By ed Richard Goodman
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Additional info for Annual review in automatic programming. . V. 3
If this element is non-positive (zero could be tolerated for A; = η — 1) the decomposition of — j / cannot be continued and must be repeated with a smaller value ofj;,. This repetition seems to be a loss of computing time, but it is actually a net gain. e. the lowest eigenvalue of W* + j / * (where / * is the n-k row unit matrix) is a lower bound for the lowest eigenvalue ofA,. Therefore, if we take this eigenvalue χ as the newjv,, then the decomposition must succeed. Practical evidence showed that this χ is often a quite narrow lower bound which may result in a tremendous acceleration of the convergence.
Of matrices such that the A^ + ^ / a r e all similar: Choose J , decompose A, — j / i n t o ÄfÄ^f 5 = 0, 1, 2, . . Compute ^ , + 1 = R,RJ^ compute Zs+i = Zs+Js It can be enforced that the last diagonal element of-4, converges to zero and Zs to the smallest eigenvalue of -4 for ^ 00, but the speed of this con vergence depends heavily upon the choice of the j ^ . For every s this value should be chosen as large as possible, but must of course lie below the smallest eigenvalue of A,, because the Choleski method can be carried out in the real domain only for positive definite matrices.
But the recursive procedure is such a neat and elegant concept that I can hardly imagine that it will not have a marked influence on the design of new machines in the near future. And this influence could quite easily be so considerable, that the possible gain in efficiency that can still be booked by excluding recursiveness, will become negligible. Personally, it will not surprise me if this will prove to be the case. ' Perhaps there are still machines in which a special multiplication of positive factors is executed faster than the general one; otherwise we can easily design such a machine.
Annual review in automatic programming. . V. 3 by ed Richard Goodman
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