Brent, R. P. (Richard P.)
Overview
Works:  106 works in 285 publications in 1 language and 1,367 library holdings 

Roles:  Author, Thesis advisor, Other, Recipient 
Classifications:  QA402.5, 515.33 
Publication Timeline
.
Most widely held works by
R. P Brent
Algorithms for minimization without derivatives by
R. P Brent(
Book
)
24 editions published between 1972 and 2013 in English and held by 672 WorldCat member libraries worldwide
Outstanding text for graduate students and research workers proposes improvements to existing algorithms, extends their related mathematical theories, and offers details on new algorithms for approximating local and global minima. Many numerical examples, along with complete analysis of rate of convergence for most of the algorithms and error bounds that allow for the effect of rounding errors
24 editions published between 1972 and 2013 in English and held by 672 WorldCat member libraries worldwide
Outstanding text for graduate students and research workers proposes improvements to existing algorithms, extends their related mathematical theories, and offers details on new algorithms for approximating local and global minima. Many numerical examples, along with complete analysis of rate of convergence for most of the algorithms and error bounds that allow for the effect of rounding errors
Modern computer arithmetic by
R. P Brent(
Book
)
12 editions published between 2010 and 2011 in English and held by 246 WorldCat member libraries worldwide
"Modern Computer Arithmetic focuses on arbitraryprecision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favorite language, while keeping a highlevel description and avoiding too lowlevel or machinedependent details. The book is intended for anyone interested in the design and implementation of efficient highprecision algorithms for computer arithmetic, and more generally efficient multipleprecision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions are available from the authors."Publisher's website
12 editions published between 2010 and 2011 in English and held by 246 WorldCat member libraries worldwide
"Modern Computer Arithmetic focuses on arbitraryprecision algorithms for efficiently performing arithmetic operations such as addition, multiplication and division, and their connections to topics such as modular arithmetic, greatest common divisors, the Fast Fourier Transform (FFT), and the computation of elementary and special functions. Brent and Zimmermann present algorithms that are ready to implement in your favorite language, while keeping a highlevel description and avoiding too lowlevel or machinedependent details. The book is intended for anyone interested in the design and implementation of efficient highprecision algorithms for computer arithmetic, and more generally efficient multipleprecision numerical algorithms. It may also be used in a graduate course in mathematics or computer science, for which exercises are included. These vary considerably in difficulty, from easy to small research projects, and expand on topics discussed in the text. Solutions are available from the authors."Publisher's website
The Complexity of computational problem solving(
Book
)
5 editions published in 1976 in English and held by 107 WorldCat member libraries worldwide
5 editions published in 1976 in English and held by 107 WorldCat member libraries worldwide
Factorizations of an̳ [plus over minus] 1, 13 [less than over minus] a [less than] 100 by
R. P Brent(
Book
)
14 editions published between 1992 and 1996 in English and held by 14 WorldCat member libraries worldwide
14 editions published between 1992 and 1996 in English and held by 14 WorldCat member libraries worldwide
Improved techniques for lower bounds for odd perfect numbers by
R. P Brent(
Book
)
9 editions published in 1989 in English and held by 12 WorldCat member libraries worldwide
Abstract: "If N is an odd perfect number, and q[superscript k] [symbol] N, q prime, k even, then it is almost immediate that N> q[superscript 2k]. We prove here that, subject to certain conditions verifiable in polynomial time, in fact N> q[superscript 5k/2]. Using this and related results, we are able to extend the computations in an earlier paper to show that N> 10[superscript 300]."
9 editions published in 1989 in English and held by 12 WorldCat member libraries worldwide
Abstract: "If N is an odd perfect number, and q[superscript k] [symbol] N, q prime, k even, then it is almost immediate that N> q[superscript 2k]. We prove here that, subject to certain conditions verifiable in polynomial time, in fact N> q[superscript 5k/2]. Using this and related results, we are able to extend the computations in an earlier paper to show that N> 10[superscript 300]."
Solving triangular systems on a parallel computer by
Ahmed Sameh(
Book
)
3 editions published in 1975 in English and held by 11 WorldCat member libraries worldwide
3 editions published in 1975 in English and held by 11 WorldCat member libraries worldwide
Algorithms for finding zeros and extrema of functions without calculating derivatives by
R. P Brent(
Book
)
9 editions published in 1971 in English and held by 11 WorldCat member libraries worldwide
Theorems are given concerning the order (i.e., rate) of convergence of a successive interpolation process for finding simple zeros of a function or its derivatives, using only function evaluations. Special cases include the successive linear interpolation process for finding zeros, and a parabolic interpolation process for finding turning points. Results on interpolation and finite differences include weakening the hypotheses of a theorem of Ralston on the derivative of the error in Lagrangian interpolation. The theoretical results are applied to given algorithms for finding zeros or local minima of functions of one variable, in the presence of rounding errors. The algorithms are guaranteed to converge nearly as fast as would bisection or Fibonacci search, and in most practical cases convergence is superlinear, and much faster than for bisection or Fibonacci search. (Author)
9 editions published in 1971 in English and held by 11 WorldCat member libraries worldwide
Theorems are given concerning the order (i.e., rate) of convergence of a successive interpolation process for finding simple zeros of a function or its derivatives, using only function evaluations. Special cases include the successive linear interpolation process for finding zeros, and a parabolic interpolation process for finding turning points. Results on interpolation and finite differences include weakening the hypotheses of a theorem of Ralston on the derivative of the error in Lagrangian interpolation. The theoretical results are applied to given algorithms for finding zeros or local minima of functions of one variable, in the presence of rounding errors. The algorithms are guaranteed to converge nearly as fast as would bisection or Fibonacci search, and in most practical cases convergence is superlinear, and much faster than for bisection or Fibonacci search. (Author)
A systolic algorithm for integer GCD computation by
R. P Brent(
Book
)
9 editions published between 1982 and 1984 in English and held by 10 WorldCat member libraries worldwide
Abstract: "We show that the greatest common divisor of two nbit integers (given in the usual binary representation) can be computed in time O(n) on a linear array of O(n) identical systolic cells, each of which is a finitestate machine with connections to its nearest neighbours."
9 editions published between 1982 and 1984 in English and held by 10 WorldCat member libraries worldwide
Abstract: "We show that the greatest common divisor of two nbit integers (given in the usual binary representation) can be computed in time O(n) on a linear array of O(n) identical systolic cells, each of which is a finitestate machine with connections to its nearest neighbours."
The areatime complexity of Binary multiplication by
R. P Brent(
Book
)
5 editions published in 1979 in English and held by 9 WorldCat member libraries worldwide
We consider the problem of performing multiplication of nbit binary numbers on a chip. Let A denote the chip area, and T the time required to perform multiplication. Using a model of computation which is a realistic approximation to current and anticipated VLSI technology, we show that (A/A sub 0) (T/T sub 0) to the 2 alpha power> or = n to the (1 + alpha) power for all alpha is an element (0, 1), where A sub 0 and T sub 0 are positive constants which depend on the technology but are independent of n. The exponent 1 + alpha is the best possible. A consequence is that binary multiplication is 'harder' than binary addition if AT to the 2 alpha power is used as a complexity measure for any alpha> or = 0. (Author)
5 editions published in 1979 in English and held by 9 WorldCat member libraries worldwide
We consider the problem of performing multiplication of nbit binary numbers on a chip. Let A denote the chip area, and T the time required to perform multiplication. Using a model of computation which is a realistic approximation to current and anticipated VLSI technology, we show that (A/A sub 0) (T/T sub 0) to the 2 alpha power> or = n to the (1 + alpha) power for all alpha is an element (0, 1), where A sub 0 and T sub 0 are positive constants which depend on the technology but are independent of n. The exponent 1 + alpha is the best possible. A consequence is that binary multiplication is 'harder' than binary addition if AT to the 2 alpha power is used as a complexity measure for any alpha> or = 0. (Author)
Analysis of the binary Euclidean algorithm by
R. P Brent(
Book
)
3 editions published in 1976 in English and held by 9 WorldCat member libraries worldwide
In this paper the author analyzes a continuous model of the binary algorithm and finds the expected number of iterations. The results agree with the observed behavior of the algorithm much better than those predicted by Knuth's 'latticepoint' model
3 editions published in 1976 in English and held by 9 WorldCat member libraries worldwide
In this paper the author analyzes a continuous model of the binary algorithm and finds the expected number of iterations. The results agree with the observed behavior of the algorithm much better than those predicted by Knuth's 'latticepoint' model
Factorizations of Cunningham numbers with bases 13 to 99 : millennium edition by
R. P Brent(
Book
)
4 editions published in 2001 in English and held by 9 WorldCat member libraries worldwide
4 editions published in 2001 in English and held by 9 WorldCat member libraries worldwide
Efficient methods for finding zeros of functions whose derivatives are easy to evaluate by
R. P Brent(
Book
)
4 editions published in 1974 in English and held by 8 WorldCat member libraries worldwide
Some multipoint iterative methods without memory for approximating simple zeros of functions of one variable are described. Explicit, nonlinear, RungeKutta methods for the solution of a special class of ordinary differential equations may be derived from the methods for finding zeros of functions. Numerical examples and some FORTRAN subroutines are given
4 editions published in 1974 in English and held by 8 WorldCat member libraries worldwide
Some multipoint iterative methods without memory for approximating simple zeros of functions of one variable are described. Explicit, nonlinear, RungeKutta methods for the solution of a special class of ordinary differential equations may be derived from the methods for finding zeros of functions. Numerical examples and some FORTRAN subroutines are given
On the zeros of the Riemann zeta function in the critical strip by
R. P Brent(
Book
)
6 editions published between 1976 and 1982 in English and held by 8 WorldCat member libraries worldwide
6 editions published between 1976 and 1982 in English and held by 8 WorldCat member libraries worldwide
On the complexity of composition and generalized composition of power series by
R. P Brent(
Book
)
3 editions published in 1978 in English and held by 7 WorldCat member libraries worldwide
Let F(x) = f1x + f2(x)(x) + ... be a formal power series over a field Delta. Let F superscript 0(x) = x and for q = 1,2 ..., define F superscript q(x) = F superscript (q1) (F(x)). The obvious algorithm for computing the first n terms of F superscript q(x) is by the composition position analogue of repeated squaring. This algorithm has complexity about log 2 q times that of a single composition. The factor log 2 q can be eliminated in the computation of the first n terms of (F(x)) to the q power by a change of representation, using the logarithm and exponential functions. We show the factor log 2 q can also be eliminated for the composition problem. F superscript q(x) can often, but not always, be defined for more general q. We give algorithms and complexity bounds for computing the first n terms of F superscript q(x) whenever it is defined
3 editions published in 1978 in English and held by 7 WorldCat member libraries worldwide
Let F(x) = f1x + f2(x)(x) + ... be a formal power series over a field Delta. Let F superscript 0(x) = x and for q = 1,2 ..., define F superscript q(x) = F superscript (q1) (F(x)). The obvious algorithm for computing the first n terms of F superscript q(x) is by the composition position analogue of repeated squaring. This algorithm has complexity about log 2 q times that of a single composition. The factor log 2 q can be eliminated in the computation of the first n terms of (F(x)) to the q power by a change of representation, using the logarithm and exponential functions. We show the factor log 2 q can also be eliminated for the composition problem. F superscript q(x) can often, but not always, be defined for more general q. We give algorithms and complexity bounds for computing the first n terms of F superscript q(x) whenever it is defined
Fast algorithms for manipulating formal power series by
R. P Brent(
Book
)
3 editions published in 1976 in English and held by 7 WorldCat member libraries worldwide
The classical algorithms require O(n sup 3) operations to compute the first n terms in the reversion of a power series or the composition of two series, and O((n sup 2) log n) operations if the fast Fourier transform is used for power series multiplication. In this paper we show that the composition and reversion problems are equivalent (up to constant factors), and we give algorithms which require only O((n log n) sup 3/2) operations. In many cases of practical importance only O(n log n) operations are required. An application to rootfinding methods which use inverse interpolation is described, some results on multivariate power series are stated, and several open questions are mentioned
3 editions published in 1976 in English and held by 7 WorldCat member libraries worldwide
The classical algorithms require O(n sup 3) operations to compute the first n terms in the reversion of a power series or the composition of two series, and O((n sup 2) log n) operations if the fast Fourier transform is used for power series multiplication. In this paper we show that the composition and reversion problems are equivalent (up to constant factors), and we give algorithms which require only O((n log n) sup 3/2) operations. In many cases of practical importance only O(n log n) operations are required. An application to rootfinding methods which use inverse interpolation is described, some results on multivariate power series are stated, and several open questions are mentioned
A regular layout for parallel adders by
R. P Brent(
Book
)
5 editions published in 1979 in English and held by 7 WorldCat member libraries worldwide
With VLSI architecture the chip area is a better measure of cost than the conventional gate count. We show that addition of nbit binary numbers can be performed on a chip in time proportional to log n and with area proportional to n log n. (Author)
5 editions published in 1979 in English and held by 7 WorldCat member libraries worldwide
With VLSI architecture the chip area is a better measure of cost than the conventional gate count. We show that addition of nbit binary numbers can be performed on a chip in time proportional to log n and with area proportional to n log n. (Author)
Numerically stable solution of dense systems of linear equations using meshconnected processors by
A Bojanczyk(
Book
)
2 editions published in 1981 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 1981 in English and held by 7 WorldCat member libraries worldwide
A Fortran multipleprecision arithmetic package by
R. P Brent(
Book
)
3 editions published in 1976 in English and held by 7 WorldCat member libraries worldwide
On a trip to the Moroccan market town of Rissani, Ali becomes separated from his father during a sandstorm
3 editions published in 1976 in English and held by 7 WorldCat member libraries worldwide
On a trip to the Moroccan market town of Rissani, Ali becomes separated from his father during a sandstorm
A systolic architecture for the singular value decomposition by
R. P Brent(
Book
)
5 editions published in 1982 in English and held by 6 WorldCat member libraries worldwide
We propose a systolic architecture for computing a singular value decomposition of an m x n matrix, where $m \geq n$. Our algorithm is stable and requires only $O(mn)$ time on a linear array of $O(n)$ processors. Extensions to algorithms for twodimensional arrays are also discussed
5 editions published in 1982 in English and held by 6 WorldCat member libraries worldwide
We propose a systolic architecture for computing a singular value decomposition of an m x n matrix, where $m \geq n$. Our algorithm is stable and requires only $O(mn)$ time on a linear array of $O(n)$ processors. Extensions to algorithms for twodimensional arrays are also discussed
Systolic array for the lineartime solution of Toeplitz systems of equations by
R. P Brent(
Book
)
4 editions published between 1982 and 1983 in English and held by 6 WorldCat member libraries worldwide
The solution of an (n+1)x(n+1) Toeplitz system of linear equations on a onedimensional systolic architecture is studied. Our implementation of an algorithm due to Bareiss is shown to require only $O(n)$ time and $O(n)$ storage, i.e. constant storage per systolic processor
4 editions published between 1982 and 1983 in English and held by 6 WorldCat member libraries worldwide
The solution of an (n+1)x(n+1) Toeplitz system of linear equations on a onedimensional systolic architecture is studied. Our implementation of an algorithm due to Bareiss is shown to require only $O(n)$ time and $O(n)$ storage, i.e. constant storage per systolic processor
more
fewer
Audience Level
0 

1  
Kids  General  Special 
Related Identities
 Zimmermann, Paul 1964
 Anderssen, R. S. Other
 Kung, H. T.
 Luk, Franklin T.
 Riele, H. J. J. te 1947
 CarnegieMellon University Computer Science Department
 Montgomery, Peter L.
 Australian National University Centre for Mathematical Analysis
 Australian National University School of Mathematical Sciences
 Bojanczyk, A. Author
Useful Links
Associated Subjects
Algorithms Approximation theory Binary system (Mathematics) Computational complexityData processing Computer arithmetic Computer engineering Computer programming Decomposition (Mathematics) Differential equations Electronic digital computersCircuits Equations, SimultaneousData processing Factorization (Mathematics)Data processing FORTRAN (Computer program language) Functions Functions, Zeta Integrated circuitsVery large scale integration Iterative methods (Mathematics) Jacobi method Linear programming Linear systemsData processing Mathematical optimization MathematicsData processing MatricesData processing Maxima and minima Multiprocessors Number theory Number theoryData processing Parallel processing (Electronic computers) Perfect numbers Polynomials Problem solving Problem solvingData processing Programming languages (Electronic computers) Roundoff errorsData processing Systolic array circuits
Covers
Alternative Names
Brent, R. 1946
Brent, R. P.
Brent, R. P. 1946
Brent, Richard
Brent, Richard 1946
Brent, Richard P.
Brent, Richard P. 1946
Brent, Richard Peirce 1946
Richard Brent matematico e informatico australiano
Richard P. Brent
Richard P. Brent Australian academic
Richard P. Brent australiensk datavetare och matematiker
Richard P. Brent Australisch wiskundige
Richard P. Brent australischer Mathematiker und Informatiker
Richard P. Brent australsk informatikar og matematikar
Richard P. Brent australsk informatiker og matematiker
Ричард Брент
Ричард Брент британский математик
Ричард Пэйрс Брент британский математик
ریچارد پیرس برنت ریاضیدان و دانشمند علوم کامپیوتر استرالیایی
リチャード・ブレント
Languages