BCMath programming hints for number theorists

Introduction

Here are some hints arising from my experience with programming number theory algorithms in BCMath.
The discussion is aimed at BC programmers who wish to convert their BC programs into BCMath, thereby making their BC programs more accessible.
My BC programs are located via http://www.numbertheory.org/gnubc/gnubc.html
Only a modicum of PHP programming knowledge is required and I found the book PHP for the world wide web by Larry Uhlman, a good introduction to PHP.
Matt Zandstra's book SAMS Teach Yourself PHP4 in 24 Hours is also quite readable.

Some remarks about BCMath

There is little online information about the use of the BCMath functions and just what features of BC programming go over to BCMath. Here are some of my observations:

Warning: Instead of the following versions of the if statement:

if($a<$b),if($a==$b), if($a>$b), if($a!=$b)

one should use

if(bccomp($a,$b)<0), if(bccomp($a,$b)==0), if(bccomp($a,$b)>0), if(bccomp($a,$b))

to be on the safe side.
There seems to be no problem for $a and $b with up to 16 decimals. However it seems that on my system, integers lie in the range -2147483647≤ n ≤ 2147483647=2³¹-1. See the PHP manual and the entry for var_dump and consider the following two php programs:
Example 1.

<?php
$large_number1 =  2147483648;
$large_number2 =  2147483647;
$large_number3 =  -2147483647;
$large_number4 =  -2147483648;
var_dump($large_number1);
echo "<br>\n";
var_dump($large_number2);
echo "<br>\n";
var_dump($large_number3);
echo "<br>\n";
var_dump($large_number4);
echo "<br>\n";
?>

The output is

float(2147483648)
int(2147483647)
int(-2147483647)
float(-2147483648)

Example 2.

<?php
$test1="10000000000000000";
$test2="10000000000000001";
if($test1>$test2){
   echo "10000000000000000 > 10000000000000001<br>\n";
}elseif($test1==$test2){
   echo "10000000000000000 = 10000000000000001<br>\n";
}else{
   echo "10000000000000000 < 10000000000000001<br>\n";
}

$t=bccomp($test1,$test2);
if($t==1){
   echo "10000000000000000 > 10000000000000001<br>\n";
}elseif($t==0){
   echo "10000000000000000 = 10000000000000001<br>\n";
}else{
   echo "10000000000000000 < 10000000000000001<br>\n";
}

$t=strcmp($test1,$test2);
if($t==1){
   echo "10000000000000000 > 10000000000000001<br>\n";
}elseif($t==0){
   echo "10000000000000000 = 10000000000000001<br>\n";
}else{
   echo "10000000000000000 < 10000000000000001<br>\n";
}
?>

The output is

10000000000000000 = 10000000000000001
10000000000000000 < 10000000000000001
10000000000000000 < 10000000000000001

Warning: One should write
$a="123456789123456";
instead of
$a=123456789123456;
There seems to be no problem if $a has less than 15 digits.
$i++ does not seem to work in the usual bc for loop
for($i="0";bccomp($i,$n)<0;$i++).
Instead this should be written as
for($i="0";bccomp($i,$n)<0;$i=bcadd($i,"1")).
Similarly, one cannot use i++ inside while.
Incrementing with ++ inside a for or while loop, using a symbol other than i++, seems to work.

Listing the members of a cartesian product of integers

See recursive construction of the cartesian product at

http://stackoverflow.com/questions/2516599/php-2d-array-output-all-combinations

preg_split() converts the string $Mvalue of integers which were entered separated by (any number of) spaces, into a normal array $a of $t=$count($a) numbers $a[0],...,$a[$n], where $n=bcsub($t,"1").

Debugging your BCMath script

Just as in debugging a Latex document, debugging a BCMath script can be challenging.

Don't forget to end statements with a ";".
One nasty, hard-to-locate error is $a="1';. The error message might state that the error lies on the last line of the file.
Writing $a[i] instead of $a[$i], is easy to do.
Forgetting to make corresponding global variable declarations in appropriate files is a trap for the forgetful programmer.
John Campbell has pointed out that including a space as part of a print statement forces the browser to emit a new line, instead of continuing to send the output off the screen. I use this trick in my prime_generator program to get the primes to fit nicely on the screen.
I use flush() after most print statements, to flush the output buffer to push all the output so far to the user's browser. But even then, I don't seem to always get the output pushed up the screen automatically as I would like, without use of the scrollbar.

13th December 2005. One nice feature of BCMath is the matrix facility. However printing a matrix needs care.

     print "matrix[$i][$j]=$matrix[$i][$j] "; /* does not work */

     $t=$matrix[$i][$j];
     print "matrix[$i][$j]=$t "; /* works */

On an IMAC, debugging your BCMath program is vastly helped by viewing MAMP logs in OSX.
Using the argument &$a in a function argument will change the value of $a.
Getting out of fullscreen in Firefox
Creating a .png from a window in a pdf using screenshot in Preview in an Imac. Create the pdf using Latex. Open the pdf and view it in preview. Go to print and select screenshot and copy a section via the select option. Then save as a .png.
This isn't BCMATH stuff, but I used the Unix command
sort -k2n -k1n ecubed_table > ecubed_table_sorted
to sort the list of the first 250,000 partial quotients of e³ into the list list, ordered with regard to increasing size.

Please email any comments or suggestions to Keith Matthews
Last modified 9th August 2017

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