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Leite and Crouch

The issue of the journal containing this article [18] arrived rather fortuitously at a time when a question about Rodriques' formula had been raised in an Internet discussion group, and the daily meetings of the summer program were discussing the representation of rotations, mainly because of the quaternion representation of matrices, its adaptation to unimodular matrices, and applications to vibrating chains, the solution of second order differential equations, and similar topics which have been a continuing source of interest.

The title of the article calls for the symbolic calculation of matrix exponentials, which was pertinent to our interests, using ``Putzer's method,'' which was unfamiliar. Reading the article, it looked rather awkward for several reasons. It did not seem to fit smoothly into any physical publication sequence, cited a handful of references, and invoked Heisenberg's theory of the charge independence of nuclear forces. That may be correct, but one has the general feeling that the innovation was due to Pauli. We lack the immediate bibliographical resources to decide the issue, which in any event was only a minor allusion in the paper.

There was an evident proofreading error whereby a certain Lie group was assigned dimension 12 one place and 21 another. There was also a mention of ``Rodrigues' formula'' but it is not clear whether the reference was to Olinde Rodrigues (most likely) to whom the formula for the three dimensional rotation group is ascribed, or to a contemporary author named W. Rodrigues, Jr., coauthor of one of the articles cited in the bibliography. Generally, we do not have access to many of the articles cited by [17], to evaluate the doubt.

Finally, a referee is thanked for a reference to Putzer, which raises the question of whether the methods of the paper were somehow worked up, and then the title adjusted once the authors were apprised of the previous art. The reason for such a concern becomes evident when one goes to find out what Putzer's method is, and why it should be efficacious in evaluating Lie exponents. The article comments on the accessibility of the method. There is also some confusion, because Putzer's method is supposed to avoid eigenvalues and Jordan forms, yet attention is given to obtaining them as a preliminary to calculating the exponential, or maybe even to get the symmetric functions which are the coefficients of the charactristic polynomial. Some trace formulas are presented, but it is not mentioned whether those are the ones which go back to Newton.

Altogether, it is hard to render a judgement on reference [17] because of the inaccessibility to us of the majority of its references, upon which one usually depends to get the full benefit and understanding from reading a scientific article. Two important references which were available in our own library were to Putzer himself, and to the survey by Moler and van Loan.


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2000-03-17