Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

An Algebraic Theory of Integration Methods

J. C. Butcher
Mathematics of Computation
Vol. 26, No. 117 (Jan., 1972), pp. 79-106
DOI: 10.2307/2004720
Stable URL: http://www.jstor.org/stable/2004720
Page Count: 28
  • Get Access
  • Read Online (Free)
  • Download ($34.00)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
An Algebraic Theory of Integration Methods
Preview not available

Abstract

A class of integration methods which includes Runge-Kutta methods, as well as the Picard successive approximation method, is shown to be related to a certain group which can be represented as the family of real-valued functions on the set of rooted trees. For each integration method, a group element is defined corresponding to it and it is shown that the numerical result obtained using the method is characterised by this group element. If two methods are given, then a new method may be defined in such a way that when it is applied to a given initial-value problem the result is the same as for the successive application of the given initial-value problem the result is the same as for the successive application of the given methods. It is shown that the group element for this new method is the product of the group elements corresponding to the given methods. Various properties of the group and certain of its subgroups are examined. The concept of order is defined as a relationship between group elements.

Page Thumbnails

  • Thumbnail: Page 
79
    79
  • Thumbnail: Page 
80
    80
  • Thumbnail: Page 
81
    81
  • Thumbnail: Page 
82
    82
  • Thumbnail: Page 
83
    83
  • Thumbnail: Page 
84
    84
  • Thumbnail: Page 
85
    85
  • Thumbnail: Page 
86
    86
  • Thumbnail: Page 
87
    87
  • Thumbnail: Page 
88
    88
  • Thumbnail: Page 
89
    89
  • Thumbnail: Page 
90
    90
  • Thumbnail: Page 
91
    91
  • Thumbnail: Page 
92
    92
  • Thumbnail: Page 
93
    93
  • Thumbnail: Page 
94
    94
  • Thumbnail: Page 
95
    95
  • Thumbnail: Page 
96
    96
  • Thumbnail: Page 
97
    97
  • Thumbnail: Page 
98
    98
  • Thumbnail: Page 
99
    99
  • Thumbnail: Page 
100
    100
  • Thumbnail: Page 
101
    101
  • Thumbnail: Page 
102
    102
  • Thumbnail: Page 
103
    103
  • Thumbnail: Page 
104
    104
  • Thumbnail: Page 
105
    105
  • Thumbnail: Page 
106
    106