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Extra info for A commentary on the continued fraction by which the illustrious La Grange has expressed the binomial powers

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X n ]'. =Xk=l. Consider the k-th equation aUxk= - Z' akjXj' j 55 The Characteristic Value Problem-Generalities This gives, since Xk = 1, lakkl ~ Z' lakillxi! i ~ Z' lakil i since IXil ~ Xk = 1 =Ak which contradicts the assumption that the diagonal is strictly dominant. , allow for the possibility of some but not all of the relations laiil>A; being equalities. For instance they allow us to conclude that As is non-singular although most of the relations laiil ~A; are equalities. , There are many other proofs of the Gerschgorin Theorem.

Summing we have a b o u t multiplications altogether. , 9. OPTIMALITY The question as to whether cheaper ways evaluating det A or of solving Ajc=6 or of inverting a matrix A exist is a natural one. A negative solution to such a question has been given by Klyuyev a n d Kokovkin—Shcherbak. Provided we use full row or full column operations the Gaussian scheme is optimal. However Strassen (cf. (/2*°^2^) operations. Winograd has shown that about operations suffice a n d that Strassen's basic result is optimal.

18. ) Find the inverse of 4 2 4 30 20 45 20 15 36 35 28 70 5 5 5 5 3 30 40 45 48 30 70 105 126 140 90 . 70 112 140 160 105 42 70 90 105 70 1 12 10 20 Can you find other (non-trivial) matrices for which the inverse is obtainable in the same way as is obtained from M^? 19. 20. , a matrix Λ = [ 0 , 7 ] where t h e elements in the main diagonal a n d in all parallel ones are constant, t h u s : ao ai a_2 Ö0. 21. 22. Find the number of operations required in the triangular decomposi­ tion of a general matrix A = L U .

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A commentary on the continued fraction by which the illustrious La Grange has expressed the binomial powers by Euler L.


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