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9.3 Calculatingthe Cartan scalars 1231991, Letniowski and McLenaghan 1988). The main sources of improvementsare as follows.First, cases can be separated according to whether or not one or more ofthe Newman–Penrose components Ψ A of the Weyl tensor vanish. In manycases this immediately gives the Petrov type. The basic idea was introducedindependently by Hon (1975) and Åman et al. (1984). Secondly,and relatedly, one arranges the tests so that any calculation of the fulldiscriminant (a sixth degree polynomial, a sextic, in the components Ψ A )is put off until it becomes unavoidable (Hon 1975, Åman et al. 1984).Thirdly, one can build up more complicated expressions in the componentsΨ A by a series of binary operations followed by simplification. Thiswill in general save both time and computer memory in the calculationsof quantities of high degree (Åman et al. 1984). Finally, one must lookvery closely at the tests required to determine the Petrov type in thoseremaining cases where a number of types are possible. The algorithms insuccessive treatments differ essentially only at this point. A deeper understandingof the origin of the special tests arises from formulating themas consequences of the Euclidean algorithm applied to finding simultaneousroots of the quartic (9.5) and its derivative (Åman et al. 1991).An alternative understanding and procedure comes from considering thecovariants (Zund 1986, Penrose and Rindler 1986, Zakhary 1994); thismay in practice involve longer calculations (Piper 1997). The ranking ofthe different versions of the improved methods now depends on rathersubjective estimates of whether the expense of a test is justified by itsprobability of success.The calculation of the Segre type (Joly and MacCallum 1990, Seixas1991, Paiva et al. 1998) makes use of the same methods for quartics asthe Petrov classification. It has two parts. The first part classifies thePlebański spinor (the coefficient in a degree-two covariant of the Riccispinor)ΦĊḊΦ(AB CD) ĊḊ (9.8)into types algebraically equivalent to the Petrov types, called Plebański–Petrov types, by the same method as for Petrov classification. Unfortunately,different Ricci tensor types give the same Plebański–Petrov type,so a second stage is usually required, in which particular methods forseparating the subcases are used. These are based on considering (9.2),by similar methods. The process aims only at separation into the differentSegre types, and does not consider refinements, e.g. those discussedby Penrose and Rindler (1986). Further improvements to the computationhave been suggested by Paiva and Skea (unpublished, 1998) andZakhary (1994).