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Author Sloate, Harry M. ♦ Bickart, Theodore A.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©1973
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract Consider the set of multistep formulas $∑\textit{l}-1\textit{j}mn-\textit{k}$ $\textit{αij}\textit{x}\textit{mn}+\textit{j}$ - $\textit{h}$ $∑\textit{l}-1\textit{j}mn-\textit{k}\textit{βij}\textit{x}\textit{mn}+\textit{j}$ = 0, $\textit{i}$ = 1, ···, $\textit{l},$ where $\textit{x}\textit{mn}+\textit{j}$ = $\textit{y}\textit{mn}+\textit{j}$ for $\textit{j}=$ $-\textit{k},$ ···, -1 and $\textit{xn}$ = $ƒ\textit{n}$ = ƒ(xn , tn). These formulas are solved simultaneously for the $\textit{x}\textit{mn}+\textit{j}$ with $\textit{j}$ = 0, ···, $\textit{l}$ - 1 in terms of the $\textit{x}\textit{mn}+\textit{j}$ with $\textit{j}$ = $-\textit{k},$ ··· , - 1, which are assumed to be known. Then $\textit{y}\textit{mn}+\textit{j}$ is defined to be $\textit{x}\textit{mn}+\textit{j}$ for $\textit{j}$ = 0, ··· , $\textit{m}$ - 1. For $\textit{j}$ = $\textit{m},$ ··· , $\textit{l}$ - 1, $\textit{x}\textit{mn}+\textit{j}$ is discarded. The set of $\textit{y}'s$ generated in this manner for successive values of $\textit{n}$ provide an approximate solution of the initial value problem: $\textit{y}$ = ƒ(y, t), $\textit{y}(\textit{t}0)$ = $\textit{y}0.$ It is conjectured that if the method, which is referred to as the composite multistep method, is $\textit{A}-stable,$ then its maximum order is $2\textit{l}.$ In addition to noting that the conjecture conforms to Dahlquist's bound of 2 for $\textit{l}$ = 1, the conjecture is verified for $\textit{k}$ = 1. A third-order $\textit{A}-stable$ method with $\textit{m}$ = $\textit{l}$ = 2 is given as an example, and numerical results established in applying a fourth-order $\textit{A}-stable$ method with $\textit{m}$ = 1 and $\textit{l}$ = 2 are described. $\textit{A}-stable$ methods with $\textit{m}$ = $\textit{l}$ offer the promise of high order and a minimum of function evaluations—evaluation of ƒ(y, t) at solution points. Furthermore, the prospect that such methods might exist with $\textit{k}$ = 1—only one past point—means that step-size control can be easily implemented
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 1973-01-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 20
Issue Number 1
Page Count 20
Starting Page 7
Ending Page 26

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Source: ACM Digital Library