Package NORMT3. February 19, 2015
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1 Package NORMT3 February 19, 2015 Title Evaluates complex erf, erfc, Faddeeva, and density of sum of Gaussian and Student's t Version Date 31/10/2012 Author Guy Nason <g.p.nason@bristol.ac.uk> Maintainer Guy Nason <g.p.nason@bristol.ac.uk> Depends R (>= 2.0) Evaluates the probability density function of the sum of the Gaussian and Student's t density on 3 degrees of freedom. Evaluates the p.d.f. of the sphered Student's t density function. Also evaluates the erf, and erfc functions on complex-valued arguments. Thanks to Krishna Myneni the function is calculates the Faddeeva function also! License GPL-2 Repository CRAN Date/Publication :08:31 NeedsCompilation yes R topics documented: dnormt dst erf erfc ic is normt3ip wofz Index 11 1
2 2 dnormt3 dnormt3 Density function of sum of Gaussian and Student s t on 3 df Computes the probability density function of the sum of the Gaussian distribution and the Student s t distribution on 3 degrees of freedom. dnormt3(x, mean = 0, sd = 1) x mean sd Where to evaluate the density function The mean of the Gaussian The standard deviation of the Gaussian Computes the probability density function of the sum of the Gaussian distribution and the Student s t distribution on 3 degrees of freedom. The appropriate pdf value. Guy Nason Nason, G.P. (2005) On the sum of the Gaussian and Student s t random variables. Technical Report, 05:01, Statistics Group, Department of Mathematics, University of Bristol. normt3ip,dst,dnorm dnormt3(0) Should be = sqrt(2)/pi x <- seq(from=-5, to=5, length=100) plot(x, dnormt3(x), type="l") Density plot
3 dst 3 dst Density function of sphered Student s t distribution Evaluates probability density function of sphered Student s t distribution on nu degrees of freedom. This is just the standard Student s t distribution rescaled to have unit variance. dst(x, nu = 3) x nu Where to evaluate the density The degrees of freedom says it all. The appropriate probability density. Guy Nason Nason, G.P. (2001) Robust projection indices. Journal of the Royal Statistical Society, Series B, 63, dnorm, dnormt3 dst(0) Should be 2/pi = x <- seq(from=-5, to=5, length=100) plot(x, dst(x), type="l") Produces a density plot
4 4 erf erf Error function Computes the error function of a (possibly) complex valued argument. This function is 1-erfc(z). erf(z) z Argument of error function Computes the error function of a (possibly) complex valued argument by computing the complementary error function and subtracting the answer from 1. The error function of z Guy P. Nason, Department of Mathematics, University of Bristol Poppe, G.P.M. and Wijers, C.M.J. (1990) More efficient computation of the complex error function. ACM Transactions on Mathematical Software, 16, erfc erf(0) Should give 0 erf(1) Should give i erf(complex(re=1, im=1))
5 erfc 5 Should give i erfc Complementary error function Computes the complementary error function of a (possibly) complex valued argument. This function is 2/ π. erfc(z) z exp t2 dt z Argument of complementary error function Computes the complementary error function of a (possibly) complex valued argument. This function is 2/ π z exp t2 dt This function actually calls FORTRAN code (algorithm TOMS 680) which computes the Faddeeva s function and then with a slight modification obtains the erfc function of a complex-valued argument. The complementary error function of z Guy P. Nason, Department of Mathematics, University of Bristol Poppe, G.P.M. and Wijers, C.M.J. (1990) More efficient computation of the complex error function. ACM Transactions on Mathematical Software, 16, erf
6 6 ic1 erfc(0) Should give 1 erfc(1) Should give i erfc(complex(re=1, im=1)) Should give i ic1 Compute IC1 formula from Nason (2005) Computes I C1 (p, d) = d cos(px) exp( x 2 )dx ic1(p, d) p d Argument for IC1 function Argument for IC1 function An intermediate function for the computation of the inner product between Gaussian and Student s t distribution. of the function IC1 Guy Nason Nason, G.P. (2005) On the sum of the Gaussian and Student s t random variables. Technical Report, 05:01, Statistics Group, Department of Mathematics, University of Bristol.
7 is1 7 normt3ip ic1(1,1) Should give is1 Compute IS1 formula from Nason (2005) Computes I S1 (p, d) = d cos(px) exp( x 2 )dx is1(p, d) p d Argument for IS1 function Argument for IS1 function An intermediate function for the computation of the inner product between Gaussian and Student s t distribution. of the function IS1 Guy Nason Nason, G.P. (2005) On the sum of the Gaussian and Student s t random variables. Technical Report, 05:01, Statistics Group, Department of Mathematics, University of Bristol.
8 8 normt3ip normt3ip is1(1,1) Should give normt3ip Compute inner product function of Gaussian and t3 distribution This function evaluates the inner product function of the (sphered) Student s t distribution on 3 df and the Gaussian distribution as defined by Theorem~1 of Nason, 2005 normt3ip(mu, sigma) mu sigma Mean of the Gaussian Standard deviation of the Gaussian No extra details The evaluated function. Guy Nason Nason, G.P. (2005) On the sum of the Gaussian and Student s t random variables. Technical Report, 05:01, Statistics Group, Department of Mathematics, University of Bristol.
9 wofz 9 normt3ip(0,1) Answer should be which is 1/pi x <- 3 wofz Faddeeva function Computes the Faddeeva function of a complex valued argument. This function is exp z 2 erfc( iz). wofz(z) z Argument of Faddeeva function Computes the Faddeeva function of a complex valued argument. This function is exp z 2 erfc( iz) This function calls FORTRAN code (algorithm TOMS 680) which computes the Faddeeva function. The Faddeeva function, w(z) Krishna Myneni, krishna.myneni@ccreweb.org Poppe, G.P.M. and Wijers, C.M.J. (1990) More efficient computation of the complex error function. ACM Transactions on Mathematical Software, 16,
10 10 wofz erf erfc options(digits=15) wofz(0) Should give 1+0i wofz(1) Should give i wofz(complex(re=1, im=1)) Should give i
11 Index Topic distribution dnormt3, 2 dst, 3 Topic math erf, 4 erfc, 5 ic1, 6 is1, 7 normt3ip, 8 wofz, 9 dnorm, 2, 3 dnormt3, 2, 3 dst, 2, 3 erf, 4, 5, 10 erfc, 4, 5, 10 ic1, 6 is1, 7 normt3ip, 2, 7, 8, 8 wofz, 9 11
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Properties of Error Function erf(z) And Complementary Error Function erfc(z) z erf (z) π e -y dy erfc (z) 1 - erf (z) erf () erf( ) 1 erf(- ) - 1 erf (z) d erf(z) dz π z for z
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Dopant Diffusion (1) Predeposition dopant gas dose control SiO Si SiO Doped Si region () Drive-in Turn off dopant gas or seal surface with oide profile control (junction depth; concentration) SiO SiO Si
Package fwsim. October 5, 2018
Version 0.3.4 Title Fisher-Wright Population Simulation Package fwsim October 5, 2018 Author Mikkel Meyer Andersen and Poul Svante Eriksen Maintainer Mikkel Meyer Andersen Description