Natural Norms in Algebraic Processes

Faddeev, D. K.; Faddeeva, V. N.

doi:10.1137/0707042

Cited by 8 publications

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“…In this section we will consider the numerical evaluation of Wilkinson numbers co(H,, HT, A,, AT) with ellipsoidal H, (for more information on ellipsoidal norms see Faddeev and Faddeeva [1]). Let us begin with the well-known For general Wilkinson numbers co~r = w(K~, KT, A~, AT) it is often advantageous to approximate the general unit balls K, with ellipsoids H,.…”

Section: Ellipsoidal Wdkinson Numbersmentioning

confidence: 99%

Computer Search for Numerical Instability

Miller

1975

J. ACM

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ABSTRac~r. It is often possible to define a measure, ~(d) ~ 0, of the effect of rounding error upon a fixed numerical algorithm given data d. One reasonable choice is for ~ to measure how much the exact computational problem must be perturbed to compensate for rounding error. In this paper such "Wilkinson numbers" ~(d) are investigated, with particular attentmn given to their numerical evaluation Also, an automatic roundoff analysis m described, which shows that the usual algorithm for inverting triangular matrices does not enjoy a certain very stringent stabdlty property KEY WORDS AND PHRASES: automatic roundoff analysis, numerical stability, matrix inversion

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Section: Ellipsoidal Wdkinson Numbersmentioning

confidence: 99%

Computer Search for Numerical Instability

Miller

1975

J. ACM

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“…Ill-conditioned systems are analyzed using the H-number as the quantitative measure of conditioning, and methods for decreasing the condition number are suggested. In [9][10][11], tools for estimating results of numerical computations are developed. A new approach to evaluating the inherent error of the solution of a computational problem based on the use of elliptic norms is suggested, and methods for constructing elliptic norms and for passing to them from norms used to describe errors in the initial data are proposed.…”

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confidence: 99%

On the occasion of the centenary of the birthday of Vera Nikolaevna Faddeeva

Kublanovskaya

Kolotilina

2007

J Math Sci

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The 20th of September 2006 was the 100th birthday of Vera Nikolaevna Faddeeva (Zamyatina), a mathematician well known to everyone who has ever dealt, directly or indirectly, with solving linear algebraic problems.Vera Nikolaevna went down in history as the leader and one of the founders of the Leningrad scientific school of numerical linear algebra.Vera Nikolaevna had won international recognition for three monographs under the common title Computational Methods of Linear Algebra [1-3], published in 1950, 1960, and 1963. The first of them, written at the dawn of the computer age, had played an important role in the formation of numerical methods as a scientific field and had been widely favored by readers in the Soviet Union and abroad. This monograph presented direct and iterative methods for solving the main problems of linear algebra and provided detailed descriptions of computational schemes supplied with illustrative examples. Much attention was paid to the notions of vector and matrix norms.One can say, without overestimation, that norms as a means for evaluating the convergence rate of iterative methods came into common use only upon publication of this monograph. The simplicity and logicality of exposition made this book comprehensible and useful for a wide audience -from students in mathematics to engineers and specialists in numerical methods and applied sciences. Quite soon the monograph became a rare book.The subsequent two monographs (written jointly with Dmitrii Konstantinovich Faddeev, the husband of Vera Nikolaevna) constitute a fundamental study of numerical methods for solving problems of linear algebra. They systematically describe numerous methods and algorithms, provide a deep analysis of the principles underlying their construction, and indicate directions for further research.These three monographs were translated into many languages, and the last of them has remained widely used and popular.The allience with D. K. Faddeev also resulted in a number of papers dedicated to numerical methods of linear algebra. Papers [4][5][6][7][8] are devoted to investigation of ill-conditioned systems of linear algebraic equations. Ill-conditioned systems are analyzed using the H-number as the quantitative measure of conditioning, and methods for decreasing the condition number are suggested. In [9][10][11], tools for estimating results of numerical computations are developed. A new approach to evaluating the inherent error of the solution of a computational problem based on the use of elliptic norms is suggested, and methods for constructing elliptic norms and for passing to them from norms used to describe errors in the initial data are proposed. Also a relation between the suggested estimate and the probabilistic error estimate is established. In [12], a new concept of error estimation in solving systems of linear algebraic equations dependent on the inaccuracy in initial data is introduced. In this context, three cases are analyzed: the so-called textbook case (where exact data are available), the regul...

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Computational methods of linear algebra

Faddeev¹,

Faddeeva²

1981

J Math Sci

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Natural Norms in Algebraic Processes

Cited by 8 publications

References 1 publication

Computer Search for Numerical Instability

Computer Search for Numerical Instability

On the occasion of the centenary of the birthday of Vera Nikolaevna Faddeeva

Computational methods of linear algebra

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