Monte Carlo Matrix Calculation with Punched Card Machines

A Retrospective and Prospective Survey of the Monte Carlo Method

“…They consider the problem of inverting a matrix: thus a matrix Neumann series of the form (74) (I H) -1 H Downloaded 11/19/14 to 129.49.23.145. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php is to be evaluated, rather than the vector Neumann series (58) above.…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php is to be evaluated, rather than the vector Neumann series (58) above. The treatments of the series (58) and (74) are completely analogous, however.…”

“…Here Prs denotes the probability of transition from index r to index s (Pro being denoted by Pr --2s=lPrs > O, with Pos 5os), with Pr-0 unless Hr 0, and each walk F Jr, 81, 82, "", Sin, 0] with r, sl, ..., Sm 4 0 corresponds to precisely one term in the series (74). Exactly as in (11) (57) arises from prior formulas, such as (61), (64) and (75)77), then the H,s and a in (80) should be replaced by estimators , and s sampled in the zeroth or (more usually) first-level manners described above.…”

“…Note that the same random walk, but now started at a random index So, sampled with probability qo, say, can be viewed in the reverse direction, to yield scores such as (85) [70]- [74], [81]- [85]. More general classes of estimators are also defined by these authors, especially by Page [73] and Muller [84], [85].…”