A new version of a Fortran multiprecision computation system, based on the Fortran 90 language, is described. With this new approach, a translator program is not required—translation of Fortran code for multiprecision is accomplished by merely utilizing advanced features of Fortran 90, such as derived data types and operator extensions. This approach results in more-reliable translation and permits programmers of multiprecision applications to utilize the full power of Fortran 90. Three multiprecision data types are supported in this system: multiprecision integer, real, and complex. All the usual Fortran conventions for mixed-mode operations are supported, and many of the Fortran intrinsics, such as SIN, EXP, and MOD, are supported with multiprecision arguments. An interesting application of this software, wherein new number-theoretic identities have been discovered by means of multiprecision computations, is included also.

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A Fortran 90-based multiprecision system

This article describes a collection of Fortran routines for multiple-precision complex arithmetic and elementary functions. The package provides good exception handling, flexible input and output, trace features, and results that are almost always correctly rounded. For best efficiency on different machines, the user can change the arithmetic type used to represent the multiple-precision numbers.

Algorithm 786: multiple-precision complex arithmetic and functions

A radix-16 divider, generally comprising two improved radix-4 dividers cascaded combinatorially, also utilizes improved overlapping quotient bit selection, together with concurrent quotient bit selection and rounding techniques, to produce four bits of quotient per divider cycle with reduced hardware requirements.

Radix-16 divider using overlapped quotient bit selection and concurrent quotient rounding and correction

The analytic representation of the exact coagulation kernel for combined Brownian motion and gravitational settling entails an infinite sum of ratios of modified Bessel functions. Heretofore, the evaluation of this sum has been limited to relatively small arguments due to computational difficulties, hindering the numerical solution of the general dynamic equation in computational aerosol transport simulations. An approximation and an asymptotic extension have been available, but their accuracy for large arguments has not been established, and they have been shown to lead to numerical diffusion and oscillation. Using multiple-precision arithmetic, exact values for the kernel are presented in this paper for large arguments, and it is shown that both the asymptotic and approximating formulae provide a poor fit in the range of practical interest. Based on the results, the asymptotic formula is rectified and it is shown that the approximation is no longer needed. Practical formulae are given to facilitate rapi...

Evaluation of the Exact Coagulation Kernel under Simultaneous Brownian Motion and Gravitational Settling

The classical algorithm for multiple-precision division normalizes digits during each step and sometimes makes correction steps when the initial guess for the quotient digit turns out to be wrong. A method is presented that runs faster by skipping most of the intermediate normalization and recovers from wrong guesses without separate correction steps.

/pdf/a-multiple-precision-division-algorithm-3ypk0eya4z.pdf

A multiple-precision division algorithm

A generalized division algorithm for use with positive integral operands is presented. Depending upon the algebraic relationship of the first two ciphers of the divisor, one or at most two adjustments to the original divisor and dividend must be performed before the division operation can be initiated. The uniqueness of this method will cause each trial cipher in the quotient to be either equal to or one greater than its final replacement.

A recurrence formula is developed that can be used to generate an intermodulation-free frequency list. A frequency list is intermodulation-free if it satisfies one or more linear inequalities. Although the list developed here satisfies a simple linear inequality, it is hoped that this presentation will provide insight into developing recurrence formulas that can be used to generate lists that satisfy more complicated linear inequalities.

Use of a Recurrence Formula in Generating Intermodulation-Free Frequency Lists

In [1] Mifsud presented a generalized division algorithm for positive integral operands. The uniqueness of the method was advertised as causing each trial cipher in the quotient to be either equal to or one greater than its final replacement. The method of describing the algorithm was intended to stress the simple mathematical facts that were the basis of the algorithm. However, some difficulty arises with the programming and implementation of the algorithm. Article [1] addressed itself to the calculation of the trial cipher by using the first two digits of the partial dividend (step 6); i.e. it formed [pr+1pr/dr], with pr+1

Charles J. Mifsud

Papers

A multiple-precision division algorithm

Use of a Recurrence Formula in Generating Intermodulation-Free Frequency Lists

Addendum to a multiple-precision division algorithm