THE PAPERS

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.

Read this Paper

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.

Read this Paper

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%.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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%.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

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.

Read this Paper

THE GAMMA FUNCTION RESEARCH TOY

Play with the Gamma Function Research Toy

Play with the Gamma Function Research Toy

Thank
You for Viewing

"Aspects of Approximation"

"Aspects of Approximation"

Compiled: | 13 March 2002 |

Last Revision: | 20 March 2019 |

Revision: | 3.1 |