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Limitations on both high-speed core available and computer running time required for solving large-scale power system problems are very serious restrictions that continually confront the power system analyst. Piecewise methods, or methods of tearing as they are also called, hold much promise for the solution of such large scale problems. In this paper, new piecewise methods and algorithms are derived. Variations of these algorithms for practical applications are developed. Both boundary iteration as well as diakoptic methods are described. The major emphasis of this paper is placed on the latter method. The use of elimination and triangular factorization techniques within the piecewise methods is also indicated.