Approximations by means of continued fractions
Toope, James Harold
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Rational approximations to real numbers have been used from ancient times, either for convenience in computation, or due to lack of more precise knowledge of the magnitude of the numbers. Better approximations may be obtained by making corrections in the form of adding aliquots, adjusting numerators, or adjusting denominators. The latter method, when extended, leads to our modern notion of continued fractions. The classical theory of continued fractions began during the Renaissance when Arabic numerals and the modern fractional notation had become common. It was studied and extended until about the end of the nineteenth century. This theory is concerned with terminating or non-terminating continued fractions whose elements are integers. The classical theory is essentially complete, being used mainly as a tool in other investigations, chiefly the solution of Diophantine equations. Advances are announced from time to time in special areas of the field. The equivalence of continued fractions to infinite products and to series expansions was recognised early in their study. Series expansions proved most convenient at that time, the other two forms being relatively neglected until more powerful computing aids were developed. Modern function theory regards the elements of a continued fraction as variables or as functions of variables. This analytic theory of continued fractions resulted from developments in the evaluation of integral functions, polynomial interpolation, rational functions and similar problems. This study has developed rapidly during the past sixty years and much research continues to be done. The development during the past decade of programmed electronic digital computers having great speed and freedom from error has stimulated the development of new numerical methods and a reexamination of old methods of computation for computer use. In particular the ability of the digital computer to iterate a series of operations ("loop") has stimulated the search for algorithms that may be used to evaluate functions. One such algorithm is the continued fraction. It is the purpose of this thesis to examine the method of evaluating functions as continued fractions with a view to its adaptability to computer operations.