Get the optimal forward difference interval by Gill83 method

nlmixrGill83(
what,
args,
envir = parent.frame(),
which,
gillRtol = sqrt(.Machine$double.eps), gillK = 10L, gillStep = 2, gillFtol = 0 ) ## Arguments what either a function or a non-empty character string naming the function to be called. a list of arguments to the function call. The names attribute of args gives the argument names. an environment within which to evaluate the call. This will be most useful if what is a character string and the arguments are symbols or quoted expressions. Which parameters to calculate the forward difference and optimal forward difference interval The relative tolerance used for Gill 1983 determination of optimal step size. The total number of possible steps to determine the optimal forward/central difference step size per parameter (by the Gill 1983 method). If 0, no optimal step size is determined. Otherwise this is the optimal step size determined. When looking for the optimal forward difference step size, this is This is the step size to increase the initial estimate by. So each iteration the new step size = (prior step size)*gillStep The gillFtol is the gradient error tolerance that is acceptable before issuing a warning/error about the gradient estimates. ## Value A data frame with the following columns: • infoGradient evaluation/forward difference information • hfForward difference final estimate • dfDerivative estimate • df22nd Derivative Estimate • errError of the final estimate derivative • aEpsAbsolute difference for forward numerical differences • rEpsRelative Difference for backward numerical differences • aEpsCAbsolute difference for central numerical differences • rEpsCRelative difference for central numerical differences The info returns one of the following: • Not AssessedGradient wasn't assessed • GoodSuccess in Estimating optimal forward difference interval • High Grad ErrorLarge error; Derivative estimate error fTol or more of the derivative • Constant GradFunction constant or nearly constant for this parameter • Odd/Linear GradFunction odd or nearly linear, df = K, df2 ~ 0 • Grad changes quicklydf2 increases rapidly as h decreases ## Author Matthew Fidler ## Examples  ## These are taken from the numDeriv::grad examples to show how ## simple gradients are assessed with nlmixrGill83 nlmixrGill83(sin, pi) #> Gill83 Derivative/Forward Difference #> (rtol=1.49011611938477e-08; K=10, step=2, ftol=0) #> #> info hf hphi df df2 err aEps #> 1 Odd/Linear Grad 2.237911e-11 1.118956e-11 -1 0 1.630865e-13 5.403504e-12 #> rEps aEpsC rEpsC f #> 1 5.403504e-12 5.403504e-12 5.403504e-12 1.224647e-16 nlmixrGill83(sin, (0:10)*2*pi/10) #> Gill83 Derivative/Forward Difference #> (rtol=1.49011611938477e-08; K=10, step=2, ftol=0) #> #> info hf hphi df df2 #> 1 Grad changes quickly 1.045337e-07 5.226686e-08 1.0000000 8.796093e+12 #> 2 Grad changes quickly 1.702142e-07 8.510710e-08 0.8090170 4.254583e+19 #> 3 Grad changes quickly 2.358947e-07 1.179473e-07 0.3090168 3.584264e+19 #> 4 Grad changes quickly 3.015752e-07 1.507876e-07 -0.3090171 2.193033e+19 #> 5 Grad changes quickly 3.672556e-07 1.836278e-07 -0.8090173 9.139274e+18 #> 6 Grad changes quickly 4.329361e-07 2.164681e-07 -1.0000004 7.692125e+11 #> 7 Grad changes quickly 4.986166e-07 2.493083e-07 -0.8090174 -4.958098e+18 #> 8 Grad changes quickly 5.642971e-07 2.821485e-07 -0.3090169 -6.263551e+18 #> 9 Grad changes quickly 6.299776e-07 3.149888e-07 0.3090172 -5.025578e+18 #> 10 Grad changes quickly 6.956580e-07 3.478290e-07 0.8090169 -2.547164e+18 #> 11 Good 2.441411e-04 7.796106e-04 1.0000000 4.583226e-08 #> err aEps rEps aEpsC rEpsC #> 1 4.597442e+05 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 2 3.620952e+12 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 3 4.227544e+12 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 4 3.306821e+12 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 5 1.678225e+12 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 6 1.665099e+05 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 7 1.236095e+12 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 8 1.767252e+12 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 9 1.583001e+12 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 10 8.859777e+11 1.045337e-07 1.045337e-07 1.045337e-07 1.045337e-07 #> 11 1.118954e-11 3.352119e-05 3.352119e-05 3.352119e-05 3.352119e-05 #> f #> 1 -4.583242e-08 #> 2 -4.583242e-08 #> 3 -4.583242e-08 #> 4 -4.583242e-08 #> 5 -4.583242e-08 #> 6 -4.583242e-08 #> 7 -4.583242e-08 #> 8 -4.583242e-08 #> 9 -4.583242e-08 #> 10 -4.583242e-08 #> 11 -4.583242e-08 func0 <- function(x){ sum(sin(x)) } nlmixrGill83(func0 , (0:10)*2*pi/10) #> Gill83 Derivative/Forward Difference #> (rtol=1.49011611938477e-08; K=10, step=2, ftol=0) #> #> info hf hphi df df2 #> 1 Grad changes quickly 7.391651e-08 3.695825e-08 1.0000000 1.203250e+17 #> 2 Grad changes quickly 1.203596e-07 6.017981e-08 0.8090168 4.538135e+16 #> 3 Grad changes quickly 1.668027e-07 8.340136e-08 0.3090166 2.362831e+16 #> 4 Grad changes quickly 2.132458e-07 1.066229e-07 -0.3090170 1.445699e+16 #> 5 Grad changes quickly 2.596889e-07 1.298445e-07 -0.8090170 9.748364e+15 #> 6 Odd/Linear Grad 3.061321e-07 1.530660e-07 -1.0000000 0.000000e+00 #> 7 High Grad Error 4.821089e-08 3.525752e-07 -0.8090170 5.876689e-01 #> 8 High Grad Error 3.788199e-08 3.990183e-07 -0.3090170 9.518254e-01 #> 9 Good 3.789588e-08 4.454614e-07 0.3090170 9.511281e-01 #> 10 Good 4.818847e-08 4.919045e-07 0.8090170 5.882158e-01 #> 11 Good 1.726341e-04 5.512680e-04 1.0000000 4.583208e-08 #> err aEps rEps aEpsC rEpsC #> 1 4.447003e+09 7.391651e-08 7.391651e-08 7.391651e-08 7.391651e-08 #> 2 2.731041e+09 7.391651e-08 7.391651e-08 7.391651e-08 7.391651e-08 #> 3 1.970634e+09 7.391651e-08 7.391651e-08 7.391651e-08 7.391651e-08 #> 4 1.541446e+09 7.391651e-08 7.391651e-08 7.391651e-08 7.391651e-08 #> 5 1.265771e+09 7.391651e-08 7.391651e-08 7.391651e-08 7.391651e-08 #> 6 2.230920e-09 7.392210e-08 7.392210e-08 7.392210e-08 7.392210e-08 #> 7 2.833204e-08 1.010729e-08 1.010729e-08 1.010729e-08 1.010729e-08 #> 8 3.605704e-08 7.017484e-09 7.017484e-09 7.017484e-09 7.017484e-09 #> 9 3.604383e-08 6.288156e-09 6.288156e-09 6.288156e-09 6.288156e-09 #> 10 2.834522e-08 7.241087e-09 7.241087e-09 7.241087e-09 7.241087e-09 #> 11 7.912182e-12 2.370311e-05 2.370311e-05 2.370311e-05 2.370311e-05 #> f #> 1 -2.291621e-08 #> 2 -2.291621e-08 #> 3 -2.291621e-08 #> 4 -2.291621e-08 #> 5 -2.291621e-08 #> 6 -2.291621e-08 #> 7 -2.291621e-08 #> 8 -2.291621e-08 #> 9 -2.291621e-08 #> 10 -2.291621e-08 #> 11 -2.291621e-08 func1 <- function(x){ sin(10*x) - exp(-x) } curve(func1,from=0,to=5) x <- 2.04 numd1 <- nlmixrGill83(func1, x) exact <- 10*cos(10*x) + exp(-x) c(numd1$df, exact, (numd1$df - exact)/exact) #> [1] 0.332398077 0.333537144 -0.003415112 x <- c(1:10) numd1 <- nlmixrGill83(func1, x) exact <- 10*cos(10*x) + exp(-x) cbind(numd1=numd1$df, exact, err=(numd1$df - exact)/exact) #> numd1 exact err #> [1,] -8.022836 -8.022836 -1.369260e-11 #> [2,] 4.216156 4.216156 -2.839580e-11 #> [3,] 1.592302 1.592302 -1.150871e-11 #> [4,] -6.651065 -6.651065 -4.125002e-11 #> [5,] 9.656398 9.656398 -5.172856e-11 #> [6,] -9.521651 -9.521651 2.985064e-10 #> [7,] 6.334104 6.334104 -8.697948e-11 #> [8,] -1.103537 -1.103537 -9.731425e-11 #> [9,] -4.480613 -4.480613 -1.320695e-10 #> [10,] 8.623852 8.623234 7.167430e-05 sc2.f <- function(x){ n <- length(x) sum((1:n) * (exp(x) - x)) / n } sc2.g <- function(x){ n <- length(x) (1:n) * (exp(x) - 1) / n } x0 <- rnorm(100) exact <- sc2.g(x0) g <- nlmixrGill83(sc2.f, x0) max(abs(exact - g$df)/(1 + abs(exact)))
#> [1] 0.0009927746