By Unknown Author, G. Meurant
Hardbound. This ebook offers with numerical tools for fixing huge sparse linear structures of equations, really these coming up from the discretization of partial differential equations. It covers either direct and iterative equipment. Direct equipment that are thought of are variations of Gaussian removal and quick solvers for separable partial differential equations in oblong domain names. The ebook experiences the classical iterative tools like Jacobi, Gauss-Seidel and alternating instructions algorithms. a selected emphasis is wear the conjugate gradient in addition to conjugate gradient -like tools for non symmetric difficulties. most productive preconditioners used to hurry up convergence are studied. A bankruptcy is dedicated to the multigrid technique and the publication ends with area decomposition algorithms which are compatible for fixing linear platforms on parallel desktops.
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P r o o f . For the proof we follow the lines of Strang . Every m a t r i x A has at least one eigenvalue, say A1 (which may be of algebraic multiplicity n) and at least one eigenvector x 1 t h a t we can assume h~s norm 1. ,U~I)) is a unitary matrix. Then, 0 9 AUI = U1 A(2) . 0 We can repeat the same process on A (2) which has at least one eigenvalue A2 and an eigenvector x 2. So there exists a unitary m a t r i x V2 of order n - 1 such t h a t ~2 A(2)V2 = V2 * 0 . * "'" A(3) 0 Let us denote 1 0 ...
0 We can repeat the same process on A (2) which has at least one eigenvalue A2 and an eigenvector x 2. So there exists a unitary m a t r i x V2 of order n - 1 such t h a t ~2 A(2)V2 = V2 * 0 . * "'" A(3) 0 Let us denote 1 0 ... 0 0 9 0 v2 7 . 12 CHAPTER 1: Introductory Material then, )k 1 9 ... , , 0 9 ... , 0 A(2) U2 A(2)V2 0 0 A1 =U2 0 0 A2 0 : 0 The result follows by induction. 20 Let A be an Hermitian matrix. Then, there exists a unitary matrix U such that U HAU = A, where A is a diagonal matrix whose diagonal entries are the eigenvalues of A.
23 Let A be a symmetric matrix. Then, IIAII = p(A). P r o o f . To prove this result, we note that the Euclidean norm is invariant under an orthogonal transformation Q. This is because IIxll 2 - ( x , x ) - (QTQx, x) = (Qx, Q x ) = IIQxll 2. Let us recall that IIA~ll iIAI[ = max ~ . 21 shows that liAxi[ = IIQAQTxl[ = IIAQTx[[, so IIAQTxll [JAil = m a x 9~o ]IQT~II- < I[All- p(A). Let us order the eigenvalues as A1 _< A2 <_ ... <_ An and )kl A2 h __. ~176 An CHAPTER 1: Introductory Material 14 Suppose An is the eigenvalue of maximum modulus.
Computer solution of large linear systems by Unknown Author, G. Meurant