Introduction to Julia Programming: For Scientists and Engineers (Open Source Computing Book 5) by Sandeep Nagar

Author:Sandeep Nagar [Nagar, Sandeep]
Language: eng
Format: epub, pdf
Published: 2017-06-07T07:00:00+00:00

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0.991414

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7.14.1 Other mathematical operators

A separate chapter (number 8) has been dedicated to explain how mathemati-

cal functions can be operated on arrays and its elements. This chapter is critical

for numerical experimentation as most of the data is converted into a matrix

(stored in computer memory) and mathematical functions are used to define a

transformation equation. This transformation equation operated on input ma-

trix and results into a new matrix (called transformed matrix). Simulating a real

system involved defining transformation equations. These transformed matrices

are converted back to original form of data for visualization and interpretation.

For this reason abilities of julia around speedy matrix transformation in a flexi-

ble manner, must be understood in an elaborate manner so that user can judge

correctly about choosing and then defining particular mathematical functions

in a right manner.

7.15 Set theory and arrays

The Array data types can be treated to equivalent to a mathematical set

too. The set operations like [ (Union) given by in-built function union(), \

(Intersection) given by in-built function intersect() and set difference can

be calculated. Union operation collects the unique occurrence of an element

of both sets. Intersection collects common elements from both sets and set-

difference (setdiff(A-B)) collects those elements which are present in A but

not in B.

julia> A = [1,2,3,4,-1,-3]

6-element Array{Int64,1}:

1

2

3

4

-1

-3

julia> B = [2,4,1,3,1,10]

6-element Array{Int64,1}:

2

4

137

1

3

1

10

julia> union(A,B)

7-element Array{Int64,1}:

1

2

3

4

-1

-3

10

julia> intersect(A,B)

4-element Array{Int64,1}:

1

2

3

4

julia> setdiff(A,B)

2-element Array{Int64,1}:

-1

-3

7.16 Summary

Arrays make the backbone of matrix computations which has enabled usage

of computers in area of mathematics. Vectorizing a problems lets computers

deal with complex tasks within a computing machine and this in-turn lets one

approximate a solution faster than achieving exact analytical solutions. Dy-

namically defining and manipulating arrays within variety of data types, makes

julia a good option for numerical computing. Fast operation is the key to julia’s

preference in this area. Ease of defining vectorization of operations lets julia

work on arrays as matrices in a flexible manner. Effectively managing copying,

sorting and generating arrays using comprehensions makes julia a good choice

for matrix based mathematical methods to solve physical problems.

138

8

Arrays for matrix operations

8.1 Defining an array

A julia array is equivalent to a mathematical matrix because just like a julia

array, a matrix is also a ordered collection of numbers. The simplest case for

matrix is the one storing components of a 3D vector. Foe example, a vector

~a = 2î + 3ˆj

4ˆ

k can also be represented as either a row matrix as:

⇥

⇤

2

3

4

or a column matrix as:

2

3

2

4 3 5

4

In both cases, the numbers 2, 3, 4 are ordered in a fashion. Now this matrix

can be represented by an array in julia as:

julia> A = [2,3,-4]

3-element Array{Int64,1}:

2

3

-4

julia> size(A)

(3,)

julia> A'

1x3 Array{Int64,2}:

2

3

-4

139

julia> size(A')

(1,3)

julia> (A')'

3x1 Array{Int64,2}:

2

3

-4

julia> size((A')')

(3,1)

• A creates a 1D array object (having only one index).

– This is not equivalent to a mathematical matrix as a matrix element

necessarily must have atleast 2 indices.

– For some practical purpose, this can be used to a vector.

– This object is mostly used to represent a sequence or series of num-

bers.

• A' creates a 1 ⇥ 3 2D array object.

– This is equivalent to a column matrix.

– Each element has two indices, one depicting row and other depicting

column.

• (A')' creates a 3 ⇥ 1 2D array object.

– This is equivalent to a row matrix.

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