Multiple Precision Computation

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


August, 2023:
For this update of version 1.4 of the FM package there have been  
a few more improvements to the way overflow, underflow and unknown  
results are handled. In addition, a general cleanup of the code  
has been done to improve its readability.  

July, 2022:
For this update of version 1.4 of the FM package I have made  
some improvements to the way overflow and underflow results are  
handled. These exceptions are rare, but in cases like expressions  
containing 1 + 2*exp(-x**2), when the exponential underflows,  
previous versions returned unknown after multiplying by 2.  
Now there is some extra information saved so that most of the  
time FM can tell that 2 times the underflowed exponential is still  
underflow. Now adding 1 to that gives 1 instead of unknown.  

September, 2021:
For this release of version 1.4 of the FM package I have  
learned of a way to modify FM to work around the gfortran bug  
mentioned in the August, 2021 release of the package (thanks,  
Luo Zhangping). All the sample and test programs below now  
work on all three compilers I have tested.  
Read more details about the standard FM 1.4 here  

August, 2021:
A new version 1.4 of the FM package is available.  
The changes in version 1.4 were made to enable a thread-safe  
special version of FM to be created, see file FM_parallel.f95.  
Read more details about the parallel FM 1.4 here  

February, 2021:
I have started making complex FM versions of some of the mathematical  
special functions. Functions in the Feb., 2021 version that now support  
complex input and results:  
erf(z), erfc(z), erfc_scaled(z) -- error function, complimentary  
                        function, scaled complimentary error function  
gamma(z), log_gamma(z), factorial(z) -- gamma, log_gamma, z!  

March, 2019:
There is a new version for the Calc-50 high-precision calculator app  
for iPhones and iPads. Version 1.3 has added a new function for  
extrapolation and the sum function has been enhanced so it can  
handle infinite series automatically.  
There are new example pages showing how these features can be used.  

January, 2018:
This multiple precision web site has been moved to a new server.  
The old site at has been shut  
down, and any reference to any of the pages at the old myweb site  
should now be automatically re-directed to this page on the new server:  

This page has some older FM news items.

There is a backup copy of this site on the Wayback Machine.
Click on the latest saved copy there.

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.

In addition to these three basic arithmetic types, multiple-precision exact rational arithmetic
and interval arithmetic are also available.

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.

Integer multiple-precision arithmetic and functions include 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 1.4: (July, 2023 version)

The actual files shown on this page are in this archive that contains all the FM files from this

The individual file pointers here display as pdf files for browsing. The longer files are cut
off after 20 pages.

Subroutine library for multiple-precision operations. 69,161 lines of code

Module for derived type interfaces. 43,371 lines of code

Module for FM internal global variables. 490 lines of code

Test program that checks a few operations for all of the FM routines. 47,550 lines of code

Small sample program using FM. 872 lines of code

Expected output file from SampleFM.f95. 157 lines

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. 2,457 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

A Multiple-Precision Interval Arithmetic Package (2014) 1 -- 13

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

Thread-safe parallel version of FM

This page has the code for the thread-safe version of FM
along with two sample programs that use the package and a user's manual.

Multiple precision rational arithmetic

This page has the code for the FM rational arithmetic package,
along with several sample programs that use the package and a user's manual.

Multiple precision interval arithmetic

This page has the code for the FM interval arithmetic package,
along with several sample programs that use the package and a user's manual.

Interface modules for user variables of double or quadruple length

This page has three modules that allow programs to work with FM when they explicitly
declare some variables to be quadruple precision real or complex, double length integer,
or quadruple length integer.
There is a test program for each module, a sample program that uses the modules and a user's
manual for the modules.

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, differential equations, derivatives for real and complex functions, etc.