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Perfect numbers: Difference between revisions
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→{{header|REXX}}: added/changed comments and whitespace, changed comments in various header and output sections.
m (→Lucas-Lehmer + other optimizations: changed indentations in the isPerfect function.) |
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===Classic REXX version of PL/I===
This version is a Classic REXX version of the PL/I program as of 14-Sep-2013, a REXX '''say''' statement
<br>was added to display the perfect numbers. Also, an epilog was written for the re-worked function.
<lang rexx>/*REXX version of the PL/I program (code was modified to run with Classic REXX). */
parse arg low high . /*obtain the specified number(s).*/
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===traditional method===
Programming note: this traditional method takes advantage of a few shortcuts:
:::* testing only goes up to the (integer) square root of '''X'''
:::* testing bypasses the test of the first and last factors
:::*
:::* testing is stopped if the sum of the factors exceeds '''X'''
<lang rexx>/*REXX program tests if a number (or a range of numbers) is/are perfect. */
parse arg low high . /*obtain optional arguments from the CL*/
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return s==x /*if the sum matches X, it's perfect! */</lang>
'''output''' when using the default inputs:
<pre>
6 is a perfect number.
28 is a perfect number.
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33550336 is a perfect number.
</pre>
For 10,000 numbers tested, this version is '''19.6''' times faster than the ooRexx program logic.<br>
For 10,000 numbers tested, this version is '''25.6''' times faster than the PL/I program logic.
<br><br>Note: For the above timings, only 10,000 numbers were tested.
===optimized using digital root===
This REXX version makes use of the fact that all ''known'' perfect numbers > 6 have a ''digital root'' of '''1'''.
<lang rexx>/*REXX program tests if a number (or a range of numbers) is/are perfect. */
parse arg low high . /*obtain the specified number(s). */
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end /*j*/ /*(above) is marginally faster. */
return s==x /*if the sum matches X, it's perfect! */</lang>
'''output''' is the same as the traditional version and is about '''5.3''' times faster (testing 34,000,000 numbers).
===optimized using only even numbers===
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end /*j*/ /*(above) is marginally faster. */
return s==x /*if sum matches X, then it's perfect!*/</lang>
'''output''' is the same as the traditional version and is about '''11.5''' times faster (testing 34,000,000 numbers).
===Lucas-Lehmer method===
This version uses memoization to implement a fast version of the Lucas-Lehmer test,
<br>
<lang rexx>/*REXX program tests if a number (or a range of numbers) is/are perfect. */
parse arg low high . /*obtain the optional arguments from CL*/
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end /*j*/ /*(above) is marginally faster. */
return s==x /*if the sum matches X, it's perfect!*/</lang>
'''output''' is the same as the traditional version and is about '''75''' times faster (testing 34,000,000 numbers).
===Lucas-Lehmer + other optimizations===
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