OPTIMIZATION IN SPLUS AND R
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deriv and deriv3
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deriv( ~ cos(x*y), c("x","y"))

deriv( ~ cos(x*y)/(x^2+y^2+1), c("x","y"))

deriv( ~ cos(x*y)/(x^2+y^2+1), c("x","y"), c("x","y"))

fn1 _ deriv( ~ cos(x*y)/(x^2+y^2+1), c("x","y"), function(x=0,y=0) NULL)



library(MASS)

fn2 _ deriv3( ~ cos(x*y)/(x^2+y^2+1),c("x","y"),function(x=0,y=0) NULL)



plotting
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fn1 _ function(x,y) { cos(x*y)/(x^2+y^2+1) }
x _ seq(-pi,pi,length=20)
y _ x
z _ outer(x,y,fn1)
persp(x,y,z)



using ms
--------

library(MASS)

fn2 _ function(x,y) { - cos(x*y)/(x^2+y^2+1) }
ms( ~ fn2(x,y), start=c(x=-2,y=-2))
ms( ~ fn2(x,y), start=c(x=-1,y=-1))
ms( ~ fn2(x,y), start=c(x=-3,y=-3))

fn2 _ function(x,y) { - cos(x*y)/(x^2+y^2+1) }
summary(ms( ~ fn2(x,y), start=c(x=-1,y=-1)))

fn2 _ deriv( ~ - cos(x*y)/(x^2+y^2+1), c("x","y"), function(x=0,y=0) NULL)
summary(ms( ~ fn2(x,y), start=c(x=-1,y=-1)))

fn2 _ deriv3( ~ - cos(x*y)/(x^2+y^2+1), c("x","y"), function(x=0,y=0) NULL)
summary(ms( ~ fn2(x,y), start=c(x=-1,y=-1)))



using nlmin
-----------

fn2 _ function(theta) {x _ theta[1]; y _ theta[2];  - cos(x*y)/(x^2+y^2+1) }
nlmin(fn2,c(-1,-1))



using nlminb
------------

fn2 _ function(theta) {x _ theta[1]; y _ theta[2];  - cos(x*y)/(x^2+y^2+1) }
fn3 _ deriv3( ~ - cos(x*y)/(x^2+y^2+1), c("x","y"), function(x=0,y=0) NULL)
fn2gh _ function(theta) {
  x _ theta[1]; y _ theta[2]
  result _ fn3(x,y)
  g _ attr(result,"gradient")
  H _ matrix(attr(result,"hessian"),ncol=2)
  list(gradient=g,hessian=H[row(H)<=col(H)])
}

fn2g _ function(theta) { fn2gh(theta)$gradient }

nlminb(c(-1,-1),fn2,lower=c(-5,-5),upper=c(4,4))

nlminb(c(-1,-1),fn2,gradient=fn2g,lower=c(-5,-5),upper=c(4,4))

nlminb(c(-1,-1),fn2,gradient=fn2gh,hessian=T,lower=c(-5,-5),upper=c(4,4))


using nlm in R
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fn2 _ function(theta) {x _ theta[1]; y _ theta[2];  - cos(x*y)/(x^2+y^2+1) }
nlm(fn2,c(-1,-1),hessian=T)

fn3 _ deriv3( ~ - cos(x*y)/(x^2+y^2+1), c("x","y"), function(x=0,y=0) NULL)
fn2 _ function(theta) { x _ theta[1]; y _ theta[2]; fn3(x,y) }
nlm(fn2,c(-1,-1),hessian=T)


using optim in R
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fn3 _ deriv3( ~ - cos(x*y)/(x^2+y^2+1), c("x","y"), function(x=0,y=0) NULL)
fn2 _ function(theta) { x _ theta[1]; y _ theta[2];  - cos(x*y)/(x^2+y^2+1) }
fn2g _ function(theta) { 
  x _ theta[1]; y _ theta[2];
  attr(fn3(x,y), "gradient")
}

optim(c(-1,-1),fn2,fn2g,hessian=T)

optim(c(-1,-1),fn2,hessian=T)

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