A Recurring Theorem on Determinants

A simple algorithm is described whi ch is well adapted to the effective solu tio n of large syste ms of linear a lgebraic eq uations by a succession of well-converge n t approximations. IntroductionIn an earlier publication [14) 2 a method was describ ed which gener ated the eigenvalues and eigenl'ecLors of a matrix by a successi ve algorithm based on minimizations by least squares. 3 The ad Irantage of thi s method consists in the fact tha t th e successive iterations arc constantly employcd with maximum efficiency which guaran tees fastcst convcrgence for a gilren number of iterations. Moreover, with th c proper care the accumul ation of rounding elTors can be avoided. The resulting high precision is of gl'ea t ad van tage if the separation of closely bunch ed eigcnvalues and eigenvectors is demanded [16) .It was pointed out in [14 , p. 256) that the inversion of a matrix, and thus th e solu tion of simultaneous systems of linear equatio ns, is contained in th e general procedure as a sp ecial casco HowC\rer , in view of the great importance associated wi th the solu tion of large systems of lineal' equations, this problem deserved more thall passing attention . It is the purpose of the present discussion to adop t the ge neral principles of t he previous imrestigation to the specifi c demands tha t arise if we are not interested in the complete analysis of a matrix but only in th e more special problem of ob taining the solution of a gi I'en set of linear equations Ay = bo (1 ) with a given matrix A and a given right side boo This is actually eq ui valent to the c valuation of one eigen vector onl~' , of a symmetric, positive definite matrix. It is clear th at this will require considerably less detailed analysis than the problem of constructing the entire set of eigenvector~ and eigenvalues assoc iated with an arbitrary matrix. . The Double Set of Vectors Associated With the Method of Minimized IterationsThe prelrious investigation [14) started out with an algorithm (see p . 261) which genera ted a double set of polynomials, later on deno ted by Pi(X) and qt(x) (see p . 274). Then a second algorithm was I '1'he preparation of this paper was sp onsored (in part) by the Office of Naval Research., Figures in brackets indicate th e literature references at the end of this paper. 3 The prescnt papcr is a natural sequel to the previous publication and depends on the prcvious find ings. The reader's fam iliarity with t he earlier development is assumed throughout t his p aper; the sym bolism of the present paper is in har mony with that used before, in particular the notatiolll'Q, if applied to vectors, shall mean the sClliar product of these two vectOl S.207064-52--3 introduced, called " minimized iter ations", which avoided th e numerical difficulties of th e fi rst algorithm (see p . 287) and had , in add ition, theoretically valuable proper ties for the solution of differential and integr al equa tions (p . 272).In this second algorithm, however, only one-half of the previous polynomials were repre ented, corr...

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“…The lar ges t eigenvalue of a m atrix can b e estimated b y the method of Gersgol'in [9], (cf. also [3] and [20] …”

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confidence: 96%

“…K ow the biorthogonaliLy of the vecLol's P i g ives, if we multiply the uppel' left equation (18) by P : the pW'po e of olving linear equations that can always be symmetrized, by transforming the originally given et (20) into Gy = g (29) ::, [oreover, the same equation shows the orthogonality of g; to all pi) except m = k and lc+ 1. In particular (21)…”

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confidence: 99%

Solution of systems of linear equations by minimized iterations

Lanczos¹

1952

J. RES. NATL. BUR. STAN.

790

356

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“…We shall need the following well-known nonsingularity result of Taussky from 1949 ( [5]), which is the special case of Theorem 1 or 2.…”

Section: The Following Results Is Proven By Huang ([4])mentioning

confidence: 99%

Characterization of diagonally dominant H-matrices

Morača

2017

Publ Inst Math (Belgr)

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We first show that sufficient conditions for a diagonally dominant matrix to be a nonsingular one (and also an H-matrix), obtained independently by Shivakumar andChew in 1974, andFarid in 1995, are equivalent. Then we simplify the characterization of diagonally dominant H-matrices obtained by Huang in 1995, and using it prove that the Shivakumar-Chew-Farid sufficient condition for a diagonally dominant matrix to be an H-matrix, is also necessary.

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“…Many sufficient conditions have been discovered (cf., for instance, [8] and [3], and their bibliographies), motivated by the fact that the negation of these conditions, applied to the matrix B -λJ, yields information about the location of the characteristic roots. We shall show that a mild generalization of the most famous conditions [2] is not only sufficient but also necessary.…”

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confidence: 99%