Numerical inverting of matrices of high order

CHAPTER I. The sources of errors in a computation 1. 1. The sources of errors. (A) Approximations implied by the mathematical model. (B) Errors in observational data. (C) Finitistic approximations to transcendental and implicit mathematical formulations. (D) Errors of computing instruments in carrying out elementary operations: "Noise. " Round off errors. "Analogy" and digital computing. The pseudo-operations 1023 1. 2. Discussion and interpretation of the errors (A) - (D). Stability 1027 1. 3. Analysis of stability. The results of Courant, Friedrichs, and Lewy 1028 1. 4. Analysis of "noise" and round off errors and their relation to high speed computing 1029 1. 5. The purpose of this paper. Reasons for the selection of its problem. . . . 1030 1. 6. Factors which influence the errors (A) - (D). Selection of the elimination method 1031 1. 7. Comparison between "analogy" and digital computing methods 1031 CHAPTER II. Round off errors and ordinary algebraical processes. 2. 1. Digital numbers, pseudo-operations. Conventions regarding their nature, size and use: (a), (b) 1033 2. 2. Ordinary real numbers, true operations. Precision of data. Conventions regarding these: (c), (d) 1035 2. 3. Estimates concerning the round off errors: (e) Strict and probabilistic, simple precision. (f) Double precision for expressions JT-1 Xiyt 1035 2. 4. The approximative rules of algebra for pseudo-operations 1038 2. 5. Scaling by iterated halving 1039 CHAPTER III. Elementary matrix relations. 3. 1. The elementary vector and matrix operations 1041 3. 2. Properties of \, |^| I and N (A) 1042 3. 3. Symmetry and definiteness 1045 3. 4. Diagonality and semi-diagonality 1046 3. 5. Pseudo-operations for matrices and vectors. The relevant estimates. . . 1047 CHAPTER IV. The elimination method. 4. 1. Statement of the conventional elimination method 1049 4. 2. Positioning for size in the intermediate matrices 1051 4. 3. Statement of the elimination method in terms of factoring A into semidiagonal factors C, W 1052 4. 4. Replacement of C, B' by, C, D 1054 4. 5. Reconsideration of the decomposition theorem. The uniqueness theorem 1055