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## Computational Complexity of Iterative Processes (1972)

Venue: | SIAM Journal Computing |

Citations: | 5 - 2 self |

### Citations

2835 |
The Art of Computer Programming
- KNUTH
- 1968
(Show Context)
Citation Context ...[41], Strassen [32], Hopcroft and Kerr [14]), polynomial evaluation (Winograd [41]), median of a set of numbers (Floyd [10]), graph planarity (Hopcroft and Tarjan [15]). Surveys may be found in Knuth =-=[19]-=-, Borodin [1], and Minsky [24]. Research on analytic computational complexity dates to the early sixties (Traub [33]-39]) and predates most of the algebraic results. More specifically, the work on ana... |

871 | The Art of Computer Programming: Volume 2, Seminumerical Algorithms - Knuth - 1969 |

694 |
Iterative solution of nonlinear equations in several variables
- Ortega, Rheinboldt
- 1970
(Show Context)
Citation Context .... A sequence of approximating iterates {x} is generated by an iteration function. We shall not give a formal definition of iteration algorithm. The interested reader may consult Ortega and Rheinboldt =-=[25]-=- and Cohen and Varaiya [4]. Our aim is to discuss optimal iteration algorithms. There are a number of measures we could optimize. For example, we could minimize the total number ofarithmetic operation... |

519 |
Gaussian elimination is not optimal
- Strassen
- 1969
(Show Context)
Citation Context ...triking developments in algebraic computational complexity; for example, the multiplication of numbers (Cook [6], Sch6nhage and Strassen [31]), the multiplication of matrices (Winograd [41], Strassen =-=[32]-=-, Hopcroft and Kerr [14]), polynomial evaluation (Winograd [41]), median of a set of numbers (Floyd [10]), graph planarity (Hopcroft and Tarjan [15]). Surveys may be found in Knuth [19], Borodin [1], ... |

287 | Seminumerical Algorithms,” - Knuth - 1981 |

216 | Iterative Methods for the Solution of Equations - Traub - 1997 |

64 | Gaussian elimination is not optimal. Numerische Mathematik - Strassen - 1969 |

29 |
Functional Analysis and Numerical Mathematics.
- Collatz
- 1966
(Show Context)
Citation Context ...ted in an abstract setting and covers partial differential equations, integral equations, boundary value problems for ordinary differential equations as well as many other important problems (Collatz =-=[5]-=-). We consider iterative algorithms for the approximation of . A sequence of approximating iterates {x} is generated by an iteration function. We shall not give a formal definition of iteration algori... |

26 | On minimizing the number of multiplications necessary for matrix multiplication - Hopcroft, Kerr - 1971 |

19 | Degree of difficulty of computing a function and a partial ordering of the recursive sets - Rabin - 1960 |

11 | private communication - Kahan, Ivory - 1997 |

11 | Solution of equations and systems of equations. Second edition - Ostrowski - 1966 |

8 | Solution of Equations and Systems of Equations, 2nd ed - Ostrowski - 1966 |

6 |
Some efficient fourth order multipoint methods for solving equations,”
- Jarratt
- 1969
(Show Context)
Citation Context ...al complexity to date has concerned optimal iteration. Recent results are due to Brent [2], Cohen [3], Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt =-=[16]-=-, King [18], Kung [21], Miller [22], [23], Paterson 27], Rissanen [30], and Winograd and Wolfe 42], [43]. (Kung and Paterson’s results are summarized at the end of 2.) In this paper we define basic co... |

6 |
Private Communication
- Winograd, Wolfe
- 1969
(Show Context)
Citation Context ...nt iterations with finite memory m, Pe,m-Pe,o < 1. Until the late sixties no progress was reported, but there have been exciting recent results. The matter has been investigated by Winograd and Wolfe =-=[42]-=- who assert a stronger result. Under weak conditions on the admissible iteration functions, interpolatory iteration Ie, is optimal among all iterations characterized by new function evaluations e and ... |

5 |
A bound on the multiplication efficiency of iteration
- KUNG
- 1972
(Show Context)
Citation Context ...has concerned optimal iteration. Recent results are due to Brent [2], Cohen [3], Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung =-=[21]-=-, Miller [22], [23], Paterson 27], Rissanen [30], and Winograd and Wolfe 42], [43]. (Kung and Paterson’s results are summarized at the end of 2.) In this paper we define basic concepts and pose some f... |

4 |
On optimum root-finding algorithms
- RISSANEN
- 1970
(Show Context)
Citation Context ...are due to Brent [2], Cohen [3], Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller [22], [23], Paterson 27], Rissanen =-=[30]-=-, and Winograd and Wolfe 42], [43]. (Kung and Paterson’s results are summarized at the end of 2.) In this paper we define basic concepts and pose some fundamental questions in optimal iteration. In th... |

3 |
analysis of algorithms
- Mathematical
- 1971
(Show Context)
Citation Context ...e 42], [43]. (Kung and Paterson’s results are summarized at the end of 2.) In this paper we define basic concepts and pose some fundamental questions in optimal iteration. In the terminology of Knuth =-=[20]-=- we perform a Type B analysis. That is, we consider a family of algorithms for solving a particular problem and select the "best possible." We survey earlier work, report recent progress, and state a ... |

3 |
On Functional Iteration and the Calculation of Roots
- Traub
- 1961
(Show Context)
Citation Context ...yd [10]), graph planarity (Hopcroft and Tarjan [15]). Surveys may be found in Knuth [19], Borodin [1], and Minsky [24]. Research on analytic computational complexity dates to the early sixties (Traub =-=[33]-=--39]) and predates most of the algebraic results. More specifically, the work on analytic computational complexity to date has concerned optimal iteration. Recent results are due to Brent [2], Cohen [... |

2 |
Private communication
- GENTLEMAN
- 1970
(Show Context)
Citation Context ... of an algorithm is the efficiency index defined by E (log2 p)/e. This measure is defined without motivation by Ostrowski [26, Chap. 3]. A derivation may be found in Traub [39, Appendix C]. Gentleman =-=[11]-=- gives an axiomatic treatment. A study of iterations with high values of the efficiency index is reported by Feldstein and Firestone [9]. When we consider algorithms for a fixed problem, it becomes me... |

2 |
Planarity Testing in V log V steps: Extended Abstract
- Hopcroft, Tarjan
- 1971
(Show Context)
Citation Context ...multiplication of matrices (Winograd [41], Strassen [32], Hopcroft and Kerr [14]), polynomial evaluation (Winograd [41]), median of a set of numbers (Floyd [10]), graph planarity (Hopcroft and Tarjan =-=[15]-=-). Surveys may be found in Knuth [19], Borodin [1], and Minsky [24]. Research on analytic computational complexity dates to the early sixties (Traub [33]-39]) and predates most of the algebraic result... |

2 | Form and Computer Science - Minsky - 1970 |

2 | Schnelle multiplication grosser zahlen - Schonhage, Strassen - 1971 |

1 | Computational complexity---a survey - BORODIN - 1970 |

1 |
On maximizing the efficiencyfor solving systems ofnonlinear equations
- BRENT
(Show Context)
Citation Context ... (Traub [33]-39]) and predates most of the algebraic results. More specifically, the work on analytic computational complexity to date has concerned optimal iteration. Recent results are due to Brent =-=[2]-=-, Cohen [3], Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller [22], [23], Paterson 27], Rissanen [30], and Winograd a... |

1 |
Rate of Convergence and Optimality Conditions of Root Finding and Optimization Algorithms
- Cohen
- 1970
(Show Context)
Citation Context ...]-39]) and predates most of the algebraic results. More specifically, the work on analytic computational complexity to date has concerned optimal iteration. Recent results are due to Brent [2], Cohen =-=[3]-=-, Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller [22], [23], Paterson 27], Rissanen [30], and Winograd and Wolfe 42... |

1 |
On the minimum computation timefor multiplication, Doctoral thesis
- COOK
- 1966
(Show Context)
Citation Context ...esses, which we call analytic computational complexity. The last few years have witnessed striking developments in algebraic computational complexity; for example, the multiplication of numbers (Cook =-=[6]-=-, Sch6nhage and Strassen [31]), the multiplication of matrices (Winograd [41], Strassen [32], Hopcroft and Kerr [14]), polynomial evaluation (Winograd [41]), median of a set of numbers (Floyd [10]), g... |

1 |
Bounds on Order and Ostrowski Efficiency for Interpolatory Iteration Algorithms
- Feldstein
- 1969
(Show Context)
Citation Context ...aic results. More specifically, the work on analytic computational complexity to date has concerned optimal iteration. Recent results are due to Brent [2], Cohen [3], Cohen and Varaiya [4], Feldstein =-=[7]-=-, Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller [22], [23], Paterson 27], Rissanen [30], and Winograd and Wolfe 42], [43]. (Kung and Paterson’s results ... |

1 |
Hermite interpolatory iteration theory andparallel numerical theory
- FELDSTEIN, FIRESTONE
- 1967
(Show Context)
Citation Context ...y, the work on analytic computational complexity to date has concerned optimal iteration. Recent results are due to Brent [2], Cohen [3], Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone =-=[8]-=-, [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller [22], [23], Paterson 27], Rissanen [30], and Winograd and Wolfe 42], [43]. (Kung and Paterson’s results are summarized at the end of ... |

1 |
Computational Complexity of Recursive Sequences
- Hartmanis, Stearns
- 1964
(Show Context)
Citation Context ...theory. 1. Introduction. Computational complexity is one of the foundations of theoretical computer science. The phrase computational complexity seems to have been first used by Hartmanis and Stearns =-=[12]-=- in 1965 although the first papers belonging to the field are those of Rabin [28], [29] in 1959 and 1960. One of its important components is the theory of optimal algorithmic processes. We distinguish... |

1 |
On minimizing the number ofmultiplications necessaryfor matrix multiplication
- HOPCROFT, KERR
- 1971
(Show Context)
Citation Context ...algebraic computational complexity; for example, the multiplication of numbers (Cook [6], Sch6nhage and Strassen [31]), the multiplication of matrices (Winograd [41], Strassen [32], Hopcroft and Kerr =-=[14]-=-), polynomial evaluation (Winograd [41]), median of a set of numbers (Floyd [10]), graph planarity (Hopcroft and Tarjan [15]). Surveys may be found in Knuth [19], Borodin [1], and Minsky [24]. Researc... |

1 |
Toward abstract numerical analysis, Doctoral thesis
- Miller
- 1969
(Show Context)
Citation Context ... optimal iteration. Recent results are due to Brent [2], Cohen [3], Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller =-=[22]-=-, [23], Paterson 27], Rissanen [30], and Winograd and Wolfe 42], [43]. (Kung and Paterson’s results are summarized at the end of 2.) In this paper we define basic concepts and pose some fundamental qu... |

1 |
problems with differentiability hypotheses
- Unsolvable
- 1970
(Show Context)
Citation Context ...al iteration. Recent results are due to Brent [2], Cohen [3], Cohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller [22], =-=[23]-=-, Paterson 27], Rissanen [30], and Winograd and Wolfe 42], [43]. (Kung and Paterson’s results are summarized at the end of 2.) In this paper we define basic concepts and pose some fundamental question... |

1 |
Form and computer science
- MINSY
- 1970
(Show Context)
Citation Context ...and Kerr [14]), polynomial evaluation (Winograd [41]), median of a set of numbers (Floyd [10]), graph planarity (Hopcroft and Tarjan [15]). Surveys may be found in Knuth [19], Borodin [1], and Minsky =-=[24]-=-. Research on analytic computational complexity dates to the early sixties (Traub [33]-39]) and predates most of the algebraic results. More specifically, the work on analytic computational complexity... |

1 |
Optimality for Square Root Algorithms. Private Communication
- Paterson
- 1971
(Show Context)
Citation Context ...ficiency index is reported by Feldstein and Firestone [9]. When we consider algorithms for a fixed problem, it becomes meaningful to optimize relative to the number of arithmetic statements. Paterson =-=[27]-=- takes for his efficiency measure Ev (log2 p)/, where p denotes the order and denotes the number of multiplications or divisions. He excludes from .M multiplication or division by a constant. Paterson... |

1 |
The Number of Multiplications Involved in Computing Certain Functions
- Winograd
- 1968
(Show Context)
Citation Context ...ave witnessed striking developments in algebraic computational complexity; for example, the multiplication of numbers (Cook [6], Sch6nhage and Strassen [31]), the multiplication of matrices (Winograd =-=[41]-=-, Strassen [32], Hopcroft and Kerr [14]), polynomial evaluation (Winograd [41]), median of a set of numbers (Floyd [10]), graph planarity (Hopcroft and Tarjan [15]). Surveys may be found in Knuth [19]... |

1 |
iterative processes, IBM Rep. RC 3511, Yorktown Heights
- Optimal
- 1971
(Show Context)
Citation Context ...ohen and Varaiya [4], Feldstein [7], Feldstein and Firestone [8], [9], Hindrnarsh [13], Jarratt [16], King [18], Kung [21], Miller [22], [23], Paterson 27], Rissanen [30], and Winograd and Wolfe 42], =-=[43]-=-. (Kung and Paterson’s results are summarized at the end of 2.) In this paper we define basic concepts and pose some fundamental questions in optimal iteration. In the terminology of Knuth [20] we per... |

1 | Computational Complexity - A Survey - Borodin - 1970 |

1 | Hermite Interpolatory Iteration Theory and Parallel Numerical Theory - Feldstein, Firestone - 1967 |

1 | A Study of Ostrowski Efficiency for Composite Iteration Functions - Feldstein, Firestone - 1969 |

1 | A Family of Fourth-Order Methods for Nonlinear Equations - Kahan, Communication |

1 | Toward Abstract Numerical Analysis - Miller - 1969 |

1 | Speed of Computation of Functions and Classification of Recursive Sets - Rabin - 1959 |

1 | Optimal m-Invariant Iteration Functions - Traub - 1962 |

1 | The Theory of Multipoint Iteration Functions - Traub - 1962 |

1 | Interpolatively Generated Iteration Functions - Traub - 1963 |