R Programming/Maths

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• Help for simple maths :

?Arithmetic

• Power function

> 10^3
[1] 1000
> 10**3
[1] 1000

• log / exp

> log(2.71)
[1] 0.9969486
> log10(10)
[1] 1
> exp(1)
[1] 2.718282

• Modulo and integer division (i.e. euclidean division)

> 5%%2
[1] 1
>5%/%2
[1] 2

• Factorial

> factorial(3)
[1] 6
> prod(1:3)
[1] 6

• The number of combination of length k within n numbers :

choose(3,2)

• Union and intersection

> union(1:10,5:7)
 [1]  1  2  3  4  5  6  7  8  9 10
> intersect(1:10,5:7)
[1] 5 6 7

Contents 1 Matrices 2 Matrix operators 2.1 Solving a linear equation 3 Numerical approximation 3.1 Derivation 3.2 Integration 4 Symbolic calculus 5 Solving polynomial equations 6 See also

[edit] Matrices

If you want to create a new matrix, one way is to use the matrix() function. You have to enter a vector of data, the number of rows and/or columns and finally you can specify if you want R to read your vector by row or by column (the default option). Here are two examples.

> matrix(data = NA, nrow = 5, ncol = 5, byrow = T)
     [,1] [,2] [,3] [,4] [,5]
[1,]   NA   NA   NA   NA   NA
[2,]   NA   NA   NA   NA   NA
[3,]   NA   NA   NA   NA   NA
[4,]   NA   NA   NA   NA   NA
[5,]   NA   NA   NA   NA   NA

> matrix(data = 1:15, nrow = 5, ncol = 5, byrow = T)
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    2    3    4    5
[2,]    6    7    8    9   10
[3,]   11   12   13   14   15
[4,]    1    2    3    4    5
[5,]    6    7    8    9   10

> v1 <- 1:5
> v2 <- 5:1
> v2
[1] 5 4 3 2 1
> cbind(v1,v2)
     v1 v2
[1,]  1  5
[2,]  2  4
[3,]  3  3
[4,]  4  2
[5,]  5  1

> rbind(v1,v2)
   [,1] [,2] [,3] [,4] [,5]
v1    1    2    3    4    5
v2    5    4    3    2    1

The dimension of a matrix can be obtained using the dim() function or alternatively nrow() and ncol()

> dim(X)
[1] 5 5
> nrow(X)
[1] 5
> ncol(X)
[1] 5

• Inner product X%*%Y
• Kronecker product X%x%Y
• Outer Product X%o%Y

[edit] Matrix operators

• Invert a matrix

> M <- cbind(c(1,0,1),c(0,1,2),c(0,0,1))
> solve(M)
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]   -1   -2    1
> solve(M)%*%M
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1

• Extract the diagonal

> diag(M)
[1] 1 1 1

• Transpose the matrix

> t(M)
     [,1] [,2] [,3]
[1,]    1    0    1
[2,]    0    1    2
[3,]    0    0    1

• Eigenvalues and eigenvectors

> eigen(M)
$values
[1] 1 1 1

$vectors
     [,1]          [,2]          [,3]
[1,]    0  2.220446e-16  0.000000e+00
[2,]    0  0.000000e+00  1.110223e-16
[3,]    1 -1.000000e+00 -1.000000e+00

> b=matrix(nrow=2,ncol=2,c(1,2,3,4))
> a=matrix(nrow=2,ncol=2,c(1,0,0,-1))
> a
     [,1] [,2]
[1,]    1    0
[2,]    0   -1
> b
     [,1] [,2]
[1,]    1    3
[2,]    2    4
> a%*%b
     [,1] [,2]
[1,]    1    3
[2,]   -2   -4
> b%*%a
     [,1] [,2]
[1,]    1   -3
[2,]    2   -4
> 

[edit] Solving a linear equation

> m=matrix(nrow=2,ncol=2,c(1,-.8,1,.2))
> m
     [,1] [,2]
[1,]  1.0  1.0
[2,] -0.8  0.2
> 
> l=matrix(c(1.0+25.0/18,25.0/18.0))
> l
         [,1]
[1,] 2.388889
[2,] 1.388889
> 
> k=solve(m,l)
> k
           [,1]
[1,] -0.9111111
[2,]  3.3000000
> 
> m%*%k          #checking the answer
         [,1]
[1,] 2.388889
[2,] 1.388889
> 

[edit] Numerical approximation

[edit] Derivation

• numDeriv package

[edit] Integration

R can perform one dimensional integration. For example we can integrate over the density of the normal distribution between and

> integrate(dnorm,-Inf,Inf)
1 with absolute error < 9.4e-05
> integrate(dnorm,-1.96,1.96)
0.9500042 with absolute error < 1.0e-11
> integrate(dnorm,-1.64,1.64)
0.8989948 with absolute error < 6.8e-14
# we can also store the result in an object
> ci90 <- integrate(dnorm,-1.64,1.64)
> ci90$value
[1] 0.8989948
> integrate(dnorm,-1.64,1.64)$value
[1] 0.8989948

see the adapt package for multivariate integration.

> library(adapt)
> ?adapt
> ir2pi <- 1/sqrt(2*pi)
> fred <- function(z) { ir2pi^length(z) * exp(-0.5 * sum(z * z))}
> 
> adapt(2, lo = c(-5,-5), up = c(5,5), functn = fred)
       value       relerr       minpts       lenwrk        ifail 
    1.039222 0.0007911264          231           73            0 
> adapt(2, lo = c(-5,-5), up = c(5,5), functn = fred, eps = 1e-4)
       value       relerr       minpts       lenwrk        ifail 
    1.000237 1.653498e-05          655          143            0 
> adapt(2, lo = c(-5,-5), up = c(5,5), functn = fred, eps = 1e-6)
      value      relerr      minpts      lenwrk       ifail 
   1.000039 3.22439e-07        1719         283           0 

[edit] Symbolic calculus

R can give the derivative of a function.

> D(expression(x^n),"x")
x^(n - 1) * n
> D(expression(exp(a*x)),"x")
exp(a * x) * a
> D(expression(1/x),"x")
-(1/x^2)

> D(expression(x^3),"x")
3 * x^2

[edit] Solving polynomial equations

To solve , where are given numbers, use the command

polyroot(c(n,...,b,a))

So, for example, to calculate the roots of the equation 2x² − 5x − 3 = 0 one would do as follows:

> polyroot(c(-3,-5,2))
[1] -0.5+0i  3.0-0i

and the solution can be read to be

[edit] See also

The following command gives help on special mathematical functions related to the beta and gamma functions.

?Special