with a lots of right-hand-side vectors • Direct factorization • as an systems of n2+n equations • Crout decomposition • Dollittle decomposition
In this slide • Special matrices • Strictly diagonally dominant matrix • Symmetric positive definite matrix • Cholesky decomposition • Tridiagonal matrix • Iterative techniques • Jacobi, Gauss-Seidel and SOR methods • conjugate gradient method • Nonlinear systems of equations • Exercise 3
3.7 Special Matrices
Special matrices • Linear systems • which arise in practice and/or in numerical methods • the coefficient matrices often have special properties or structure • Strictly diagonally dominant matrix • Symmetric positive definite matrix • Tridiagonal matrix
Symmetric positive definiteRelations to • Eigenvalues • Leading principal sub-matrix
Cholesky decomposition • For symmetric positive definite matrices • greater efficiency can be obtained • consider the symmetric of the matrix • Rather than LU form, we factor the matrix into the form • A=LLT
Tridiagonal • Only 8n-7 operations • factor step 3n-3 • solve step 5n-4
Before entering 3.8 • So far, we have learnt three methods algorithms in Chapter 3 • Gaussian elimination • LU decomposition • direct factorization • Are they algorithms? • What’s the differences to those algorithms in Chapter 2? • they report exact solutions rather than approximate solutions question further question answer
3.8 Iterative Techniques for Linear Systems
Iterative techniques • Analytic techniques is slow • O(n3) • Especially for systems with large but sparse coefficient matrices • As an added bonus, iterative techniques are less insensitive to roundoff error
Iteration matrixImmediate questions • When does T guarantee a unique solution? • When does T guarantee convergence? • How quick does {x(k)} converge? • How to generate T?
Assume that I-T is singular, there exists a nonzero vector x such that (T-1I)x=0 • 1 is a eigenvalue of T • but ρ(T)<1, contradiction
(in section 2.3 with proof) Recall that http://www.dianadepasquale.com/ThinkingMonkey.jpg
Recall that http://www.dianadepasquale.com/ThinkingMonkey.jpg
Iteration matrixFor these questions • We know that when ρ(T)<1, {x(k)} from x(k+1)=Tx(k)+c will converge linearly to a unique solution with any initial vector x(0) • What is missing? • remember the problem is to solve Ax=b • How to generate T? • like f(x) and g(x), different algorithms construct different iteration matrix question hint answer
Splitting methods • Ax=b (M–N)x=b Mx=Nx+b • x=M-1Nx+M-1b • T=M-1N and c=M-1b • A class of iteration methods • Jacobi method • Gauss-Seidel method • SOR method
3.9 Conjugate Gradient Method 37
Conjugate gradient method • Not all iterative methods are based on the splitting concept • The minimization of an associated quadratic functional
http://fuzzy.cs.uni-magdeburg.de/~borgelt/doc/somd/parabola.gifhttp://fuzzy.cs.uni-magdeburg.de/~borgelt/doc/somd/parabola.gif
Choose the search direction d(m) • as the tangent line in Newton’s method • the gradient of f at x(m) • Choose the step size • as the root of the tangent line
http://www.mathworks.com/cmsimages/op_main_wl_3250.jpg Global optimization problem
3.10 Nonlinear Systems of Equations