Newton-Raphson method
Arguments
- start
Starting value
- objective
Optional objective function (used for selecting step length)
- gradient
gradient
- hessian
hessian (if NULL a numerical derivative is used)
- control
optimization arguments (see details)
- args
Optional list of arguments parsed to objective, gradient and hessian
- ...
additional arguments parsed to lower level functions
Details
control should be a list with one or more of the following components:
trace integer for which output is printed each 'trace'th iteration
iter.max number of iterations
stepsize: Step size (default 1)
nstepsize: Increase stepsize every nstepsize iteration (from stepsize to 1)
tol: Convergence criterion (gradient)
epsilon: threshold used in pseudo-inverse
backtrack: In each iteration reduce stepsize unless solution is improved according to criterion (gradient, armijo, curvature, wolfe)
Examples
# Objective function with gradient and hessian as attributes
f <- function(z) {
x <- z[1]; y <- z[2]
val <- x^2 + x*y^2 + x + y
structure(val, gradient=c(2*x+y^2+1, 2*y*x+1),
hessian=rbind(c(2,2*y),c(2*y,2*x)))
}
NR(c(0,0),f)
#> $par
#> [1] -0.7324166 0.6825751
#>
#> $iterations
#> [1] 12
#>
#> $method
#> [1] "NR"
#>
#> $gradient
#> [1] 2.451187e-07 7.301897e-07
#>
#> $iH
#> [,1] [,2]
#> [1,] -0.3054540 -0.2849143
#> [2,] -0.2849143 0.4172596
#> attr(,"det")
#> [1] 1.937717
#> attr(,"pseudo")
#> [1] FALSE
#> attr(,"minSV")
#> [1] -2.473621
#>
# Parsing arguments to the function and
g <- function(x,y) (x*y+1)^2
NR(0, gradient=g, args=list(y=2), control=list(trace=1,tol=1e-20))
#>
#> Iter=0 ;
#> p= 0
#> [1] "Numerical Hessian"
#> Iter=1 ;
#> D= 1
#> p= -0.25
#> [1] "Numerical Hessian"
#> Iter=2 ;
#> D= 0.25
#> p= -0.375
#> [1] "Numerical Hessian"
#> Iter=3 ;
#> D= 0.0625
#> p= -0.4375
#> [1] "Numerical Hessian"
#> Iter=4 ;
#> D= 0.01562
#> p= -0.4688
#> [1] "Numerical Hessian"
#> Iter=5 ;
#> D= 0.003906
#> p= -0.4844
#> [1] "Numerical Hessian"
#> Iter=6 ;
#> D= 0.0009766
#> p= -0.4922
#> [1] "Numerical Hessian"
#> Iter=7 ;
#> D= 0.0002441
#> p= -0.4961
#> [1] "Numerical Hessian"
#> Iter=8 ;
#> D= 6.104e-05
#> p= -0.498
#> [1] "Numerical Hessian"
#> Iter=9 ;
#> D= 1.526e-05
#> p= -0.499
#> [1] "Numerical Hessian"
#> Iter=10 ;
#> D= 3.815e-06
#> p= -0.4995
#> [1] "Numerical Hessian"
#> Iter=11 ;
#> D= 9.537e-07
#> p= -0.4998
#> [1] "Numerical Hessian"
#> Iter=12 ;
#> D= 2.384e-07
#> p= -0.4999
#> [1] "Numerical Hessian"
#> Iter=13 ;
#> D= 5.96e-08
#> p= -0.4999
#> [1] "Numerical Hessian"
#> Iter=14 ;
#> D= 1.49e-08
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=15 ;
#> D= 3.725e-09
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=16 ;
#> D= 9.313e-10
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=17 ;
#> D= 2.328e-10
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=18 ;
#> D= 5.821e-11
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=19 ;
#> D= 1.455e-11
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=20 ;
#> D= 3.638e-12
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=21 ;
#> D= 9.095e-13
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=22 ;
#> D= 2.274e-13
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=23 ;
#> D= 5.684e-14
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=24 ;
#> D= 1.421e-14
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=25 ;
#> D= 3.553e-15
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=26 ;
#> D= 8.882e-16
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=27 ;
#> D= 2.22e-16
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=28 ;
#> D= 5.551e-17
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=29 ;
#> D= 1.388e-17
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=30 ;
#> D= 3.469e-18
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=31 ;
#> D= 8.674e-19
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=32 ;
#> D= 2.168e-19
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=33 ;
#> D= 5.421e-20
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=34 ;
#> D= 1.355e-20
#> p= -0.5
#> [1] "Numerical Hessian"
#> Iter=35 ;
#> D= 3.388e-21
#> p= -0.5
#> $par
#> [1] -0.5
#>
#> $iterations
#> [1] 35
#>
#> $method
#> [1] "NR"
#>
#> $gradient
#> [1] 3.388132e-21
#>
#> $iH
#> [,1]
#> [1,] -4294965233
#> attr(,"det")
#> [,1]
#> [1,] -2.328308e-10
#>