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An Alternating Sequence Iteration's Method for Computing Largest Real Part Eigenvalue of Essentially Positive Matrices

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An Alternating Sequence Iteration's Method for Computing Largest Real Part Eigenvalue of Essentially Positive Matrices Computa & tio d n ie a l l Oepomo, J Appl Computat Math 2016, 5:6 p M p a Journal of A t h f DOI: 10.4172/2168-9679.1000334 e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access An Alternating Sequence Iteration’s Method for Computing Largest Real Part Eigenvalue of Essentially Positive Matrices: Collatz and Perron- Frobernius’ Approach Tedja Santanoe Oepomo* Mathematics Division, Los Angeles Harbor College/West LA College, and School of International Business, California International University, USA Abstract This paper describes a new numerical method for the numerical solution of eigenvalues with the largest real part of essentially positive matrices. Finally, a numerical discussion is given to derive the required number of mathematical operations of the new method. Comparisons between the new method and several well know ones, such as Power and QR methods, were discussed. The process consists of computing lower and upper bounds which are monotonically approximating the eigenvalue. Keywords: Collatz’s theorem; Perron-Frobernius’ theorem; Background Eigenvalue Let A be an n x n essentially positive matrix. The new method can AMS subject classifications: 15A48; 15A18; 15A99; 65F10 and 65F15 be used to numerically determine the eigenvalue λA with the largest real part and the corresponding positive eigenvector x[A] for essentially Introduction positive matrices. This numerical method is based on previous manuscript [16]. A matrix A=(a ) is an essentially positive matrix if a A variety of numerical methods for finding eigenvalues of non- ij ij negative irreducible matrices have been reported over the last decades, ≥ 0 for all i ≠ j, 1 ≤ i, j ≤ n, and A is irreducible. and the mathematical and numerical aspects of most of these methods Let x > 0 be any positive n components vector [19]. Let are well reported [1-24]. In recent article of Tedja [19], it was presented the mathematical aspects of Collatz’s eigenvalue inclusion theorem for = zi () x∑ axir r ; (1) non-negative irreducible matrices. It is the purpose of this manuscript r =1 to present the numerical implementation of [19]. Indeed, there is the ri≠ hope that developing new numerical method could lead to discovering ()Ax n ax i ij, j ; (2) properties that might be responsible for better numerical method fxi ()=≡∈∑ ()i N xX= in finding and estimating eigenvalues of non-negative irreducible iij 1 matrices. Birkhoff and Varga [2] observed that the results of the mx()= f () x min i Perron-Frobernius theory and consequently Collatz’s theorem could iN∈ [6, 8, 20, 23]* (3) be slightly generalized by allowing the matrices considered to have Mx()= f() x max i negative diagonal elements. They introduced the terms “essentially iN∈ non-negative for matrices, the off-diagonal elements of which are ∆=()x M () x − mx (); (4) non-negative, and “essentially positive matrices” for essentially non- n negative, irreducible matrices. The only significant changes is that xx= . * In Ostrowsky [25] m(x) is defined as: whenever Perron-Frobernius theory and Collatz’s theorem refer to the ∑ i i=1 spectral radius of a non-negative matrix A, the corresponding quantity for an essentially non-negative matrix à is the (real) eigenvalue of the ∑axµυ υ υ (5) maximal real part in the spectrum of Ã, also to be denoted by Λ[Ã]. min µ x Of course Λ[Ã] need not be positive and it is not necessary dominant µ The following theorem is an application of corollary 2.3 from [16] among the eigenvalues in the absolute value sense. Definition *Corresponding author: Oepomo TS, Science, Technology, Engineering, and Incidentally, if A is what you call an essentially positive matrix (so Mathematics Division, Los Angeles Harbor College/West LA College, and School it is a real matrix with positive off-diagonal entries), then A + aI has of International Business, California International University. 1301 Las Riendas positive entries for sufficiently large positive a, so A+ aI has a Perron Drive Suite: 15, Las Habra, CA 90631, USA, Tel: 310-287-4216; E-mail: [email protected]; [email protected]; [email protected] eigenvalue, p say, with corresponding eigenvector v, say, having positive entries. But p is the only eigenvalue of A + aI for which there is Received November 10, 2016; Accepted December 23, 2016; Published December 27, 2016 a corresponding eigenvector with positive entries. Thus p - a is the only eigenvalue of A with a corresponding eigenvector having all its entries Citation: Oepomo TS (2016) An Alternating Sequence Iteration’s Method nonnegative, but p - a is real and need not be positive (since a could for Computing Largest Real Part Eigenvalue of Essentially Positive Matrices: Collatz and Perron-Frobernius’ Approach. J Appl Computat Math 5: 334. doi: be greater than p). In this manuscript the eigenvalue corresponding to 10.4172/2168-9679.1000334 a positive eigenvector is real. There probably is a term already in the literature for "essentially positive" For example: Z-matrix, tau-matrix, Copyright: © 2016 Oepomo TS. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted M-matrix, and Metzler matrix all refer to matrices with off-diagonal use, distribution, and reproduction in any medium, provided the original author and entries all of the same sign, but having extra conditions. source are credited. J Appl Computat Math, an open access journal Volume 5 • Issue 6 • 1000334 ISSN: 2168-9679 Citation: Oepomo TS (2016) An Alternating Sequence Iteration’s Method for Computing Largest Real Part Eigenvalue of Essentially Positive Matrices: Collatz and Perron-Frobernius’ Approach. J Appl Computat Math 5: 334. doi: 10.4172/2168-9679.1000334 Page 2 of 7 p to the design of numerical method using the Perron-Frobenius-Collatz necessarily implies that infp||x || < ∞ is satisfied. mini-max principle for the calculation of x[A] [10]. Note: In any case, if we have an increasing bounded sequence, Let {xp} (p=0, 1, 2,……) be a sequence of positive vectors and the limit always exists and is finite (the sup is an upper bound/ the T p= pp p . sequence is bounded from above by a supremum). It does converge x xx12, ,........., xn to the supremum. This criterion is also true for a decreasing bounded Theorem 1 If the sequence {xp} (p=0, 1, 2,…) of positive unit sequence and is bounded below by an infimum. It will converge to the p p infimum. But we need to be very careful to define the term bounded, vectors is such that either mx()→Λ[ A] ≡λA or Mx()→Λ[ A] ≡λA it means bounded from above and bounded from below. These as p → ∞ then xp → x[A] ξ. Moreover, the sequence {m(xi)}, {xi} are statements hold for sets of real numbers. We will now define a second equi-convergent in the sense that an index υ and a constant K > 0 exist ≡ group of sequences, the “Increasing-sequence”. ii such hat x−<ξλ K − mx( ) if I ≥ υ. Similar statements can be i Increasing-sequence expressed if Mx( ) →Λ[] A ≡λA is known. ([19], theorem 2.4) Let yr(x) (r=1, 2, 3,…, n) be an n component vector valued function Numerical Implication of Theorem 1 such that the following equation is valid: We will now define a group of sequences, the “decreasing- r yi (x) = xi if i ≠ r and ωr(x) if i=r. (10) sequence.” Here ωr(x) (r=1, 2, 3,…, n) are scalar valued functions which are Decreasing-sequence having properties as follows: Let Yr(x) (r=1, 2, 3,…,n) be an n component vector valued function ωr(x) is a continuous positive valued function and bounded in such that the following equation is valid: ∑=(x½ > 0; x³0) . r r Yi (x) = xi if i ≠ r and Ωr(x) if i=r. (6) ωr(x) ≥ xr Here Ωr(x) (r=1, 2, 3,…, n) are scalar valued functions which are having properties as follows: fr(yr(x)) ≥ m(x) (11) Equality in c) may be applicable only if yr(xr-1) converges to x; this • Ωr(x) is a continuous positive valued function which maps the yields that fr(x)=m(x). set of positive vectors V+ into a set of real numbers R. If for some x > 0 ω (x)=x for each r (N=1, 2, 3,…, n) then • Ωr(x) ≤ xr. r r fx( )= fx ( ) = ............ = f ( x ) = λ . This will imply that • Fr(Yr(x)) M(x). (7) 12 n−1∈ f (x)=λ according to Collatz, hence x=x[A]. In Faddeev [25], • Equality in c) may be applicable only if Yr(xr-1) converges to x; n 1 xxii this yields that fr(x)=M (x). = Min Max = Max Min 11≤≤in nn≤≤in λ xxii≥=0,∑∑ 1 xxii≥=0, 1 . ax ax • If for some x > 0 Ω (x)=x for each r (N=1, 2, 3,…, n) then ∑∑ij j ij j r r jj=11= fx( )= fx ( ) = ............ = f ( x ) = λ 12 n−1 . This∈ will imply that Then n-component vector valued function Yr(x) defined in equation p fn(x)=λ according to Collatz, hence x=x[A]. In Faddeev [25], (9) will be referred to as the Decreasing-functions. A sequence {x } 1 xx = Min Maxii= Max Min (p=0, 1, 2, 3,…,) of positive n-vectors is constructed which satisfy the ≥=11≤≤in nn≥=≤≤in . p λ xxii0,∑∑ 1 xxii0, 1 conditions of theorem 1. The terms of the sequence {x } are generated ∑∑axij j axij j jj=11= by the following recursive formula: r Then n-component vector valued function Y (x) defined in equation xp + 1=yp + 1(xp) (12).
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