非負(fù)矩陣譜半徑估計(jì)的研究

摘　要

本文目標(biāo)為討論非負(fù)矩陣譜半徑估計(jì)1類方法。在蓋爾圓盤定理及Frobenius界值定理基礎(chǔ)上，對(duì)這類方法給出不同程度的改進(jìn)，使新界值更精確。
利用Perron補(bǔ)的概念，提出非負(fù)不可約矩陣譜半徑界值的1個(gè)新的估計(jì)算法。該算法利用Perron補(bǔ)保持原矩陣的非負(fù)不可約性及譜半徑的性質(zhì)，使新得到的矩陣最大行和變小，最小行和變大，從而得到比Frobenius界值定理更精確的界。詳細(xì)論述算法思想并給予嚴(yán)格證明。給出適當(dāng)?shù)臄?shù)值例子，比較新算法相對(duì)于Frobenius界值定理的改進(jìn)效果，最后簡(jiǎn)要評(píng)價(jià)各算法，并討論矩陣特征問題的研究方法。

關(guān)鍵詞  非負(fù)矩陣；譜半徑；界；估計(jì)；Perron補(bǔ)

Abstract

This paper focuses on discussion of a class of estimation methods for spectral radius of nonnegative Matrix.based on Gerschgorin Disk theory and  Frobenius’theory,these methods improve the former theories and provide sharper bounds.
Furthermore,the concept of  Perron complement is introduced a new estimating method for spectral radius of nonnegative irreducible matrix is proposed and explained in detail.A new matrix dereved preserves the spectral radius while its minimun row sum increases and its minimun row sum decreases.Detail designing method and strict proof are provided with illustration of numerical examples.Finally,these algorithms’characters and the studying methods for matrix eigenproblems are also briefly discussed.
Keywords   nonnegative Matrix；spectral radius；bounds；estimation；Perron complement

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