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A method of gradients for the calculation of the characteristic roots and vectors of a real symmetric matrix

1951; US government; Volume: 47; Issue: 1 Linguagem: Inglês

10.6028/jres.047.008

2376-5305

Magnus R. Hestenes, William Karush,

Advanced Theoretical and Applied Studies in Material Sciences and Geometry

A be a real symmetric mat rid,'(X) .. J! he Rayleigh quot ient form ed wit h a vec Lor x, and Hx) t he gradie nt vector of I' (x).The met hod of gradients consists in an infini te itera tion of t he operat ion x =x-a (x) H x). The convergence of t he procedure is proved for se veral choices for a (x), and t he rate of convergence is studied extensi vely for one part icula r a (x).The directions of t he vectors in t he sequence are seen t o tend to that of t he cha racteri stic vector belon ging t o t he lowest characteristic value.The method can be used for a numeri cal d eterminat ion of all chara cteristic vectors and values.