****************
Basic Operations
****************

.. highlight:: r

.. _basicOps:

::
  
  > a <- c(7+4,7-4,7*4,7/4) 	# elemental arithmetic operations
  > a
  [1] 11.00  3.00 28.00  1.75

  > length(a)			# return vector length
  [1] 4
  
  > c(min(a),max(a))		# calculate minimum and maximum value of the vector
  [1]  1.75 28.00

  > which.min(a)		# determine the location (index) of the minimum
  [1] 4

  > which.max(a)		# determine the location (index) of the maximum
  [1] 3

  > sort(a)			# sort vector values
  [1]  1.75  3.00 11.00 28.00

  > sum(a)			# calculate sum of all vector values
  [1] 43.75

  > cumsum(1:10)                # calculate cumulative sum
  [1]  1  3  6 10 15 21 28 36 45 55

  > cumprod(1:5)                # calculate cumulative product
  [1]     1     2     6    24   120   720  5040 40320
  
  > mean(a)			# calculate the mean value
  [1] 10.9375

  > median(a)			# calculate the median value
  [1] 7

  > var(a)			# calculate the variance
  [1] 146.1823

  > sd(a)			# calculate the standard deviation
  [1] 12.09059

  > quantile(a, 0.25)		# calculate first quantile (prob=25%)
     25% 
  2.6875 

There is a command to get basic statistical information in a simple way:

::

  > summary(a)
      Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    1.750   2.688   7.000  10.940  15.250  28.000 

Some important mathematical functions are ``exp()``, ``sin()``, ``cos()``, ``tan()``, 
``log()``, ``log10()``,...

::

  > ?Trig			# show information about trigonometric functions
  > ?exp			# help about 'exp()' function

R also includes Special functions of Mathematics: ``beta(a,b)``, ``gamma(x)``, ...

::

  > ?Special			# help about Special mathematical functions
  

Operations in R can be *vectorized* helping to improve the code readability and efficiency:

::

  > a <- seq(10,30,10)
  > b <- seq(1:3)
  > a + b			# makes the sum of two vectors
  [1] 11 22 33
  
  > a * b			# vector product
  [1] 10 40 90
  > a / b                       # vector division
  [1] 10 10 10

  > a > 5			# logical operations
  [1] TRUE TRUE TRUE
  > b == 2
  [1] FALSE  TRUE FALSE

The vectorization can be also performed over matrices:

::

  > m1 <- matrix(1:9, 3, 3)	# 3 x 3 matrix definition
  > m1
      [,1] [,2] [,3]
  [1,]    1    4    7
  [2,]    2    5    8
  [3,]    3    6    9
  
  > m2 <- matrix(11:19, 3, 3)	# 3 x 3 matrix definition
  > m2
      [,1] [,2] [,3]
  [1,]   11   14   17
  [2,]   12   15   18
  [3,]   13   16   19

  > m1 * m2 			# element-wise matrix multiplication
       [,1] [,2] [,3]
  [1,]   11   56  119
  [2,]   24   75  144
  [3,]   39   96  171
  
  > m1 %*% m2 			# true matrix multiplication
       [,1] [,2] [,3]
  [1,]  150  186  222
  [2,]  186  231  276
  [3,]  222  276  330

  
**Examples:**

.. hidden-code-block:: r
    :starthidden: True
    :label: --- SHOW/HIDE ---

    # Some additional examples of matrix algebra:

    > v1 <- c(1,3,5)
    > v2 <- c(2,-1,3)

    > v1%*%v2                   # inner product
         [,1]
    [1,]   14

    > sqrt(v1%*%v1))            # vector modulus
            [,1]
    [1,] 5.91608

    > v1%o%v2                   # outer product
         [,1] [,2] [,3]
    [1,]    2   -1    3
    [2,]    6   -3    9
    [3,]   10   -5   15
    
    
    # Solving the system:
    #  4 x +   y  -2 z = 0
    #  2 x - 3 y + 3 z = 9
    # -6 x - 2 y +   z = 0

    > A <- matrix(c(4,2,-6,1,-3,-2,-2,3,1),nrow=3)  # system matrix
    > A
         [,1] [,2] [,3]
    [1,]    4    1   -2
    [2,]    2   -3    3
    [3,]   -6   -2    1

    > b <- c(0,9,0)
    > b
    [1] 0 9 0

    > x <- solve(A,b)           # computing vector x in A x = b
    > x
    [1]  0.75 -2.00  0.50

    > A%*%x                     # checking the solution
                  [,1]
    [1,]  1.110223e-16          # = 0 (within rounding error)
    [2,]  9.000000e+00          # = 9
    [3,] -5.551115e-17          # = 0 (within rounding error)