SciRuby

Tools for Scientific Computing in Ruby

Benchmarks for Integration Gem

Rajat Kapoor •

The Integration gem now includes quite a few functions from which to choose. To make this choice easier, I have done some benchmarking to see how fast and acuurate each of these functions are. The results of these tests for some randomly chosen functions are given below. The speed test reports shows the user CPU time, system CPU time, the sum of the user and system CPU times, and the elapsed real time to perform 100 iterations of the code. The unit of time is seconds.

For function f(x)= x2+3*x+7 on interval (0,1)

Speed Tests

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 
Benchmarking with 100 iterations  user system total real rectangle 72.040000 0.010000 72.050000 ( 72.058014)  user system total real trapezoid 105.500000 0.000000 105.500000 (105.489225)  user system total real simpson 0.020000 0.000000 0.020000 ( 0.017601)  user system total real romberg 0.000000 0.000000 0.000000 ( 0.002219)  user system total real adaptive_quadrature 0.010000 0.000000 0.010000 ( 0.002757)  user system total real gauss 0.010000 0.000000 0.010000 ( 0.006439)  user system total real gauss_kronrod 0.000000 0.000000 0.000000 ( 0.008615)  user system total real simpson3by8 0.080000 0.000000 0.080000 ( 0.074791)  user system total real boole 0.100000 0.000000 0.100000 ( 0.100300)  user system total real open_trapezoid 181.470000 0.050000 181.520000 (181.572745)  user system total real milne 0.060000 0.000000 0.060000 ( 0.058677)  user system total real qng 0.000000 0.000000 0.000000 ( 0.003788)  user system total real qag 0.010000 0.000000 0.010000 ( 0.010407)

Accuracy Tests

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 
+---------------------+-------------------+-------------------+------------------------+-------------------+ | Method | Result | Actual Result | Error | Accuracy | +---------------------+-------------------+-------------------+------------------------+-------------------+ | rectangle | 8.833333324123249 | 8.833333333333334 | 9.210085138988688e-09 | 99.99999989573489 | | trapezoid | 8.83333334502252 | 8.833333333333334 | 1.1689186507624072e-08 | 99.99999986766959 | | simpson | 8.833333333333332 | 8.833333333333334 | 1.7763568394002505e-15 | 99.99999999999997 | | romberg | 9.0 | 8.833333333333334 | 0.16666666666666607 | 98.11320754716982 | | adaptive_quadrature | 8.833333333333334 | 8.833333333333334 | 0.0 | 100.0 | | gauss | 8.750000000006125 | 8.833333333333334 | 0.08333333332720905 | 99.05660377365425 | | gauss_kronrod | 8.75 | 8.833333333333334 | 0.08333333333333393 | 99.0566037735849 | | simpson3by8 | 8.833333333333336 | 8.833333333333334 | 1.7763568394002505e-15 | 99.99999999999997 | | boole | 8.833333333333336 | 8.833333333333334 | 1.7763568394002505e-15 | 99.99999999999997 | | open_trapezoid | 8.833333325264693 | 8.833333333333334 | 8.068640866554233e-09 | 99.9999999086569 | | milne | 8.833333333333334 | 8.833333333333334 | 0.0 | 100.0 | | qng | 8.833333333333334 | 8.833333333333334 | 0.0 | 100.0 | | qag | 8.833333333333336 | 8.833333333333334 | 1.7763568394002505e-15 | 99.99999999999997 | +---------------------+-------------------+-------------------+------------------------+-------------------+

For function f(x) = x5+7*x2+2*cos(x) on interval (0,1)

Speed Tests

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 
Benchmarking with 100 iterations  user system total real rectangle 534.790000 0.510000 535.300000 (538.238675)  user system total real trapezoid 835.950000 0.540000 836.490000 (839.233313)  user system total real simpson 0.650000 0.000000 0.650000 ( 0.652694)  user system total real romberg 0.010000 0.000000 0.010000 ( 0.002073)  user system total real adaptive_quadrature 0.170000 0.000000 0.170000 ( 0.167520)  user system total real gauss 0.000000 0.000000 0.000000 ( 0.007621)  user system total real gauss_kronrod 0.020000 0.000000 0.020000 ( 0.010610)  user system total real simpson3by8 0.820000 0.000000 0.820000 ( 0.831039)  user system total real boole 0.120000 0.000000 0.120000 ( 0.119929)  user system total real open_trapezoid 1019.100000 0.150000 1019.250000 (1020.034828)  user system total real milne 0.980000 0.000000 0.980000 ( 0.970671)  user system total real qng 0.000000 0.000000 0.000000 ( 0.008263)  user system total real qag 0.020000 0.000000 0.020000 ( 0.020050)

Accuracy Tests

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 
+---------------------+--------------------+--------------------+------------------------+-------------------+ | Method | Result | Actual Result | Error | Accuracy | +---------------------+--------------------+--------------------+------------------------+-------------------+ | rectangle | 4.182941950501391 | 4.1829419696157935 | 1.9114402505238104e-08 | 99.99999954303927 | | trapezoid | 4.182941993679488 | 4.1829419696157935 | 2.406369414842402e-08 | 99.99999942471844 | | simpson | 4.182941969798875 | 4.1829419696157935 | 1.8308110583120651e-10 | 99.99999999562314 | | romberg | 5.54030230586814 | 4.1829419696157935 | 1.3573603362523468 | 67.5501035847021 | | adaptive_quadrature | 4.182941969702552 | 4.1829419696157935 | 8.675815621472793e-11 | 99.9999999979259 | | gauss | 3.5364151237832213 | 4.1829419696157935 | 0.6465268458325721 | 84.54372901826423 | | gauss_kronrod | 3.5364151237807455 | 4.1829419696157935 | 0.6465268458350479 | 84.54372901820506 | | simpson3by8 | 4.182941969676288 | 4.1829419696157935 | 6.049472034419523e-11 | 99.99999999855378 | | boole | 4.182941969615793 | 4.1829419696157935 | 8.881784197001252e-16 | 99.99999999999997 | | open_trapezoid | 4.182941952971977 | 4.1829419696157935 | 1.6643816103112385e-08 | 99.99999960210263 | | milne | 4.18294196954598 | 4.1829419696157935 | 6.981348832368894e-11 | 99.99999999833099 | | qng | 4.182941969615793 | 4.1829419696157935 | 8.881784197001252e-16 | 99.99999999999997 | | qag | 4.1829419696157935 | 4.1829419696157935 | 0.0 | 100.0 | +---------------------+--------------------+--------------------+------------------------+-------------------+

If you are keen to test any function of your choice you can pull from my Integration gem fork and change the files in the benchmark folder. For both the files, accuracy.rb and speed.rb, change the line func = lambda{|x| f(x)} with f(x) as your desired function. In case you want to perform accuracy benchmarks, make sure that you know the exact result of the integration operation before hand and assign this value to the variable actual_result Feel free to comment if you face any problem running the benchmarks with your desired functions.

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