The.Algorithm.Design.Manual.Springer-Verlag.1998

Calendrical Calculations
designing efficient shortcuts for the computation.
The basic approach behind calendar systems is to start with a particular reference date and count from
there. The particular rules for wrapping the count around into months and years is what distinguishes a
given calendar system from another. To implement a calendar, we need two functions, one that given a
date returns the integer number of days that have elapsed since the reference start date, the other of which
takes an integer n and returns the calendar date exactly n days from the reference date. This is analogous
to the ranking and unranking rules for combinatorial objects, such as permutations (see Section ).
The major source of complications in calendar systems is that the solar year is not an integer number of
days long. Thus if a calendar seeks to keep its annual dates in sync with the seasons, leap days must be
added at both regular and irregular intervals. Since a solar year is 365 days and 5:49:12 hours long, an
extra 10:48 minutes would have to be accounted for at the end of each year if we were simply to add a
leap day every four years.
The original Julian calendar (from Julius Caesar) did not account for these extra minutes, which had
accumulated to ten days by 1582 when Pope Gregory XIII proposed the Gregorian calendar used today.
Gregory deleted the ten days and eliminated leap days in years that are multiples of 100 but not 400.
Supposedly, riots ensued because the masses feared their lives were being shortened by ten days. Outside
the Catholic church, resistance to change slowed the reforms. The deletion of days did not occur in
England and America until September 1752, and not until 1927 in Turkey.
The rules for most calendrical systems are sufficiently complicated and pointless that you should lift
code from a reliable place rather than attempt to write your own. We identify suitable implementations
below.
There are a variety of ``impress your friends'' algorithms that enable you to compute in your head on
which day of the week a particular date occurred. Such algorithms often fail to work reliably outside the
given century and should be avoided for computer implementation.
Implementations: Dershowitz and Reingold provide a uniform algorithmic presentation [DR90,
RDC93] for a variety of different calendar systems, including the Gregorian, ISO, Julian, Islamic, and
Hebrew calendars, as well as other calendars of historical interest. Further, they provide Common Lisp
and C++ routines to convert dates between calendars, day of the week computations, and the
determination of secular and religious holidays. These are likely to be the most comprehensive and
reliable calendrical routines you will be able to get your hands on, and are available at
http://emr.cs.uiuc.edu:80/reingold/calendars.html.
A nice package for calendrical computations in Mathematica by Ilan Vardi is available in the
Packages/Miscellaneous directory of the standard Mathematica distribution, and also from MathSource.
Vardi's book [Var91] discusses the theory behind the implementation, which provides support for the
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