Srinivas Vasudevan
6e215cf109
Add Bessel functions to SpecialFunctions.
...
- Split SpecialFunctions files in to a separate BesselFunctions file.
In particular add:
- Modified bessel functions of the second kind k0, k1, k0e, k1e
- Bessel functions of the first kind j0, j1
- Bessel functions of the second kind y0, y1
2019-09-14 12:16:47 -04:00
Srinivas Vasudevan
facdec5aa7
Add packetized versions of i0e and i1e special functions.
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- In particular refactor the i0e and i1e code so scalar and vectorized path share code.
- Move chebevl to GenericPacketMathFunctions.
A brief benchmark with building Eigen with FMA, AVX and AVX2 flags
Before:
CPU: Intel Haswell with HyperThreading (6 cores)
Benchmark Time(ns) CPU(ns) Iterations
-----------------------------------------------------------------
BM_eigen_i0e_double/1 57.3 57.3 10000000
BM_eigen_i0e_double/8 398 398 1748554
BM_eigen_i0e_double/64 3184 3184 218961
BM_eigen_i0e_double/512 25579 25579 27330
BM_eigen_i0e_double/4k 205043 205042 3418
BM_eigen_i0e_double/32k 1646038 1646176 422
BM_eigen_i0e_double/256k 13180959 13182613 53
BM_eigen_i0e_double/1M 52684617 52706132 10
BM_eigen_i0e_float/1 28.4 28.4 24636711
BM_eigen_i0e_float/8 75.7 75.7 9207634
BM_eigen_i0e_float/64 512 512 1000000
BM_eigen_i0e_float/512 4194 4194 166359
BM_eigen_i0e_float/4k 32756 32761 21373
BM_eigen_i0e_float/32k 261133 261153 2678
BM_eigen_i0e_float/256k 2087938 2088231 333
BM_eigen_i0e_float/1M 8380409 8381234 84
BM_eigen_i1e_double/1 56.3 56.3 10000000
BM_eigen_i1e_double/8 397 397 1772376
BM_eigen_i1e_double/64 3114 3115 223881
BM_eigen_i1e_double/512 25358 25361 27761
BM_eigen_i1e_double/4k 203543 203593 3462
BM_eigen_i1e_double/32k 1613649 1613803 428
BM_eigen_i1e_double/256k 12910625 12910374 54
BM_eigen_i1e_double/1M 51723824 51723991 10
BM_eigen_i1e_float/1 28.3 28.3 24683049
BM_eigen_i1e_float/8 74.8 74.9 9366216
BM_eigen_i1e_float/64 505 505 1000000
BM_eigen_i1e_float/512 4068 4068 171690
BM_eigen_i1e_float/4k 31803 31806 21948
BM_eigen_i1e_float/32k 253637 253692 2763
BM_eigen_i1e_float/256k 2019711 2019918 346
BM_eigen_i1e_float/1M 8238681 8238713 86
After:
CPU: Intel Haswell with HyperThreading (6 cores)
Benchmark Time(ns) CPU(ns) Iterations
-----------------------------------------------------------------
BM_eigen_i0e_double/1 15.8 15.8 44097476
BM_eigen_i0e_double/8 99.3 99.3 7014884
BM_eigen_i0e_double/64 777 777 886612
BM_eigen_i0e_double/512 6180 6181 100000
BM_eigen_i0e_double/4k 48136 48140 14678
BM_eigen_i0e_double/32k 385936 385943 1801
BM_eigen_i0e_double/256k 3293324 3293551 228
BM_eigen_i0e_double/1M 12423600 12424458 57
BM_eigen_i0e_float/1 16.3 16.3 43038042
BM_eigen_i0e_float/8 30.1 30.1 23456931
BM_eigen_i0e_float/64 169 169 4132875
BM_eigen_i0e_float/512 1338 1339 516860
BM_eigen_i0e_float/4k 10191 10191 68513
BM_eigen_i0e_float/32k 81338 81337 8531
BM_eigen_i0e_float/256k 651807 651984 1000
BM_eigen_i0e_float/1M 2633821 2634187 268
BM_eigen_i1e_double/1 16.2 16.2 42352499
BM_eigen_i1e_double/8 110 110 6316524
BM_eigen_i1e_double/64 822 822 851065
BM_eigen_i1e_double/512 6480 6481 100000
BM_eigen_i1e_double/4k 51843 51843 10000
BM_eigen_i1e_double/32k 414854 414852 1680
BM_eigen_i1e_double/256k 3320001 3320568 212
BM_eigen_i1e_double/1M 13442795 13442391 53
BM_eigen_i1e_float/1 17.6 17.6 41025735
BM_eigen_i1e_float/8 35.5 35.5 19597891
BM_eigen_i1e_float/64 240 240 2924237
BM_eigen_i1e_float/512 1424 1424 485953
BM_eigen_i1e_float/4k 10722 10723 65162
BM_eigen_i1e_float/32k 86286 86297 8048
BM_eigen_i1e_float/256k 691821 691868 1000
BM_eigen_i1e_float/1M 2777336 2777747 256
This shows anywhere from a 50% to 75% improvement on these operations.
I've also benchmarked without any of these flags turned on, and got similar
performance to before (if not better).
Also tested packetmath.cpp + special_functions to ensure no regressions.
2019-09-11 18:34:02 -07:00
Srinivas Vasudevan
18ceb3413d
Add ndtri function, the inverse of the normal distribution function.
2019-08-12 19:26:29 -04:00
Michael Figurnov
4bd158fa37
Derivative of the incomplete Gamma function and the sample of a Gamma random variable.
...
In addition to igamma(a, x), this code implements:
* igamma_der_a(a, x) = d igamma(a, x) / da -- derivative of igamma with respect to the parameter
* gamma_sample_der_alpha(alpha, sample) -- reparameterization derivative of a Gamma(alpha, 1) random variable sample with respect to the alpha parameter
The derivatives are computed by forward mode differentiation of the igamma(a, x) code. Although gamma_sample_der_alpha can be implemented via igamma_der_a, a separate function is more accurate and efficient due to analytical cancellation of some terms. All three functions are implemented by a method parameterized with "mode" that always computes the derivatives, but does not return them unless required by the mode. The compiler is expected to (and, based on benchmarks, does) skip the unnecessary computations depending on the mode.
2018-06-06 18:49:26 +01:00
Michael Figurnov
f216854453
Exponentially scaled modified Bessel functions of order zero and one.
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The functions are conventionally called i0e and i1e. The exponentially scaled version is more numerically stable. The standard Bessel functions can be obtained as i0(x) = exp(|x|) i0e(x)
The code is ported from Cephes and tested against SciPy.
2018-05-31 15:34:53 +01:00
Gael Guennebaud
2f7e2614e7
bug #1232 : refactor special functions as a new SpecialFunctions module, currently in unsupported/.
2016-07-08 11:13:55 +02:00
