Woźniakowski, H.
Overview
Works:  75 works in 195 publications in 2 languages and 1,657 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author, Editor 
Classifications:  QA297, 511.4028542 
Publication Timeline
.
Most widely held works by
H Woźniakowski
A general theory of optimal algorithms by
J. F Traub(
Book
)
13 editions published in 1980 in English and Undetermined and held by 475 WorldCat member libraries worldwide
13 editions published in 1980 in English and Undetermined and held by 475 WorldCat member libraries worldwide
Information, uncertainty, complexity by
J. F Traub(
Book
)
16 editions published between 1983 and 1988 in 3 languages and held by 368 WorldCat member libraries worldwide
16 editions published between 1983 and 1988 in 3 languages and held by 368 WorldCat member libraries worldwide
Informationbased complexity by
J. F Traub(
Book
)
12 editions published between 1988 and 1990 in English and Undetermined and held by 313 WorldCat member libraries worldwide
12 editions published between 1988 and 1990 in English and Undetermined and held by 313 WorldCat member libraries worldwide
Tractability of multivariate problems by
Erich Novak(
Book
)
in English and held by 99 WorldCat member libraries worldwide
in English and held by 99 WorldCat member libraries worldwide
Tractability of multivariate problems by
Erich Novak(
Book
)
27 editions published between 2008 and 2012 in English and held by 82 WorldCat member libraries worldwide
This threevolume set is a comprehensive study of the tractability of multivariate problems. Volume I covers algorithms using linear information consisting of arbitrary continuous linear functionals. Volumes II and III are devoted to algorithms using standard information consisting of function values. Approximation of linear and selected nonlinear functionals is dealt with in volume II, and linear and selected nonlinear operators are studied in volume III. To a large extent, volume III can be read independently of volumes I and II. The most important example studied in volume III is the approximation of multivariate functions. It turns out that many other linear and some nonlinear problems are closely related to the approximation of multivariate functions. While the lower bounds obtained in volume I for the class of linear information also yield lower bounds for the standard class of function values, new techniques for upper bounds are presented in volume III. One of the main issues here is to verify when the power of standard information is nearly the same as the power of linear information. In particular, for the approximation problem defined over Hilbert spaces, the power of standard and linear information is the same in the randomized and average case (with Gaussian measures) settings, whereas in the worst case setting this is not true. The book is of interest to researchers working in computational mathematics, especially in approximation of highdimensional problems. It may be well suited for graduate courses and seminars. The text contains 58 open problems for future research in tractability
27 editions published between 2008 and 2012 in English and held by 82 WorldCat member libraries worldwide
This threevolume set is a comprehensive study of the tractability of multivariate problems. Volume I covers algorithms using linear information consisting of arbitrary continuous linear functionals. Volumes II and III are devoted to algorithms using standard information consisting of function values. Approximation of linear and selected nonlinear functionals is dealt with in volume II, and linear and selected nonlinear operators are studied in volume III. To a large extent, volume III can be read independently of volumes I and II. The most important example studied in volume III is the approximation of multivariate functions. It turns out that many other linear and some nonlinear problems are closely related to the approximation of multivariate functions. While the lower bounds obtained in volume I for the class of linear information also yield lower bounds for the standard class of function values, new techniques for upper bounds are presented in volume III. One of the main issues here is to verify when the power of standard information is nearly the same as the power of linear information. In particular, for the approximation problem defined over Hilbert spaces, the power of standard and linear information is the same in the randomized and average case (with Gaussian measures) settings, whereas in the worst case setting this is not true. The book is of interest to researchers working in computational mathematics, especially in approximation of highdimensional problems. It may be well suited for graduate courses and seminars. The text contains 58 open problems for future research in tractability
Optimal recovery : proceedings of the Second International Symposium on Optimal Algorithms, Varna, May 29June 2, 1989 by International Symposium on Optimal Algorithms(
Book
)
3 editions published in 1992 in English and held by 58 WorldCat member libraries worldwide
3 editions published in 1992 in English and held by 58 WorldCat member libraries worldwide
Standard information for functionals by
Erich Novak(
Book
)
2 editions published in 2010 in English and held by 11 WorldCat member libraries worldwide
2 editions published in 2010 in English and held by 11 WorldCat member libraries worldwide
Roundoff error analysis of iterations for large linear systems by
H Woźniakowski(
Book
)
3 editions published in 1977 in English and held by 7 WorldCat member libraries worldwide
This document deals with the rounding error analysis of successive approximation iterations for the solution of large linear systems Ax = b
3 editions published in 1977 in English and held by 7 WorldCat member libraries worldwide
This document deals with the rounding error analysis of successive approximation iterations for the solution of large linear systems Ax = b
Convergence and complexity of Newton iteration for operator equations by
J. F Traub(
Book
)
3 editions published in 1977 in English and held by 7 WorldCat member libraries worldwide
An optimal convergence condition for Newton iteration in a Banach space is established. There exist problems for which the iteration converges but the complexity is unbounded. It is shown which stronger condition must be imposed to also assure good complexity
3 editions published in 1977 in English and held by 7 WorldCat member libraries worldwide
An optimal convergence condition for Newton iteration in a Banach space is established. There exist problems for which the iteration converges but the complexity is unbounded. It is shown which stronger condition must be imposed to also assure good complexity
Numerical stability of iterations for solution of nonlinear equations and large linear systems by
H Woźniakowski(
Book
)
3 editions published in 1975 in English and held by 7 WorldCat member libraries worldwide
Some recent results are discussed concerning the problem of numerical stability of iterations for the solution of nonlinear equations F(x) = 0 and large linear systems Ax+g = 0 where A = A* is positive definite. For systems of nonlinear equations it is assumed that the function F depends on a so called data vector F(x) = F(x;d). One defines the condition number cond(F;d), numerical stability and wellbehavior of iterations for the solution of F(x) = 0. Necessary and sufficient conditions for a stationary iteration to be numerically stable and wellbehaved are presented. It is shown that Newton iteration for the multivariate case and secant iteration for the scalar case are wellbehaved. For large linear systems the author presents the rounding error analysis for the Chebyshev iteration and for the successive approximation iterations. It is shown that these iterations are numerically stable and that the condition number of A is a crucial parameter
3 editions published in 1975 in English and held by 7 WorldCat member libraries worldwide
Some recent results are discussed concerning the problem of numerical stability of iterations for the solution of nonlinear equations F(x) = 0 and large linear systems Ax+g = 0 where A = A* is positive definite. For systems of nonlinear equations it is assumed that the function F depends on a so called data vector F(x) = F(x;d). One defines the condition number cond(F;d), numerical stability and wellbehavior of iterations for the solution of F(x) = 0. Necessary and sufficient conditions for a stationary iteration to be numerically stable and wellbehaved are presented. It is shown that Newton iteration for the multivariate case and secant iteration for the scalar case are wellbehaved. For large linear systems the author presents the rounding error analysis for the Chebyshev iteration and for the successive approximation iterations. It is shown that these iterations are numerically stable and that the condition number of A is a crucial parameter
Optimal radius of convergence of interpolatory iterations for operator equations by
J. F Traub(
Book
)
3 editions published in 1976 in English and held by 7 WorldCat member libraries worldwide
The convergence of the class of direct interpolatory iterations I sub n for a simple zero of a nonlinear operator F in a Banach space of finite or infinite dimension is studied. A general convergence result is established and used to show that if F is entire the radius of convergence goes to infinity with n while if F is analytic in a ball of radius of convergence increases to at least R/2 with n. (Author)
3 editions published in 1976 in English and held by 7 WorldCat member libraries worldwide
The convergence of the class of direct interpolatory iterations I sub n for a simple zero of a nonlinear operator F in a Banach space of finite or infinite dimension is studied. A general convergence result is established and used to show that if F is entire the radius of convergence goes to infinity with n while if F is analytic in a ball of radius of convergence increases to at least R/2 with n. (Author)
Maximal order of multipoint iterations using n evaluations by
H Woźniakowski(
Book
)
3 editions published in 1975 in English and held by 7 WorldCat member libraries worldwide
This paper deals with multipoint iterations without memory for the solution of the nonlinear scalar equation f(m) (x) = 0, m> or = p sub n(m) be the maximal order of iterations which use n evaluations of the function or its derivatives per stop. We prove the Kung and Traub conjecture p sub n(m) (0) = 2(n1) for Hermitian information. We show p sub n(m + 1)> or = p sub n(m) and conjecture p sub n(m) = 2(n1). The problem of the maximal order is connected with Birkhoff interpolation. Under a certain assumption we prove that the Polya conditions are necessary for maximal order
3 editions published in 1975 in English and held by 7 WorldCat member libraries worldwide
This paper deals with multipoint iterations without memory for the solution of the nonlinear scalar equation f(m) (x) = 0, m> or = p sub n(m) be the maximal order of iterations which use n evaluations of the function or its derivatives per stop. We prove the Kung and Traub conjecture p sub n(m) (0) = 2(n1) for Hermitian information. We show p sub n(m + 1)> or = p sub n(m) and conjecture p sub n(m) = 2(n1). The problem of the maximal order is connected with Birkhoff interpolation. Under a certain assumption we prove that the Polya conditions are necessary for maximal order
Numerical stability for solving nonlinear equations by
H Woźniakowski(
Book
)
4 editions published in 1975 in English and held by 7 WorldCat member libraries worldwide
The concepts of the condition number, numerical stability and wellbehavior for solving systems of nonlinear equations F(x) = 0 are introduced. Necessary and sufficient conditions for numerical stability and wellbehavior of a stationary iteration are given. The author proves numerical stability and wellbehavior of the Newton iteration for solving systems of equations and of some variants of secant iteration for solving a single equation under a natural assumption on the computed evaluation of F. Furthermore it is shown that the Steffensen iteration is unstable and it is shown how to modify it to have wellbehavior and hence stability
4 editions published in 1975 in English and held by 7 WorldCat member libraries worldwide
The concepts of the condition number, numerical stability and wellbehavior for solving systems of nonlinear equations F(x) = 0 are introduced. Necessary and sufficient conditions for numerical stability and wellbehavior of a stationary iteration are given. The author proves numerical stability and wellbehavior of the Newton iteration for solving systems of equations and of some variants of secant iteration for solving a single equation under a natural assumption on the computed evaluation of F. Furthermore it is shown that the Steffensen iteration is unstable and it is shown how to modify it to have wellbehavior and hence stability
Convergence and complexity of interpolatoryNewton iteration in a banach space by
J. F Traub(
Book
)
3 editions published in 1977 in English and held by 6 WorldCat member libraries worldwide
The class of InterpolatoryNewton iterations is defined and analyzed for the computation of a simple zero of a nonlinear operator in a Banach space of finite or infinite dimension. Convergence of the class is established. The concepts of informationally optimal class of algorithms and optimal algorithm are formalized. For the multivariate case, the optimality of Newton iteration is established in the class of onepoint iterations under an equal cost assumption. (Author)
3 editions published in 1977 in English and held by 6 WorldCat member libraries worldwide
The class of InterpolatoryNewton iterations is defined and analyzed for the computation of a simple zero of a nonlinear operator in a Banach space of finite or infinite dimension. Convergence of the class is established. The concepts of informationally optimal class of algorithms and optimal algorithm are formalized. For the multivariate case, the optimality of Newton iteration is established in the class of onepoint iterations under an equal cost assumption. (Author)
Roundoff error analysis of a new class of conjugate gradient algorithms by
H Woźniakowski(
Book
)
3 editions published in 1978 in English and held by 6 WorldCat member libraries worldwide
We perform the rounding error analysis of the conjugate gradient algorithms for the solution of a large system of linear equations Ax = b where A is an hermitian and positive definite matrix. We propose a new class of conjugate gradient algorithms and prove that in the spectral norm the relative error of the computed sequence (x sub k) (in floating point arithmetic) depends at worst on zeta eta to the 3/2 power where zeta is the relative computer precision and eta is the condition number of A. We show that the residual vectors r sub k  Ax sub kb are at worst of order eta (A) abs. val. x sub k. We point out that with iterative refinement these algorithms are numerically stable. If zeta etasquared is at most of order unity, then they are also wellbehaved. (Author)
3 editions published in 1978 in English and held by 6 WorldCat member libraries worldwide
We perform the rounding error analysis of the conjugate gradient algorithms for the solution of a large system of linear equations Ax = b where A is an hermitian and positive definite matrix. We propose a new class of conjugate gradient algorithms and prove that in the spectral norm the relative error of the computed sequence (x sub k) (in floating point arithmetic) depends at worst on zeta eta to the 3/2 power where zeta is the relative computer precision and eta is the condition number of A. We show that the residual vectors r sub k  Ax sub kb are at worst of order eta (A) abs. val. x sub k. We point out that with iterative refinement these algorithms are numerically stable. If zeta etasquared is at most of order unity, then they are also wellbehaved. (Author)
Optimal linear information for the solution of nonlinear operator equations by
J. F Traub(
Book
)
4 editions published in 1976 in English and held by 6 WorldCat member libraries worldwide
In this paper the authors explore the space of all onepoint iterative algorithms which use 'linear information' to solve nonlinear operator equations. To do this, the authors pose a different question than the type usually asked in a paper on algorithms: What information is relevant to the solution of a problem. The authors provide a complete answer to this question for onepoint algorithms with linear information
4 editions published in 1976 in English and held by 6 WorldCat member libraries worldwide
In this paper the authors explore the space of all onepoint iterative algorithms which use 'linear information' to solve nonlinear operator equations. To do this, the authors pose a different question than the type usually asked in a paper on algorithms: What information is relevant to the solution of a problem. The authors provide a complete answer to this question for onepoint algorithms with linear information
Numerical stability of the Chebyshev method for the solution of large linear systems by
H Woźniakowski(
Book
)
4 editions published in 1975 in English and held by 6 WorldCat member libraries worldwide
This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systems Ax+g = 0 where A = A* is positive definite
4 editions published in 1975 in English and held by 6 WorldCat member libraries worldwide
This paper contains the rounding error analysis for the Chebyshev method for the solution of large linear systems Ax+g = 0 where A = A* is positive definite
Optimality of spline algorithms by
G. W Wasilkowski(
Book
)
3 editions published in 1978 in English and held by 6 WorldCat member libraries worldwide
Algorithms were studied for approximating Sf where S is a linear operator and f is an element of a set. The concept of spline algorithms is introduced and optimality properties of these algorithms are established. This unifies and generalizes many known results
3 editions published in 1978 in English and held by 6 WorldCat member libraries worldwide
Algorithms were studied for approximating Sf where S is a linear operator and f is an element of a set. The concept of spline algorithms is introduced and optimality properties of these algorithms are established. This unifies and generalizes many known results
Essays on the complexity of continuous problems by
Erich Novak(
Book
)
2 editions published in 2009 in English and held by 4 WorldCat member libraries worldwide
"This book contains five essays on the complexity of continuous problems, written for a wider audience. The first four essays are based on talks presented in 2008 when Henryk Wozniakowski received an honorary doctoral degree from the Friedrich Schiller University of Jena. The focus is on the introduction and history of the complexity of continuous problems, as well as on recent progress concerning the complexity of highdimensional numerical problems. The last essay provides a brief and informal introduction to the basic notions and concepts of informationbased complexity addressed to a general readership"Publisher's description
2 editions published in 2009 in English and held by 4 WorldCat member libraries worldwide
"This book contains five essays on the complexity of continuous problems, written for a wider audience. The first four essays are based on talks presented in 2008 when Henryk Wozniakowski received an honorary doctoral degree from the Friedrich Schiller University of Jena. The focus is on the introduction and history of the complexity of continuous problems, as well as on recent progress concerning the complexity of highdimensional numerical problems. The last essay provides a brief and informal introduction to the basic notions and concepts of informationbased complexity addressed to a general readership"Publisher's description
Tractability of multivariate problems by
Erich Novak(
)
in English and held by 0 WorldCat member libraries worldwide
in English and held by 0 WorldCat member libraries worldwide
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Related Identities
 Traub, J. F. (Joseph Frederick) 19322015 Author
 Wasilkowski, G. W.
 Wasilkowski, G. W. Author
 Novak, Erich 1953 Author
 Bojanov, B. Editor
 Bŭlgarska akademii︠a︡ na naukite
 CarnegieMellon University Computer Science Department
 Novak, Erich Author
 CARNEGIEMELLON UNIV PITTSBURGH PA Dept. of COMPUTER SCIENCE
 European Mathematical Society Publisher
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Associated Subjects
Algorithms Approximation theory Approximation theoryData processing Banach spaces Chebyshev approximationComputer programs Complexes Computational complexity Computational complexityData processing Convergence Differential equations, Nonlinear EquationsNumerical solutions Floatingpoint arithmetic Hilbert space Iterative methods (Mathematics) Linear systems Mathematical optimization Mathematical optimizationData processing Mathematics Multivariate analysis Nonlinear theories Numerical analysis Operator equations Roots, Numerical Roundoff errors