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 474 WorldCat member libraries worldwide
13 editions published in 1980 in English and Undetermined and held by 474 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 367 WorldCat member libraries worldwide
16 editions published between 1983 and 1988 in 3 languages and held by 367 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 312 WorldCat member libraries worldwide
12 editions published between 1988 and 1990 in English and Undetermined and held by 312 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 86 WorldCat member libraries worldwide
Multivariate problems occur in many applications. These problems are defined on spaces of dvariate functions and d can be huge  in the hundreds or even in the thousands. Some highdimensional problems can be solved efficiently to within [epsilon], i.e., the cost increases polynomially in [epsilon]1 and d. However, there are many multivariate problems for which even the minimal cost increases exponentially in d. This exponential dependence on d is called intractability or the curse of dimensionality. This is the first of a threevolume set comprising a comprehensive study of the tractability of multivariate problems. It is devoted to algorithms using linear information consisting of arbitrary linear functionals. The theory for multivariate problems is developed in various settings: worst case, average case, randomized and probabilistic. A problem is tractable if its minimal cost is not exponential in [epsilon]1 and d. There are various notions of tractability, depending on how we measure the lack of exponential dependence. For example, a problem is polynomially tractable if its minimal cost is polynomial in [epsilon]1 and d. The study of tractability was initiated about 15 years ago. This is the first research monograph on this subject. Many multivariate problems suffer from the curse of dimensionality when they are defined over classical (unweighted) spaces. But many practically important problems are solved today for huge d in a reasonable time. One of the most intriguing challenges of theory is to understand why this is possible. Multivariate problems may become tractable if they are defined over weighted spaces with properly decaying weights. In this case, all variables and groups of variables are moderated by weights. The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability. The book is of interes
27 editions published between 2008 and 2012 in English and held by 86 WorldCat member libraries worldwide
Multivariate problems occur in many applications. These problems are defined on spaces of dvariate functions and d can be huge  in the hundreds or even in the thousands. Some highdimensional problems can be solved efficiently to within [epsilon], i.e., the cost increases polynomially in [epsilon]1 and d. However, there are many multivariate problems for which even the minimal cost increases exponentially in d. This exponential dependence on d is called intractability or the curse of dimensionality. This is the first of a threevolume set comprising a comprehensive study of the tractability of multivariate problems. It is devoted to algorithms using linear information consisting of arbitrary linear functionals. The theory for multivariate problems is developed in various settings: worst case, average case, randomized and probabilistic. A problem is tractable if its minimal cost is not exponential in [epsilon]1 and d. There are various notions of tractability, depending on how we measure the lack of exponential dependence. For example, a problem is polynomially tractable if its minimal cost is polynomial in [epsilon]1 and d. The study of tractability was initiated about 15 years ago. This is the first research monograph on this subject. Many multivariate problems suffer from the curse of dimensionality when they are defined over classical (unweighted) spaces. But many practically important problems are solved today for huge d in a reasonable time. One of the most intriguing challenges of theory is to understand why this is possible. Multivariate problems may become tractable if they are defined over weighted spaces with properly decaying weights. In this case, all variables and groups of variables are moderated by weights. The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability. The book is of interes
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 57 WorldCat member libraries worldwide
3 editions published in 1992 in English and held by 57 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
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)
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 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
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 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
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)
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
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
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)
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
 CARNEGIEMELLON UNIV PITTSBURGH PA Dept. of COMPUTER SCIENCE
 European Mathematical Society Publisher
 Association for Computing Machinery
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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