Professor of Mathematics (Emeritus)

Loyola Marymount University

Los Angeles, CA

** News:
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** 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. | |

Calc-50 |

** January, 2018: **

This multiple precision web site has been moved to a new server. | |

The old site at http://myweb.lmu.edu/dmsmith/FMLIB.html 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: | |

http://dmsmith.lmu.build |

This page has some older FM news items.

**
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There is a
backup copy
of this site on the Wayback Machine.

Click on the latest saved copy there.

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.

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

website: FM_files.zip

The individual file pointers here display as pdf files for browsing. The longer files are cut

off after 20 pages.

** **

FM.f95

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

FMZM90.f95

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

FMSAVE.f95

Module for FM internal global variables. 490 lines of code

TestFM.f95

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

SampleFM.f95

Small sample program using FM. 872 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. 2,457 lines

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

http://dmsmith.lmu.build/FMinterval2014.pdf (2014) 1 -- 13

Using Multiple-Precision Arithmetic

Computing in Science and Engineering 5 (July, 2003) 88 -- 93

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.

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.

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.

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.

This page has a table of domains for FM functions.

This page has a table of times for FM functions.

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.