> round(sqrt(2),4)
[1] 1.4142

expression

> D(expression(sinx+cosy),"y")

[1] 0

> D(expression(sin(x)+cos(y)),"y")

-sin(y)

> D(expression(log(sin(y)),"y")

+ D(expression(log(sin(y)),"y")

Error: unexpected symbol in:

"D(expression(log(sin(y)),"y")

D"

> D(expression(log(sin(y))),"y")

cos(y)/sin(y)

> 

library(animation)

ani.options(nmax = 20)

par(mar = c(1, 1, 1, 1))

vi.lilac.chaser()

set.seed(20111105)
x = rbind(matrix(rnorm(10000 * 2), ncol = 2), local({
  r = runif(10000, 0, 2 * pi)
  0.5 * cbind(sin(r), cos(r))
}))
x = as.data.frame(x[sample(nrow(x)), ])
plot(x,pch=".")

> polyroot(c(40,0,0,5,20,10))
[1]  0.7747767+0.7263645i -0.7747767+1.0830293i -0.7747767-1.0830293i
[4]  0.7747767-0.7263645i -2.0000000+0.0000000i
> 

integrate( )

> f<-function(x){1/log(x)}

> integrate(f,lower=0,upper=1)

Error in integrate(f, lower = 0, upper = 1) : non-finite function value

> 

> f<-function(x){1/log(x)}

> integrate(f,lower=0,upper=1)

Error in integrate(f, lower = 0, upper = 2) : non-finite function value

> 

> 

> p<-function(x){log(x)}

> integrate(p,lower=0,upper=1)

-1 with absolute error < 1.1e-15

> integrate(dnorm,-Inf,Inf)
1 with absolute error < 9.4e-05
> 

library(cubature)

adaptIntegrate

install.packages("cubature")
library(cubature)

> testFn0 <- function(x) {
+   prod(cos(x))
+ }
> 
> adaptIntegrate(testFn0, rep(0,2), rep(1,2), tol=1e-4)
$integral
[1] 0.7080734

$error
[1] 1.709434e-05

$functionEvaluations
[1] 17

$returnCode
[1] 0



> M_2_SQRTPI <- 2/sqrt(pi)
> testFn1 <- function(x) {
+   scale = 1.0
+   val = 0
+   dim = length(x)
+   val = sum (((1-x) / x)^2)
+   scale = prod(M_2_SQRTPI/x^2)
+   exp(-val) * scale
+ }
> 
> adaptIntegrate(testFn1, rep(0, 3), rep(1, 3), tol=1e-4)
$integral
[1] 1.00001

$error
[1] 9.677977e-05

$functionEvaluations
[1] 5115

$returnCode
[1] 0




> testFn2 <- function(x) {
+   ## discontinuous objective: volume of hypersphere
+   radius = as.double(0.50124145262344534123412)
+   ifelse(sum(x*x) < radius*radius, 1, 0)
+ }
> 
> adaptIntegrate(testFn2, rep(0, 2), rep(1, 2), tol=1e-4)
$integral
[1] 0.19728

$error
[1] 1.972614e-05

$functionEvaluations
[1] 166141

$returnCode
[1] 0

> adaptIntegrate(testFn3, rep(0,3), rep(1,3), tol=1e-4)
$integral
[1] 1

$error
[1] 2.220446e-16

$functionEvaluations
[1] 33

$returnCode
[1] 0


> testFn4 <- function(x) {
+   a = 0.1
+   s = sum((x-0.5)^2)
+   (M_2_SQRTPI / (2. * a))^length(x) * exp (-s / (a * a))
+ }
> 
> adaptIntegrate(testFn4, rep(0,2), rep(1,2), tol=1e-4)
$integral
[1] 1.000003

$error
[1] 9.843987e-05

$functionEvaluations
[1] 1853

$returnCode
[1] 0



> testFn5 <- function(x) {
+   a = 0.1
+   s1 = sum((x-1/3)^2)
+   s2 = sum((x-2/3)^2)
+   0.5 * (M_2_SQRTPI / (2. * a))^length(x) * (exp(-s1 / (a * a)) + exp(-s2 / (a * a)))
+ }
> 
> adaptIntegrate(testFn5, rep(0,3), rep(1,3), tol=1e-4)
$integral
[1] 0.9999937

$error
[1] 9.980147e-05

$functionEvaluations
[1] 59631

$returnCode
[1] 0



> testFn6 <- function(x) {
+   a = (1+sqrt(10.0))/9.0
+   prod(a/(a+1)*((a+1)/(a+x))^2)
+ }
> 
> adaptIntegrate(testFn6, rep(0,4), rep(1,4), tol=1e-4)
$integral
[1] 0.9999984

$error
[1] 9.996851e-05

$functionEvaluations
[1] 18753

$returnCode
[1] 0



> testFn7 <- function(x) {
+   n <- length(x)
+   p <- 1/n
+   (1+p)^n * prod(x^p)
+ }
> adaptIntegrate(testFn7, rep(0,3), rep(1,3), tol=1e-4)
$integral
[1] 1.000012

$error
[1] 9.966567e-05

$functionEvaluations
[1] 7887

$returnCode
[1] 0

> I.1d <- function(x) {

+   sin(4*x) *

+     x * ((x * ( x * (x*x-4) + 1) - 1))

+ }

> 

> adaptIntegrate(I.1d, -2, 2, tol=1e-7)

$integral

[1] 1.635644

$error

[1] 4.024021e-09

$functionEvaluations

[1] 105

$returnCode

[1] 0

> adaptIntegrate(I.2d, rep(-1, 2), rep(1, 2), maxEval=10000)

$integral

[1] -0.01797993

$error

[1] 7.845607e-07

$functionEvaluations

[1] 10013

$returnCode

[1] 0

> dmvnorm <- function (x, mean, sigma, log = FALSE) {
+     if (is.vector(x)) {
+         x <- matrix(x, ncol = length(x))
+     }
+     if (missing(mean)) {
+         mean <- rep(0, length = ncol(x))
+     }
+     if (missing(sigma)) {
+         sigma <- diag(ncol(x))
+     }
+     if (NCOL(x) != NCOL(sigma)) {
+         stop("x and sigma have non-conforming size")
+     }
+     if (!isSymmetric(sigma, tol = sqrt(.Machine$double.eps),
+         check.attributes = FALSE)) {
+         stop("sigma must be a symmetric matrix")
+     }
+     if (length(mean) != NROW(sigma)) {
+         stop("mean and sigma have non-conforming size")
+     }
+     distval <- mahalanobis(x, center = mean, cov = sigma)
+     logdet <- sum(log(eigen(sigma, symmetric = TRUE, only.values = TRUE)$values))
+     logretval <- -(ncol(x) * log(2 * pi) + logdet + distval)/2
+     if (log)
+         return(logretval)
+     exp(logretval)
+ }
> 
> m <- 3
> sigma <- diag(3)
> sigma[2,1] <- sigma[1, 2] <- 3/5 ; sigma[3,1] <- sigma[1, 3] <- 1/3
> sigma[3,2] <- sigma[2, 3] <- 11/15
> adaptIntegrate(dmvnorm, lower=rep(-0.5, m), upper=c(1,4,2),
+                         mean=rep(0, m), sigma=sigma, log=FALSE,
+                maxEval=10000)
$integral
[1] 0.3341125

$error
[1] 4.185435e-06

$functionEvaluations
[1] 10065

$returnCode
[1] 0

MC.simple.est <- function(g, a, b, n=1e4) {

  xi <- runif(n,a,b)      # step 1

  g.mean <- mean(g(xi))   # step 2

  (b-a)*g.mean            # step 3

}

runif(n,a,b)»n random sampling from uniform(a,b) distribution 

g <- function(x) exp(-x)

MC.simple.est(g, 2, 4)

 [1] 0.1161926

> MC.simple.est <- function(g, a, b, n=1e4) {
+   xi <- runif(n,a,b)      # step 1
+   g.mean <- mean(g(xi))   # step 2
+   (b-a)*g.mean            # step 3
+ }
> g <- function(x) 1/log(x)
> 
> MC.simple.est(g, 2, 4)
[1] 1.922819
> integrate(g,2,4)
1.922421 with absolute error < 7.2e-14
>