This note is a detailed explanation of Shanks-Atkin NUCOMP -composition and reduction carried out "simultaneously" - for all quadratic fields, that is, including real quadratic fields. That explanation incidentally deals with various "exercises" left for confirmation by the reader in standard texts. Extensive testing in both the numerical and function field cases by Michael J Jacobson, Jr, reported elsewhere, confirms that NUCOMP as here described is in fact efficient for composition both of indefinite and of definite forms once the parameters are large enough to compensate for NUCOMP's extra overhead. In the numerical indefinite case that efficiency is a near doubling in speed already exhibited for discriminants as small as 10 7.
- Binary quadratic form