Iterative Solution of Large Linear Systems by David M. Young

By David M. Young

This self-contained remedy deals a scientific improvement of the idea of iterative tools. Its point of interest is living in an research of the convergence homes of the successive overrelaxation (SOR) process, as utilized to a linear approach with a regularly ordered matrix. The textual content explores the convergence homes of the SOR strategy and comparable innovations when it comes to the spectral radii of the linked matrices in addition to when it comes to sure matrix norms. Contents comprise a overview of matrix thought and basic houses of iterative equipment; SOR procedure and desk bound transformed SOR technique for continuously ordered matrices; nonstationary tools; generalizations of SOR concept and versions of procedure; second-degree tools, alternating direction-implicit equipment, and a comparability of tools. 1971 edition.

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Let A' — ε~χΑ and let V be any matrix which reduces A' to Jordan canonical form. 43) where J(e) is the same as the Jordan canonical form J of A except that the off-diagonal elements of J are multiplied by ε. Thus a typical block of/(e> is 'λ ε 0 . 44) λ ε 0 λ, Since EE1 has the form 1 0 0 0 1 0 0 ON 0 0 ΕΕτ = ■(145) 0 0 0 ,0 0 0 1 0 0 0> it follows that 5(££:T) = 1 and \\E\\ = 1. 46) This theorem is given by Householder [1964]. The proof is based on that of House­ holder but with the use of a technique of Ortega and Rheinboldt [1970].

Since 5 is similar to ß , μ is an eigenvalue of 5 and  is not positive definite. 8,  must be positive definite. This contradiction proves that μ < 1 and shows that A is an M-matrix. The class of M-matrices is a subclass of the class of monotone matrices. 3. A matrix A is a monotone matrix, if A is nonsingular and A-1 > 0. 4. A matrix A is monotone if and only if Ax > 0 implies *>0. Proof. Suppose A is monotone. If Ax = y > 0, then x = ^4 -1 y > 0. On the other hand, suppose Ax > 0 implies x > 0, and let # be any vector such that Az = 0.

29) (We note that w φ 0; otherwise, 0^· = 0 for ally and A = 0). 5) follows. 6) is the induced matrix norm corresponding to the vector norm || · l^. 2 that the last expression is equal to (S(A*A))i. 7). This latter norm is sometimes referred to as the spectral norm. When no confusion will arise we shall omit the norm labels. 32) A vector norm || · \\x and a matrix norm || · ||^ are said to be consistent, or compatible, if for all v e EN we have || Av ||. 33) Evidently, any vector norm and the induced matrix norm are consistent.

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