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THE PAPERS

NAME OF PAPER: A Logistic Model for the
Approximation of The Gaussian Probability Integral
Modified by a Skewed Sinusoid
DATE OF WRITING: 8 June 1991
LENGTH: 9 A4 Pages at Times New Roman 14
ILLUSTRATIONS: 0
EQUATIONS: 26
TABLES: 0
REFERENCES: 3
APPENDIX AVAILABILITY: 5 Supplements not available
in the InterNet Presentation
FILENAME: gausmodla.pdf
ABSTRACT:

A finite logistic function can be evolved to approximate the essentially indeterminate integral of Gauss'es Probability Distribution Function ( i.e. the Normal Distribution Function ).
This logistic function may be resolved to Linear, Sinusoidal and Line Dip components whose empirical coefficients may be optimised to sculpt a useful closed-form estimator of the PDF integral.
It is shown that the logistic estimator is computationally-competitive with four classical methods.


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NAME OF PAPER: Some Improved Methods for
The Determination of a Student's t-Quantile
DATE OF WRITING: 27 October 1996
LENGTH: 54 A4 Pages at Times New Roman 14
ILLUSTRATIONS: 4
EQUATIONS: 18
TABLES: 2
REFERENCES: 3
APPENDIX AVAILABILITY: Includes 37 Pages of Appendices
in the InterNet Presentation
FILENAME: ttimer9.pdf
NOTES: The prior paper "A Description of Program BELEM.BAS"
is not available on the InterNet.
ABSTRACT:

The Author presents a method twice as fast and seven orders of magnitude more accurate
than Hill's Process. The new algorithm involves a procedure the Author calls "kedging"
which furnishes a convegent solution for f(x) based on Lagrangian Interpolation.
The speed and power of kedging requires some refinement of the computational control to
pre-empt small number errors.
The new method is compared with classical estimation techniques such as Romberg Integration,
Hasting's Polynomials and power series options.
Computation timing tariff experiments are undertaken to identify algorithmic improvements
in solutions engineering.


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NAME OF PAPER: A Enhanced Estimator for Factorials
and for
The Gamma Function
DATE OF WRITING: 9 June 1998
LENGTH: 4 A4 Pages at Times New Roman 14
ILLUSTRATIONS: 0
EQUATIONS: 13
TABLES: 0
REFERENCES: 1
APPENDIX AVAILABILITY: No Appendices
FILENAME: gammaaest.pdf
ABSTRACT:

An enhanced Gamma Function estimator may readily be established as the
medial point between the error bounds of Stirling's Formula.
This medial point is the Central Expectation of the Factorial.
When the relative error due to this approximation is correlated with
the argument n a simple power law of determination coefficient 0.99987222
fits the data. This power law may in turn be approximated as 1/(8.pi.n)
and used to correct the Central Expectation.
The method is generalised for any Gamma Function, and is exact near to f=4.5.
The error between f=0.1 and f=0.9 is about 4%.


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NAME OF PAPER: Some Further Enhancement of a
Gamma Function Estimator
DATE OF WRITING: 7 February 1999
LENGTH: 5 A4 Pages at Times New Roman 14
ILLUSTRATIONS: 0
EQUATIONS: 6
TABLES: 3
REFERENCES: 1
APPENDIX AVAILABILITY: No Appendices
FILENAME: gammaesa.pdf
ABSTRACT:

The Enhanced Central Expectation of the Error Bounds of Stirling's Formula
is subjected to further experimentation to improve its behavior as a
goodenough Gamma Function estimator.
Two cubic equation multpliers are regressed and respectively applied as
correctors in the range z=1 to 2 and the ( overlapping ) range z=1 to 12.
The RMS errors of the enhanced estimator bettered all methods except Hasting's
Eight-Degree Polynomial in the range z=1 to 2, and was dramatically better than
anything other than a five-term Stirling Formula in the range z=1 to 12.


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NAME OF PAPER: An Approximation of
The Complete Gamma Function for
Very Small Arguments
DATE OF WRITING: 11 January 2005
LENGTH: 4 A4 Pages at Times New Roaman 14
ILLUSTRATIONS: 0
EQUATIONS: 18
TABLES: 0
REFERENCES: 1
APPENDICES: No Appendices
FILENAME: gamtiny.pdf
ABSTRACT:

The Power Series of the Napierian Logarithm ( Abramowitz and Stegun: 6.1.33 ) is substituted into the Reflection Formula for Gamma(z). For very small arguments the Sine of pi.z is very nearly pi.z and accordingly we are able to approximate Gamma(z) as the argument's reciprocal minus The Euler Constant. The approximation improves as z decreases, and is almost exact at the sub-atto level.

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NAME OF PAPER: A Closed-Form Approximation to the
Arithmetic-Geometric Mean
DATE OF WRITING: 21 November 2002
LENGTH: 12 A4 Pages at Times New Roman 12
ILLUSTRATIONS: 0
EQUATIONS: 45
TABLES: 2
REFERENCES: 2
APPENDICES: Includes 2 Pages of Appendices
FILENAME: agmappb.pdf
ABSTRACT:

The Arithmetic-Geometric Mean of Gauss is classically determined to an arbitrary precision by successive substitution of the arithmetic and the geometric means of two values, as the twin arguments of successor iterates.
In this paper, the Author, ably assisted by MATHCAD Student Edition, develops an algebraic series that approximates the usual approximation.
The series structure resolves to term groups in which two power groups sandwich a nested root group.
These term groups are simplified to a large but finite estimator.


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NAME OF PAPER: The Volumetry of the Bronshtein Obelisk
and its implications for
Approximation of The Square Root
DATE OF WRITING: 15 April 2005
LENGTH: 7 A4 Pages at Times New Roman 12
ILLUSTRATIONS: 8
EQUATIONS: 23
TABLES: 0
REFERENCES: 2
APPENDICES: No Appendices
FILENAME: obelroot.pdf
ABSTRACT:

Bronshtein obelisks that have a different length and breadth are volumetrically very well approximated by square-pyramidal frusta that exhibit equivalent areas for their upper and basal surfaces. This property sometimes allows a good estimate of the square-root of a number that can be resolved into four suitable and convenient factors. The estimate is a determinant-like semi-sum of the factor cross-products, and offers a potentially useful vicinitation function to start root-finding algorithms.

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NAME OF PAPER: An Approximate Square Root
by means of the
Partition of the Obelisk
DATE OF WRITING: 25 April 2005
LENGTH: 11 A4 Pages at Times New Roaman 12
ILLUSTRATIONS: 7
EQUATIONS: 28
TABLES: 1
REFERENCES: 1
APPENDICES: No Appendices
FILENAME: obpartrt.pdf
ABSTRACT:

In a variation upon the analysis of the Bronshtein obelisk we note that we may resolve its structure into that of two component obelisks, one of which can be set to be, or very closely approach, a pyramidal frustum. Specifically, the volume of the complete obelisk is virtually the sum of the volumes of the two component ones, less the volume of the pentahedral frog where they interpenetrate. Algebraic simplification of this near equivalence allows us to approximate a square-root as the sum of the argument and an arbitrary number; divided by twice the square-root of that latter number. Accordingly, we have a square-root estimator that is convenient when we have an exact root for some other value. The root to be estimated should not be an integer.

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NAME OF PAPER: A Simple but Accurate Approximator of
The Complete Elliptic Integral of the Fist Kind
DATE OF WRITING: 31 October 2013
LENGTH: 16 A4 Pages at Times New Roman 12
ILLUSTRATIONS: 2
EQUATIONS: 32
TABLES: 5
REFERENCES: 5
APPENDICES: 1 Appendix included in the Internet Version
FILENAME: ceif-compressed.pdf
ABSTRACT:

Estimation of The Complete Elliptic Integral of the Fist Kind
By Approxiamting the Arithmetic-Geometric Mean
and then using the well-known formula that relates
the CIEF to the AGM.
The estimator's fidelity is assessed by reference to
Classical Solution and the Hasting's Formula,
as well as simple Simpson's Rule integration.


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Compiled: 13 March 2002
Last Revision: 20 March 2019
Revision: 3.1