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Exact Generalized Inverses and Solution to Linear Least Squares Problems Using Multiple Modulus Residue Arithmetic Sallie Ann Keller Mcnulty Iowa State University

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Exact Generalized Inverses and Solution to Linear Least Squares Problems Using Multiple Modulus Residue Arithmetic Sallie Ann Keller Mcnulty Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1983 Exact generalized inverses and solution to linear least squares problems using multiple modulus residue arithmetic Sallie Ann Keller McNulty Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation McNulty, Sallie Ann Keller, "Exact generalized inverses and solution to linear least squares problems using multiple modulus residue arithmetic " (1983). Retrospective Theses and Dissertations. 8420. https://lib.dr.iastate.edu/rtd/8420 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This reproduction was made from a copy of a document sent to us for microfilming. While the most advanced technology has been used to photograph and reproduce this document, the quality of the reproduction is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help clarify markings or notations which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure complete continuity. 2. When an image on the film is obliterated with a round black mark, it is an indication of either blurred copy because of movement during exposure, duplicate copy, or copyrighted materials that should not have been filmed. For blurred pages, a good image of the page can be found in the adjacent frame. If copyrighted materials were deleted, a target note will appear listing the pages in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photographed, a definite method of "sectioning" the material has been followed. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again—beginning below the first row and continuing on until complete. 4. For illustrations that cannot be satisfactorily reproduced by xerographic means, photographic prints can be purchased at additional cost and inserted into your xerographic copy. These prints are available upon request from the Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases the best available copy has been filmed. Universi^ MiCTOTlms International 300 N. Zeeb Road Ann Arbor. Ml 48106 8323301 McNulty, Sallie Ann Keller EXACT GENERALIZED INVERSES AND SOLUTION TO LINEAR LEAST SQUARES PROBLEMS USING MULTIPLE MODULUS RESIDUE ARITHMETIC Iowa State University PH.D. 1983 University Microfilms I ntsrnsitionâl SOO N. zeeb Road, Ann Arbor. Ml 48106 Exact generalized inverses and solution to linear least squares problems using multiple modulus residue arithmetic by Saille Ann Keller McNulty A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Statistics Approved: Signature was redacted for privacy. In Charge of Major Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. For the Gradate College Iowa State University Ames, Iowa 1983 ii TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. RESIDUE ARITHMETIC AND NONSINGULAR MATRIX INVERSION USING MULTIPLE MODULUS RESIDUE ARITHMETIC 6 2.1 Residue Arithmetic for Integers 6 2.2 Residue Arithmetic for Matrices 22 2.3 Error-Free Nonsingular Matrix Inversion 33 2.4 Alternative Solution to the Error-Free Nonsingular Matrix Inversion Problem 38 3. GENERALIZED INVERSES USING MDLTIPLE MODULUS RESIDUE ARITHMETIC 45 3.1 Existence of Generalized Inverses Over Various Fields and Rings 46 3.2 lAiltiple Modulus Residue Arithmetic Generalized Matrix Inversion 61 3.3 The t&iltiple Modulus Base and Corresponding Product Modulus 74 3.4 Algorithms for Finding Generalized Diverses Using %iltiple Modulus Residue Arithmetic 84 4. MOORE-PENROSE INVERSE USING MULTIPLE MODULUS RESIDUE ARITHMETIC 105 4.1 Existence of the Maore-Penrose Inverse Over Various Fields and Rings 106 4.2 Multiple Modulus Residue Arithmetic Ifoore-Penrose Inverse 110 5. LINEAR LEAST SQUARES SOLUTIONS USING MULTIPLE MODULUS RESIDUE ARITHMETIC 123 5.1 The Multiple Modulus Residue Arithmetic Least Squares Solution b, = (X*X) X'y 126 J. 5.2 The Miltiple Modulus Residue Arithmetic Least Squares Solution bg = jd'y 131 iii Page 6. RATIONAL MATRICES 136 6.1 Scaling a Rational Matrix to be Used for Multiple Modulus Residue Arithmetic 136 6.2 Scaling Rational Matrices on the Computer 141 7. COMPUTER IMPLEMENTATION 149 7.1 Product Modulus Bound and the lAiltiple Modulus Base 150 7.2 In^lementing Multiple Modulus Residue Arithmetic and the Symmetric Mixed Radix Representation 154 7.3 Examples 166 8. BIBLIOGRAPHY 178 9. ACKNOWLEDGMENTS 183 10. APPENDIX 184 10.1 The Gaussian Elimination Method FORTRAN Program 184 10.2 The Bordering Method FORTRAN Program 200 10.3 The Moore-Penrose Inversion Method FORTRAN Program 216 10.4 The Minimum Euclidean Norm Least Squares Mkthod 227 FORTRAN Program iv LIST OF NOTATIONS Z - set of all Integers z"*" - set of positive integers G(p) - Galois field generated by a prime number p R(M) - commutative ring generated by a composite number M |x|^ - residue of the scalar x modulo m /x/ - symmetric residue of the scalar x modulo m m IA|^ - residue of the matrix A modulo m /A/ - symmetric residue of the matrix A modulo m m X ^(m) - inverse of the scalar x modulo m or the Inverse of X over R(m) X (p) - inverse of the scalar x modulo p, p-prlme, or the inverse of x over G(p) A (m) - inverse of the matrix A modulo m or the inverse of I A| over R(m) m A ^(p) - inverse of the matrix A modulo p, p-prime, or the inverse of |A|^ over G(p) A^*^^ (p) - adjoint of the matrix 1^!^, p-prlme, over G(p) A - generalized inverse of the matrix A over the field of real or rational numbers (specified in context.) A (m) - generalized inverse of the matrix |A|^ over R(m) A (p) - generalized inverse of the matrix |A)^, p-prime, over G(p) V - Mbore-Penrose inverse of the matrix A over the field of real or rational numbers (specified in context.) A^(m) - Mbore-Penrose inverse of the matrix |A|^ over R(m) A^(p) - Mbore-Penrose inverse of the matrix IaI^, p-prime, over G(p) rank(A) - maximum number of linearly independent columns in the matrix A rank(|A|^) - maximum number of linearly independent columns in the matrix |A|^, linearly independent over G(p) {pi* Pg, •••» Pg} - multiple modulus base, p^-prime, i = 1, 2, ...» s s M = n p. - product modulus for the multiple modulus base i=l •fpl» P2» • • • » Pg} 1 1. INTRODUCTION A problem ^Ich is frequently present in statistical computing is generalized matrix inversion and the use of these generalized inverses in the solution to linear least square problems. Many different numerical methods have been proposed and used for obtaining generalized inverses and computing linear least squares parameter estimates. Most of these methods require the aid of the computer. Since floating-point arithmetic is normally used in computer solutions, the accuracy of solutions obtained for differing data conditions and algorithm stability has been extensively studied. The result of this effort is that there are several good algorithms available which provide reasonably accurate generalized inverses for most matrices, but none is entirely satisfactory. In fact, disastrously inaccurate results are sometimes obtained from all of the algorithms which utilize the standard floating-point arithmetic provided by the computer hardware. This is because the numerical theory behind these algorithms is based on the real number system but the computer's floating-point number system does not accurately model the real number system. This research will develop generalized matrix inversion methods and linear least squares solutions, based on the rational number system, which can be implemented on the computer to generate error-free solutions. Implementation of these methods in floating-point arithmetic will once again cause inaccurate results because the floating-point number system cannot accurately model the rational number system either. A multiple 2 modulus number system, however. Is capable of exactly modeling the rational number system. Therefore, with only the assumption that the problem has rational entries, using a multiple modulus number system (implemented in fixed-point arithmetic) an exact solution can be sought. This assumption poses no difficulty in problems requiring con^uter generated solutions because computers can only deal with rational numbers. Available in the published literature, there are methods for obtaining exact solutions (<» - precision) to full-rank systems of linear equations which use congruence techniques (i.e., residue arithmetic). The reader is referred to Borosh and Fraenkel (1966), Newman (1967), Earless (1968 and 1972), Howell and Gregory (1969a, 1969b, and 1970), Fraenkel and Loewenthal (1971), Cabay (1971), Cabay and Lam (1977a), and Gregory (1980) for discussion of the theory and application of such methods. The works of Newman, Cabay and Lam, and Howell and Gregory will be discussed in detail in Chapter 2. There has also been some research done by Stallings and Bouillon (1972), Rao et al. (1976), Adegbeyeni and Krlshnamurthy (1977) and Sen and Shamin (1978) on finding error-free generalized inverses of rational matrices using a single modulus number system.
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