Parallel computations in linear algebra

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

doi:10.1007/bf01068849

Cited by 14 publications

(4 citation statements)

References 28 publications

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“…The %eigenvalues of the smallest absolute value were found by the power method (Faddeev and Faddeeva 1963) modified for this case (see section 2.2). In order to find the approximate eigenvalue of H ( r e'") for a given value of 8, one should minimize Ihl with respect to E, and E,.…”

Section: Further Computational Detailsmentioning

confidence: 99%

The Hermitian representation of the complex-rotation method and its application to the 1s2s²²S resonance of He^-

Bylicki¹

1991

J. Phys. B: At. Mol. Opt. Phys.

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The spectrum of the 2 operator. fundamental for the Hermitian representation of the complex-coordinate method, is proven tp be symmetric with respen to zero. Thjs properly makes the effective application of Z eigenproblems possible, instead of Z' eigenproblems. The Hermitian representation complex-coordinate method modified in this way is applied, with a superposition of correlated configurations as a trial function, to the Hel ~2 5 ~' S resonance and Rives the position at 19.367 eV above the He ground state and a width of 8.6 meV.

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Section: Further Computational Detailsmentioning

confidence: 99%

The Hermitian representation of the complex-rotation method and its application to the 1s2s²²S resonance of He^-

Bylicki¹

1991

J. Phys. B: At. Mol. Opt. Phys.

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“…Developer of shell methods for reconstructing images of piecewise homogeneous objects with a finite number of different homogeneous parts. Developer of new parallel algorithms using GPUs and using the MPI system for image reconstruction.ZOLOTAREV C.A 1 ,. TARUAT A.T 2 ,.…”

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confidence: 99%

Iterative reconstruction of medical phantoms on the computing cluster of eagle

Zolotarev,

Taruat,

Bilenko

2023

Sist. anal.i prikl. inform.

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Now one of the most important tasks is development and adaptation of iterative methods for the solution of the superbig rarefied systems of the algebraic equations. The problem of iterative parallel reconstruction of three-dimensional images of industrial facilities leads to such computing tasks. The fact that iterative methods of the solution of computing problems of big dimension are implemented on parallel structures much more effectively, than direct methods of their decision is important.

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“…In this context, three cases are analyzed: the so-called textbook case (where exact data are available), the regular case (where the data are inexact but their variation is far from critical), and the irregular case (where the variation of initial data is either critical or close to it), and, in each case, information on the solution is extracted.Papers [13,14] (written jointly with V. N. Kublanovskaya) consider the solution of general systems of linear algebraic equations, including the cases of rectangular and singular coefficient matrices. An algorithm for computing the so-called normalized decomposition of a matrix into the product of a left triangular matrix (with a special ordering of the diagonal entries) and an orthogonal matrix is suggested, an interrelation between the diagonal entries of the triangular (trapezoidal) factor and the singular values of the original matrix is established, and bounds for the largest and smallest singular values are obtained.Survey papers [15][16][17][18][19] give a comprehensive idea of the evolution of numerical linear algebra in the 50-70s of the 20th century.…”

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confidence: 99%

“…Survey papers [15][16][17][18][19] give a comprehensive idea of the evolution of numerical linear algebra in the 50-70s of the 20th century.…”

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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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Parallel computations in linear algebra

Cited by 14 publications

References 28 publications

The Hermitian representation of the complex-rotation method and its application to the 1s2s²²S resonance of He^-

The Hermitian representation of the complex-rotation method and its application to the 1s2s²²S resonance of He^-

Iterative reconstruction of medical phantoms on the computing cluster of eagle

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

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Parallel computations in linear algebra

Cited by 14 publications

References 28 publications

The Hermitian representation of the complex-rotation method and its application to the 1s2s22S resonance of He-

The Hermitian representation of the complex-rotation method and its application to the 1s2s22S resonance of He-

Iterative reconstruction of medical phantoms on the computing cluster of eagle

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

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Product

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The Hermitian representation of the complex-rotation method and its application to the 1s2s²²S resonance of He^-

The Hermitian representation of the complex-rotation method and its application to the 1s2s²²S resonance of He^-