Get the optimal forward difference interval by Gill83 method
Source:R/nlmixrGrad.R
nlmixr2Gill83.RdGet the optimal forward difference interval by Gill83 method
Usage
nlmixr2Gill83(
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.
- args
a list of arguments to the function call. The
namesattribute ofargsgives the argument names.- envir
an environment within which to evaluate the call. This will be most useful if
whatis a character string and the arguments are symbols or quoted expressions.- which
Which parameters to calculate the forward difference and optimal forward difference interval
- gillRtol
The relative tolerance used for Gill 1983 determination of optimal step size.
- gillK
Max steps to determine the optimal forward/central difference step size per parameter (Gill 1983). `0` = no optimal step size determined.
- gillStep
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
- gillFtol
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:
- info Gradient evaluation/forward difference information
- hf Forward difference final estimate
- df Derivative estimate
- df2 2nd Derivative Estimate
- err Error of the final estimate derivative
- aEps Absolute difference for forward numerical differences
- rEps Relative Difference for backward numerical differences
- aEpsC Absolute difference for central numerical differences
- rEpsC Relative difference for central numerical differences
The info returns one of the following:
- "Not Assessed" Gradient wasn't assessed: the parameter is left out
by which, or gillK = 0, which determines no interval and
reports the one the search starts from as hf
- "Good Success" in Estimating optimal forward difference interval
- "High Grad Error" Large error; Derivative estimate error fTol or more of the derivative
- "Constant Grad" Function constant or nearly constant for this parameter
- "Odd/Linear Grad" Function odd or nearly linear, df = K, df2 ~ 0
- "Grad changes quickly" df2 increases rapidly as h decreases
Examples
## These are taken from the numDeriv's grad examples to show how
## simple gradients are assessed with nlmixr2Gill83
nlmixr2Gill83(sin, pi)
#> Gill83 Derivative/Forward Difference
#> (rtol=1.49011611938477e-08; K=10, step=2, ftol=0)
#>
#> info hf hphi df df2 err aEps
#> 1 Good 0.002022262 0.0161781 -0.9999993 0 1.473712e-05 0.0004882812
#> rEps aEpsC rEpsC f
#> 1 0.0004882812 0.0004882812 0.0004882812 1.224647e-16
nlmixr2Gill83(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 Odd/Linear Grad 0.0004882812 0.0002441406 1.0000000 0
#> 2 Grad changes quickly 0.0007950774 0.0003975387 0.8090170 1949978254364
#> 3 Grad changes quickly 0.0011018736 0.0005509368 0.3090170 1642754654097
#> 4 Grad changes quickly 0.0014086697 0.0007043349 -0.3090170 1005119906032
#> 5 Grad changes quickly 0.0017154659 0.0008577329 -0.8090170 418874984766
#> 6 Odd/Linear Grad 0.0020222620 0.0010111310 -0.9999993 0
#> 7 Grad changes quickly 0.0023290582 0.0011645291 -0.8090170 -227241639381
#> 8 Grad changes quickly 0.0026358544 0.0013179272 -0.3090170 -287073706156
#> 9 Grad changes quickly 0.0029426505 0.0014713253 0.3090170 -230334431768
#> 10 Grad changes quickly 0.0032494467 0.0016247233 0.8090170 -116742709124
#> 11 Good 0.0035562428 0.0142249713 0.9999979 0
#> err aEps rEps aEpsC rEpsC
#> 1 6.103516e-05 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 2 7.751918e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 3 9.050540e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 4 7.079410e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 5 3.592829e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 6 1.473712e-05 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 7 2.646295e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 8 3.783422e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 9 3.388969e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 10 1.896746e+08 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> 11 8.380283e-06 0.0004882812 0.0004882812 0.0004882812 0.0004882812
#> f
#> 1 -2.449294e-16
#> 2 -2.449294e-16
#> 3 -2.449294e-16
#> 4 -2.449294e-16
#> 5 -2.449294e-16
#> 6 -2.449294e-16
#> 7 -2.449294e-16
#> 8 -2.449294e-16
#> 9 -2.449294e-16
#> 10 -2.449294e-16
#> 11 -2.449294e-16
func0 <- function(x){ sum(sin(x)) }
nlmixr2Gill83(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 err
#> 1 Good 0.0004882812 0.125000000 1.0000000 0.0000000 6.103516e-05
#> 2 Good 0.0003184423 0.001590155 0.8089234 -0.5877851 1.871756e-04
#> 3 Good 0.0002503438 0.001101874 0.3088979 -0.9510564 2.380911e-04
#> 4 Good 0.0002503438 0.001408670 -0.3091360 -0.9510564 2.380911e-04
#> 5 Good 0.0003184423 0.001715466 -0.8091106 -0.5877851 1.871756e-04
#> 6 Good 0.0020222620 0.016178096 -0.9999993 0.0000000 1.473712e-05
#> 7 Good 0.0003184423 0.002329058 -0.8089234 0.5877850 1.871756e-04
#> 8 Good 0.0002503439 0.002635854 -0.3088979 0.9510560 2.380911e-04
#> 9 Good 0.0002503439 0.002942651 0.3091360 0.9510558 2.380910e-04
#> 10 Good 0.0003184424 0.003249447 0.8091106 0.5877847 1.871756e-04
#> 11 Good 0.0035562428 0.028449943 0.9999979 0.0000000 8.380283e-06
#> aEps rEps aEpsC rEpsC f
#> 1 4.882812e-04 4.882812e-04 4.882812e-04 4.882812e-04 -1.224145e-16
#> 2 1.955651e-04 1.955651e-04 1.955651e-04 1.955651e-04 -1.224145e-16
#> 3 1.109367e-04 1.109367e-04 1.109367e-04 1.109367e-04 -1.224145e-16
#> 4 8.677563e-05 8.677563e-05 8.677563e-05 8.677563e-05 -1.224145e-16
#> 5 9.063976e-05 9.063976e-05 9.063976e-05 9.063976e-05 -1.224145e-16
#> 6 4.882812e-04 4.882812e-04 4.882812e-04 4.882812e-04 -1.224145e-16
#> 7 6.676064e-05 6.676064e-05 6.676064e-05 6.676064e-05 -1.224145e-16
#> 8 4.637518e-05 4.637518e-05 4.637518e-05 4.637518e-05 -1.224145e-16
#> 9 4.154018e-05 4.154018e-05 4.154018e-05 4.154018e-05 -1.224145e-16
#> 10 4.785106e-05 4.785106e-05 4.785106e-05 4.785106e-05 -1.224145e-16
#> 11 4.882812e-04 4.882812e-04 4.882812e-04 4.882812e-04 -1.224145e-16
func1 <- function(x){ sin(10*x) - exp(-x) }
curve(func1,from=0,to=5)
x <- 2.04
numd1 <- nlmixr2Gill83(func1, x)
exact <- 10*cos(10*x) + exp(-x)
c(numd1$df, exact, (numd1$df - exact)/exact)
#> [1] 0.332315771 0.333537144 -0.003661882
x <- c(1:10)
numd1 <- nlmixr2Gill83(func1, x)
exact <- 10*cos(10*x) + exp(-x)
cbind(numd1=numd1$df, exact, err=(numd1$df - exact)/exact)
#> numd1 exact err
#> [1,] -8.022836 -8.022836 -1.166840e-10
#> [2,] 4.216156 4.216156 6.611818e-11
#> [3,] 1.592302 1.592302 -2.766382e-10
#> [4,] -6.651065 -6.651065 -9.572443e-11
#> [5,] 9.656398 9.656398 -4.522570e-12
#> [6,] -9.521651 -9.521651 -2.119119e-10
#> [7,] 6.334104 6.334104 -3.813201e-10
#> [8,] -1.103537 -1.103537 7.983780e-11
#> [9,] -4.480613 -4.480613 -2.876500e-10
#> [10,] 8.624103 8.623234 1.007135e-04
sc2.f <- function(x){
n <- length(x)
sum((1:n) * (exp(x) - x)) / n
}
sc2.g <- function(x){
n <- length(x)
(1:n) * (exp(x) - 1) / n
}
x0 <- rnorm(100)
exact <- sc2.g(x0)
g <- nlmixr2Gill83(sc2.f, x0)
max(abs(exact - g$df)/(1 + abs(exact)))
#> [1] 0.001120989