BOOK REVIEW PAUL S. DWYER. Linear Pp. xi + 344.

Comt~tations. New

York: John Wiley and Sons, Inc., 1951.

Linear Computo~ions is addressed to the theoretical and practical considerations of the general problem of finding numerical solutions for systems of linear equations. The author hoped that this volume might serve as a textbook for a course in linear computations and also as a reference book--and from all appearances it seems that both these objectives willbe attained. The organization of the material follows sound pedagogical principles which, with the m a n y illustrativeproblems and the exercisesat the close of each chapter, make it especially suitable as a textbook. At the same time, the orderly grouping of methods, the illustrative problems, the annotated references at the end of each chapter, and the detailed index wilt be appreciated by the research worker who uses it as a reference work in computing techniques. It would be very presumptuous of the reviewer to attempt to summarize the breadth of the theory and technique covered in Linear Computalions in a few short pages. Instead, he can but hope that a brief outline of its contents will indicate the scope of the volume. Introductory remarks, defining and delimiting the problem of linear computations, and general principles of computational design are covered in Chapters 1 and 3. Special emphasis is given in this book (Chapters 2 and 17) to the question of errors in the solutions of simultaneous linear equations when the coefficients themselves are recognized as being subject to error. While some of the classical material on calculations with significant numbers is presented, what is much more important is the introduction of "range numbers" or "approximation-error numbers" for precise results. However, extensive computations with these numbers would be required in determining bounds for the error in the basic linear problems. Therefore, the author recommends the use of "incomplete numbers" with separate estimates of the maximum errors in linear computations such as evaluation of determinants, solutions of simultaneous equations, and determination of elements of the inverse matrix. An orderly grouping of various methods for solving systems of linear equations is carried out in Chapters 4, 5, and 6, and the relationships among these methods are brought out in Chapter 7. Then applications are made (Chapter 8) to the solution of many sets of simultaneous linear equations derived from the original set by deletion of certain variables or by replacement of the constant terms by new values. The different computational methods are exhibited by schematic arrangement, in general algebraic form, and then followed by numerical illustrations to assist the reader in understanding the development. Next, an elementary exposition of the theory of determinants is presented (Chapter 9), followed by the application of t~e various reduction methods to the evaluation of determinants (Chapter 10). Then the reader is introduced to the basic algebra of matrices in Chapter 12. Matrix theory enables the author to develop a more powerful exposition of the solution of simultaneous linear equations, employing the calculation of the inverse matrix (Chapters 13 and 14). Also, certain problems involving the characteristic equation are solved by the method of determinants (Chapter 15). The Hotelling-Bingham-Girshick method is presented for obtaining the inverse of a matrix based on the characteristic equation; and an excellent computational modification of the recently developed recursion formula of J. S. Frame is presented for the determination of the ad]oint, the determinant, and the inverse as well as the characteristic equation. Because of the many diverse approaches to the problem of solving simultaneous

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linear equations (and related problems), the author frankly admits that it would be an impossible task to review all of them. He devotes Chapter 16 to the treatment of other methods (e.g., extension and iterative methods) which he does not recommend for use with desk calculators. While the general problem of finding numerical solutions to systems of linear equations arises in a variety of ways, least squares theory and related statistical studies immediately lead to such mathematical problems. The presentation in this book is of a general mathematical nature rather than in specific statistical terms, although problems in statistics are mentioned. Then in Chapter 18 the author exhibits a number of statistical applications to assist the reader in translating the mathematical results to appropriate statistical results, with emphasis on improvement of the computational design. Finally, in the concluding chapter, the author looks ahead to procedures for generalizing the techniques presented to non-linear problems; and points to the continued major role that the desk calculator will play (high-speed digital computers notwithstanding) in solving linear problems involving 15 to 20 variables by the elimination methods featured in this book. The general organization of the text, designation of sections and tables, and over-all typography is such as to enhance the usefulness of this book as a reference work. However, in a volume of this type (with such extensive numerical and algebraic illustrations) it is almost inevitable that errors will appear in the initial printing. A list of such corrections has been prepared by the author and will be distributed by the publishers. Linear Computations is the outgrowth of Professor Dvv~yer'smany years of experience and interest in both the development of complex theory in mathematical statistics and the design of appropriate computational techniques for the application of such theory. The practicability of the book was assured by the author's years of experience in applying the methods to problems in the Statistical Research Laboratory at the University of Michigan, and more recently as a consultant on such matters to the Department of the Army. The use of Linear Computations will extend to many fields. It should provide an excellent foundation in numerical analysis--a subject of increasing importance and interest since the rapid development of high-speed electronic digital computers. Psychological statisticians and research psychologists, in particular, will welcome the many computational techniques applicable to their special problems. However, the important implications will not be arrived at by scanning, but diligent work with the material presented can produce very profitable results in psychometrics.

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