ROUNDING-OFF ERRORS IN MATRIX PROCESSES

A number of methods of solving sets of linear equations and inverting matrices are discussed. The theory of the rounding-off errors involved is investigated for some of the methods. In all cases examined, including the well-known 'Gauss elimination process', it is found that the errors are normally quite moderate: no exponential build-up need occur. Included amongst the methods considered is a generalization of Choleski's method which appears to have advantages over other known methods both as regards accuracy and convenience. This method may also be regarded as a rearrangement of the elimination process.

[1]  F. S. Prout.  Philosophical Transactions of the Royal Society of London , 2009, The London Medical Journal.

[2]  George Cboom,et al.  Mind: A Quarterly Review of Psychology and Philosophy , 1876, Journal of Psychological Medicine and Mental Pathology (London, England : 1875).

[3]  J. Cremona,et al.  Proceedings of the London Mathematical Society , 1893 .

[4]  Nellie Clarke Brown.  Trees , 1896, Savage Dreams.

[5]  A. Turing.  Equivalence of Left and Right almost Periodicity , 1935 .

[6]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[7]  Alan M. Turing,et al.  Computability and λ-definability , 1937, Journal of Symbolic Logic.

[8]  Alan M. Turing,et al.  Systems of Logic Based on Ordinals , 2012, Alan Turing's Systems of Logic.

[9]  A. Turing.  Finite Approximations to Lie Groups , 1938 .

[10]  Alan M. Turing,et al.  The use of dots as brackets in Church's system , 1942, Journal of Symbolic Logic.

[11]  H. Hotelling.  Some New Methods in Matrix Calculation , 1943 .

[12]  J. Neumann,et al.  Numerical inverting of matrices of high order , 1947 .

[13]  H. D. Huskey,et al.  NOTES ON THE SOLUTION OF ALGEBRAIC LINEAR SIMULTANEOUS EQUATIONS , 1948 .

[14]  Alan M. Turing,et al.  Practical forms of type theory , 1948, Journal of Symbolic Logic.

[15]  A. Turing.  THE WORD PROBLEM IN SEMI-GROUPS WITH CANCELLATION , 1950 .

[16]  Logic based on Inclusion and Abstraction WV Quine , 2022 .

[17]  Universal Turing,et al.  The chemical basis of morphogenesis reprinted from Philosophical Transactions of the Royal Society ( Part B ) 237 37-72 ( 1953 ) Universal Turing Machine , 2022 .

[18]  Michael O. Rabin,et al.  Lecture to the london mathematical society on 20 february 1947 Universal Turing Machine , 2022 .

[19]  Proposed electronic calculator ; reprinted in ( Copeland 2005 ) Universal Turing Machine , 2022 .

[20]  Proof That Every Typed Formula Has a Normal Form Universal Turing Machine , 2022 .

[21]  Collected Works: Mathematical Logic (RO Gandy and CEM Yates Editors) , 2022 .

[22]  Intelligent Machinery 1948 Report for National Physical Laboratory Universal Turing Machine , 2022 .

[23]  Programmers ’ handbook for Manchester electronic computer Universal Turing Machine , 2022 .

[24]  Proposed electronic calculator report for National Physical Laboratory Teddington , 2022 .

[25]  Universal Turing,et al.  Can Digital Computers Think?; Reprinted in (copeland 2004) , 2022 .