, Volume 7, Issue 1–2, pp 17–24 | Cite as

A note on the ε-algorithm

  • J. B. McLeod


This paper contains the proof of a fundamental algebraic results in the theory of the vector ε-algorithm. The relationships of this algorithm involve the addition, subtraction and inversion of vectors of complex numbers: the first two operations are defined by component-wise addition and subtraction; the inverse of the vectorz=(z1 ...,z N ) is taken to be
$$z^{ - 1} = \frac{{(\bar z_1 ,...,\bar z_N )}}{{\sum\limits_{i = 1}^N {\left| {z_i } \right|^2 } }}$$
where the bar denotes a complex conjugate. It is proved that if vectorsε s (m) can be constructed from the initial valuesε −1 (m) =0, (m=1,2,...),ε 0 (m) =s m , (m=0,1, ...) by means of the relationshipsε s+1 (m) =ε s-1 (m+1) +(ε s (m+1) -ε s (m) )−1, (m, s=0,1, ...); and if the recursion relations\(\sum\limits_{i = 0}^n {\beta _i s_{m + i} = \left( {\sum\limits_{i = 0}^n {\beta _i } } \right)} a,(m = 0,1,...)\) hold for the initial values, where the coefficients β i (i=0,1,...,n) are real and βn≠0, then form=0,1, ...,ε 2s (m) =a, if\(\sum\limits_{i = 0}^n {\beta _i } \ne 0\) andε 2s (m) =0, if\(\sum\limits_{i = 0}^n {\beta _i } = 0\).

Bemerkung zum ε-Algorithmus


Diese Arbeit beinhaltet ein fundamentales algebraisches Ergebnis der Theorie des vektoriellen ε-Algorithmus. Als Verknüpfungen dieses Algorithmus werden verwendet die Addition, die Subtraktion und der inverse Vektor mit komplexen Komponenten. Die ersten beiden Operationen sind definiert durch komponentenweise Addition beziehungsweise Subtraktion. Seiz=(z1, ...,z N ) ein vorgegebener Vektor, so soll der inverse Vektor auf folgende Weise gebildet werden.
$$z^{ - 1} = \frac{{(\bar z_1 ,...,\bar z_N )}}{{\sum\limits_{i = 1}^N {\left| {z_i } \right|^2 } }},$$
wobei der Querstrich die konjugiert komplexe Zahl bedeutet. Unter der Voraussetzung, daß der Vektorε s (m) aus den Anfangsbedingungenε −1 (m) =0, (m=1, 2, ...),ε 0 (m) =s m , (m=0,1, ...) mittels der Beziehungenε s+1 (m) =ε s-1 (m+1) +(ε s (m+1) -ε s (m) )−1, (m, s=0,1,...) gebildet werden kann und unter der weiteren Voraussetzung, daß die Rekursionsformel\(\sum\limits_{i = 0}^n {\beta _i s_{m + 1} = \left( {\sum\limits_{i = 0}^n {\beta _i } } \right)} a,(m = 0,1,...)\) (m=0,1,...) auch für die Anfangsbedingungen gilt, wobei die Koeffizienten β i (i=0,1,...,n) reell und ungleich Null sein sollen, wird fürm=0,1, ... bewiesen, daß die Beziehungenε 2n (m) =a gilt für\(\sum\limits_{i = 0}^n {\beta _i } \ne 0\) undε 2n (m) =0 gilt, wenn\(\sum\limits_{i = 0}^n {\beta _i } = 0\).


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Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • J. B. McLeod
    • 1
    • 2
  1. 1.Madison
  2. 2.Wadham CollegeOxfordUK

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