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Recursive methods for generating conjugate directions with respect to an arbitrary matrix are investigated. There are three basic techniques to achieve this aim: (i) minimizing a quadratic form, (ii) generation by projections, and (iii) use of matrix equations. These techniques are equivalent to each other, however, the third one is stressed in this paper because of its versatility. Among matrix equation forms Hestenes -Stiefel type recursions and L~nczos type recursions are mentioned, wheredoi:10.1016/0898-1221(91)90232-s fatcat:nsqvkm5it5httdqtad2aforbfu