Part of the Operator Theory: Advances and Applications book series (OT, volume 27)
After introducing algorithms of type 1 and 2 in chapter 3, we now give a third class of algorithms that are of rhombus type. This type of algorithm was first introduced by Rutishauser [RUT] and called the qd algorithm. We shall also give the variant of it which is in a sense more natural. It was introduced by Gragg [GR2] and called πζ algorithm. A third type of rhombus algorithm which we shall consider was given by McCabe [MC1] (see also [JOT]). We shall call it the FG algorithm because of the notation used by Jones and his coworkers. The qd and πζ algorithms are in a sense dual algorithms. This duality is the same duality as we have used for the quantities with and without a hat. To make this duality explicit, we shall not use theqd or πζ notation, but use the notation ab and â
instead. Also the indexing of these numbers will be more adapted to our previous approach where the algorithms progressed along horizontal lines in a Padé table. At the end of this chapter we shall give explicit correspondences between our notation and the original notation for these algorithms. With these ab, â
, FG and
Ĝ numbers, it will be possible to give a lot of recurrence relations between three neighbouring elements in a Padé table. These relations will also be given for further reference.
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© Birkhäuser Verlag Basel 1987