Multiple Precision Computation

Dr. David M. Smith
Professor of Mathematics (Emeritus)
Loyola Marymount University
Los Angeles, CA


News:

January, 2013:
A few minor changes have been made to FM.f95.  
Some unused variables have been removed, and some new code for  
exception-handling in the complex arithmetic versions of exp, sin,  
cos, sinh, and cosh can now return overflowed results where the  
previous version gave unknown.  

July, 2011:
Some new routines and programs for multiple-precision integration  
have been added to the sample program page.  

May, 2011:
Version 1.3b of the FM package (April, 2011) is now available here.  
This is a minor update of version 1.3 as described in the TOMS 2011 paper.  
New features in this version:  
Many of the elementary and special functions are faster than the versions in 1.2 or 1.3.  
New Fortran-08 functions have been added:  
acosh(x), asinh(x), atanh(x) -- real and complex  
bessel_j0(x), bessel_j1(x), bessel_jn(n,x) -- standard names for existing functions  
bessel_y0(x), bessel_y1(x), bessel_yn(n,x)  
bessel_jn(n1,n2,x), bessel_yn(n1,n2,x)  
erfc_scaled(x)  
atan(x,y) -- name for atan2(x,y)  
hypot(x,y) -- for sqrt( x^2 + y^2 )  
norm2(a) -- for sqrt( a(1)^2 + a(2)^2 + ... + a(n)^2 )  

February, 2011:
This is a major revision of the FM package.  
The code on the ACM web site is version 1.3 of FM (June, 2010).  
This version of the FM package is described in a paper appearing in  
Transactions on Mathematical Software 37 (February, 2011) 1 -- 18.  
New functions in this version:  
bessel_j(n,x), bessel_y(n,x)  
cos_integral(x), sin_integral(x)  
cosh_integral(x), sinh_integral(x)  
erf(x), erfc(x), log_erfc(x)  
exp_integral_ei(x), exp_integral_en(n,x)  
fresnel_c(x), fresnel_s(x)  
log_integral(x)  
New features in this version:  
Array syntax for the three multiple precision derived types.  
Support for much higher precision.  
Perfect rounding is now default.  
Use of undefined multiple-precision variables is detected.  


Overview of the FM package

The package performs multiple-precision real, complex, and integer arithmetic.
It provides the intrinsic Fortran numerical functions, as well as many special functions that are
not included in the Fortran standard.

One of the primary uses of the package is to study the accuracy and stability of numerical
algorithms by comparing results computed with several different levels of precision.

An existing Fortran program can be checked by converting it from double precision
to FM arithmetic. The package is designed to make this conversion fairly easy.

The precision, base, and rounding mode for the arithmetic can be set by the user.

Functions are available for conversion between multiple-precision numbers and machine-precision
numbers (single and double precision real and complex, and integer types). Mixed-mode operations
involving multiple-precision numbers and machine-precision numbers are handled automatically in
statements such as a = b - 1 or y = x/3.

There are functions for input/output operations and output formatting. Formatting functions are
very similar to Fortran's format specifications.

The package also has routines for integer multiple-precision arithmetic and functions, including
GCD, modular products and powers, and a random number generator based on 49-digit primes.

Array syntax works like Fortran's array operations in statements like v = 1, a = b + c, and
y = cos(x) when these variables are vectors or matrices of multiple precision numbers.

The program SampleFM.f95 below shows several examples of using FM for high-precision real,
complex, and integer calculations.


List of files for FMLIB 1.3: (1-7-2013 version)

Click a given file. If you directly download it, there may be an extra ".txt" filetype at the
end of the file name. Delete it. (You might also have to change "f95" to "f90" or something
else, depending on your compiler.)

FM.f95
Subroutine library for multiple-precision operations. 63,326 lines of code

FMZM90.f95
Module for derived type interfaces. 47,541 lines of code

FMSAVE.f95
Module for FM internal global variables. 505 lines of code

TestFM.f95
Test program that checks a few operations for all of the FM routines. 43,792 lines of code

SampleFM.f95
Small sample program using FM. 892 lines of code

SampleFM.chk
Expected output file from SampleFM.f95. 157 lines

FM_User_Manual.txt
User's Guide for the package, along with a list of the files, some troubleshooting advice,
and an example set of compiler/linker commands for building the programs. 2,354 lines


Papers Online (in pdf format)

All of these are technical papers explaining the algorithms used by the multiple-precision
package, except the last one.

"Using Multiple-Precision Arithmetic" gives some samples and discusses the package
from a user's point of view.

Efficient Multiple-Precision Evaluation of Elementary Functions
Mathematics of Computation 52 (1989) 131 -- 134

A Fortran Package For Floating-Point Multiple-Precision Arithmetic
Transactions on Mathematical Software 17 (1991) 273 -- 283

A Multiple-Precision Division Algorithm
Mathematics of Computation 66 (1996) 157 -- 163

Multiple Precision Complex Arithmetic and Functions
Transactions on Mathematical Software 24 (1998) 359 -- 367

Multiple-Precision Gamma Function and Related Functions
Transactions on Mathematical Software 27 (2001) 377 -- 387

Multiple-Precision Exponential Integral and Related Functions
Transactions on Mathematical Software 37 (2011) 1 -- 18

Using Multiple-Precision Arithmetic
Computing in Science and Engineering 5 (July, 2003) 88 -- 93


Papers Online (individual pages can be viewed with a browser)

Efficient Multiple-Precision Evaluation of Elementary Functions
Mathematics of Computation 52 (1989) 131 -- 134
Page 1, 2, 3, 4

A Fortran Package For Floating-Point Multiple-Precision Arithmetic
Transactions on Mathematical Software 17 (1991) 273 -- 283
Page 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

A Multiple-Precision Division Algorithm
Mathematics of Computation 66 (1996) 157 -- 163
Page 1, 2, 3, 4, 5, 6, 7, 8

Multiple Precision Complex Arithmetic and Functions
Transactions on Mathematical Software 24 (1998) 359 -- 367
Page 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Multiple-Precision Gamma Function and Related Functions
Transactions on Mathematical Software 27 (2001) 377 -- 387
Page 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Multiple-Precision Exponential Integral and Related Functions
Transactions on Mathematical Software 37 (2011) 1 -- 18
Page 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Using Multiple-Precision Arithmetic
Computing in Science and Engineering 5 (July, 2003) 88 -- 93
Page 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11


Automatic conversion of Fortran programs to use FM

This page has a program that tries to automate most of the changes
needed to convert a normal Fortran program to a multiple-precision version using FM.


Function domains

This page has a table of domains for FM functions.


Function timing

This page has a table of times for FM functions.


More sample programs

This page has some example programs solving problems in root-finding, linear systems of equations,
least-squares fitting, integrals, derivatives for real and complex functions, etc.