New PDF release: Computing Methods

By I. S. Berezin and N. P. Zhidkov (Auth.)

ISBN-10: 0080100112

ISBN-13: 9780080100111

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R(fe). These non-zero vectors are t h e basis for t h e lowest linear manifold containing t h e vectors r (0) , Ar^0\ . . , Amf^\ Hence t h e confirmation. Side by side with t h e ordinary scalar product, suppose we now consider t h e scalar product as defined b y equality (27) in t h e previous section. The series of vectors r (0) , Af^\ A2r^°\ . , Akr{0\ . . can be orthogonalized in t h e sense of this scalar product. We get a new series of vectors j? , p^kK . . The enumerated properties 1 t o 5 hold good for these vectors.

B y repeating this reasoning a sufficient number of times we finally come t o t h e case when the index of r will be one unit greater t h a n t h a t of p. Consequently, if i>j, (f<*>, pW) = 0. (37) Finally, we have t o consider (f(t), f(j)) when ij^j. To be specific, suppose t h a t i>j. Then, (r(i),fö>) = ( f ^ p W - f t - ^ - t f ) = (r^fp^)-ßHl(r(i\p^-^) = 0. (38) Since there can be no more t h a n τι mutually orthogonal vectors in ^-dimensional space, a t a certain step Ic^n we get r(Ä) = 0.

This condition is theoretically valid since it accurately reflects t h e position of things. B u t it is inconvenient in practical use since t h e eigen values are generally unknown a n d it is a more complicated problem t o find t h e m t h a n t o solve t h e set of linear algebraic equations. I n Chapter 8 we shall describe various methods of estimating the maximum eigen value t a k e n absolutely. Meanwhile, we use t h e matrix norms of Section 7 and the inequality max | λ{ | =^ || B\\. Here we obtain the following three sufficient conditions: (9) (10) (11) Only the last condition needs an explanation.

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Computing Methods by I. S. Berezin and N. P. Zhidkov (Auth.)

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