Functional Programming In Mathematica

Functional Programming
in Mathematica
An Introduction

Hossam Karim
ITWorx

2 FunctionalProgramming-Egypt-2010.nb

About
Work for an Egyptian Software Services Company
Design software for the Bioinformatics and Telecommunications industries
Use Mathematica every day
Implement prototypes and proof of concepts
Design and implement solutions and algorithms
Validate implementations' feasibility, performance and correctness

Code, concepts and algorithms provided in this presentation are not related or affiliated by any means to my employer nor my clients. All the material in this presentations
are samples not suitable for production, and are provided for the sole purpose of demonstration.

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FunctionalProgramming-Egypt-2010.nb 3

The Function

Functions in Mathematics

Function Definition

A Function f is a mapping from the domain X to the co-domain Y and is defined by

f : X Y

The curve y 2 x4 x2 1 can be defined as function by the triple notation

, , x, x4 x2 1 : x (1)

where is the domain of real numbers and also the co-domain, alternatively,

f: , x x4 x2 1 (2)

A more common notation is

f x x4 x2 1 (3)

Plotting a Function
Mathematica supports both function and curve plotting


Function plot

In[1]:= Plot x4 x2 1 , x4 x2 1 , x, 3, 3 , PlotRange 3,
AspectRatio 1, PlotStyle Black, Thick , AxesLabel x, y

y
3

2

1

Out[1]=
x
3 2 1 1 2 3

1

2

3


Curve Plot

In[2]:= ContourPlot y2 x4 x2 1, x, 3, 3 , y, 3, 3 ,
Axes True, Frame False, ContourStyle Thick , AxesLabel x, y

y

3

2

1

Out[2]=
x
3 2 1 1 2 3

1

2

3

Functional Programming Languages
Functional Programming Languages has always been the natural choice for implementing computer derived solutions for mathematical
problems including Numerical Analysis, Combinatorics and Algorithms

Haskell

Implementation of the Quick Sort algorithm in Haskell. Demonstrates polymorphic types, pattern matching, list comprehension and
immutability

qsort :: Ord a => [a] -> [a]
qsort [] = []
qsort (p:xs) = qsort lesser ++ [p] ++ qsort greater
where
lesser = [ y | y <- xs, y < p ]
greater = [ y | y <- xs, y >= p ]

Scala

Implementation of the Quick Sort algorithm in Scala. Demonstrates polymorphic types, pattern matching, object-oriented support and partial
functions application

def qsort[T <% Ordered[T]](list:List[T]):List[T] = {
list match {
case Nil Nil
case p::xs
val (lesser,greater) = xs partition (_ <= p)
qsort(lesser) ++ (p :: qsort(greater))
}
}


def qsort[T <% Ordered[T]](list:List[T]):List[T] = {
list match {
case Nil Nil
case p::xs
val (lesser,greater) = xs partition (_ <= p)
qsort(lesser) ++ (p :: qsort(greater))
}
}

Mathematica

Implementation of the Quick Sort algorithm in Mathematica. Demonstrates pattern matching, pattern guards and rule based programming

In[3]:= ClearAll qsort ;
qsort : ;
qsort p_, xs___ :
Cases xs , y_ ; y p , Cases xs , y_ ; y p .
lesser_, greater_
qsort lesser , p, qsort greater Flatten

In[6]:= qsort 0, 9, 1, 8, 3, 6, 2, 7, 5, 4

Out[6]= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

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Functions in Mathematica

Numeric Computations

Numeric computations through function calls

In[210]:= Plus 1, Exp Times I , Pi

Out[210]= 0

Or through Mathematical notation

Π
In[8]:= 1

Out[8]= 0

Arbitrary precision

In[9]:= N Π, 100

Out[9]= 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068

Calculus

Symbolic Calculus

1
In[10]:= Θ
Sin Θ

1 Π
Out[10]= 2 EllipticF Θ , 2
2 2

Accurate results

In[11]:= Θ FullSimplify

1
Out[11]=
Sin Θ

Algorithms

Optimized algorithms


In[12]:= Timing Sort RandomInteger 1, 10 000 000 , 100 000 Short

Out[12]//Short=
0.04, 9, 400, 418, 657, 719, 859, 947, 1057, 1061, 99 982 ,
9 998 951, 9 998 953, 9 999 009, 9 999 077, 9 999 123, 9 999 227, 9 999 236, 9 999 314, 9 999 497

Algorithm analysis

2n
In[13]:= T n . RSolve T n T 1, T 1 1 , T n , n
3

Log n
Out[13]= 1
3
Log
2

Graphics

Stunning graphics

In[236]:=
ParametricPlot3D Cos Φ Sin Θ , Sin Φ Sin Θ , Cos Θ , Φ, 0, 2 Π ,
Θ, 0, Π , PlotPoints 100, Mesh None, ColorFunction x, y, z, Φ, Θ Hue ,
ColorFunctionScaling False, Boxed False, Axes False Magnify , .5 & &
Θ Φ Θ Φ
Sin Θ Cos Θ , Sin Φ Cos Φ , Sin Θ Φ Cos Θ Φ , , ,
Partition , 3 & Grid

Out[236]=

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Define Your Own Function

A Function as a Rule

Functions can be defined as a delayed assignment rule

In[15]:= f x_ : x3 x 1

Functions can accept multiple parameters

In[16]:= f x_, a_, b_ : x3 ax b


Calling a Function

Default Form

Default Form, notice the square brackets

In[17]:= f 2

Out[17]= 7

Prefix Form

Prefix Form, using the @ sign

In[18]:= f 2

Out[18]= 7

Postfix Form

Postfix Form, using the // sign

In[19]:= 2Π Sin

Out[19]= 0

Infix Form

Infix Form, function surrounded by ~ sign

In[20]:= 1, 2, 3 Join 4, 5, 6

Out[20]= 1, 2, 3, 4, 5, 6

Multiple Arguments

Function call with multiple actual arguments


In[223]:= Magnify ChemicalData "Ethanol", "MoleculePlot" , 1

Out[223]=

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Domains and Patterns

Domains as Patterns

Domains can be specified as patterns

In[22]:= ClearAll square ;
square i_Integer : i2
2
square i_Real : Floor i
2
square i_Complex : Re i
2
square a_Symbol : a
square SomeDataType a_ : a2

In[28]:= square 2.1 , square 2 , square 2 3 , square x , square SomeDataType x

Out[28]= 4, 4, 4, x2 , x2

Patterns and Lists

Expressive patterns on lists, notice how ordering is important

One or more __ Zero or more ___
2 3

In[29]:= ClearAll listOp ;
listOp :
listOp x_, y_ : y, x flip a tuple
listOp x_, xs__ : xs drop the first element
listOp l : __ ... : Reverse l match a list of lists

In[34]:= listOp 1, 2, 3 , listOp 1, 2 , listOp 1, 2 , 3, 4

Out[34]= 2, 3 , 2, 1 , 3, 4 , 1, 2

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Guards on Patterns

Guards on Domains

Guards can be specified using the Pattern ? Predicate notation

In[35]:= ClearAll collatz ;
collatz n_Integer ? EvenQ : n 2 Matches only even integers
collatz n_Integer : 3 n 1 Matches all integers

In[38]:= collatz 3 , collatz 4

Out[38]= 10, 2

Arbitrary Conditions

A condition can be specified on a pattern using Pattern /; Predicate notation

Example : Bubble Sort

By Dr Jon D Harrop, 2010 (Modified)

Using pattern matching and conditions on patterns, bubble sort can then be defines as

In[39]:= bubbleSort xs___, x_, y_, ys___ : bubbleSort xs, y, x, ys ; x y

For example

In[40]:= bubbleSort 3, 2, 1

Out[40]= bubbleSort 1, 2, 3

We can simply extract the sorted list using the rule

In[211]:= bubbleSort 3, 2, 1 . _ sorted_ sorted

Out[211]= 1, 2, 3

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Pure Functions

Defining a Pure Function

The Notation

A function that squares its argument

In[41]:= x x2

2
Out[41]= Function x, x

Square function applied at 3

In[42]:= x x2 3

Out[42]= 9

A function with a list as its argument

In[43]:= x, y x2 y2 3, 4

Out[43]= 5

The (# &) Notation

The notation is programmatically equivalent to the notation

Square function applied at 3

2
In[44]:= & 3

Out[44]= 9

A pure function with 2 arguments, notice the numbering after #

In[45]:= 12 22 & 3, 4

Out[45]= 5


Higher-Order Functions

Map

Map as a function call

In[46]:= Map x x2 , a, b, c, d

Out[46]= a2 , b2 , c2 , d2

Using the ( /@ ) Notation

In[47]:= x x2 a, b, c, d

Out[47]= a2 , b2 , c2 , d2

The mapped function can be composed of any type of expression

In[48]:= Mean , Variance , PDF , x &
NormalDistribution Μ, Σ , MaxwellDistribution Σ , GammaDistribution Α, Β

x2
x Μ 2 2 x
2 Σ2 x2
2 Σ2 2 8 3 Π Σ2 Π Β x 1 Α Β Α
2 2
Out[48]= Μ, Σ , , 2 Σ, , , Α Β, Α Β ,
2Π Σ Π Π Σ3 Gamma Α

Or a composed function

In[215]:= ChemicalData "Caffeine", Magnify , .5 & &
"CHColorStructureDiagram", "CHStructureDiagram", "ColorStructureDiagram", "StructureDiagram",
"MoleculePlot", "SpaceFillingMoleculePlot" Partition , 3 & Grid , Frame All &

H H
O H O H
C C O
H H H H
H H
C C C C
N N
N C N C N
H H N
C H C H
C C C C
N N
O N O N N
O N

H C H H C H

H H

Out[215]=
O

N
N

N
O N

Or used inside a manipulation


Or used inside a manipulation

In[50]:= ClearAll tux, browsers ;

tux, browsers , , , ;

Manipulate
Map
ImageCompose tux, ImageResize , Scaled scale , horizontal, vertical &, browsers
GraphicsRow,
scale, 0.5, 1 ,
horizontal, 60, 200, 10 ,
vertical, 60, 200, 10

scale

horizontal

vertical

Out[52]=

Select

In[53]:= Select 1, 3, 2, 5, 0 , n 0 n

Out[53]= 3, 5

Fold

In[54]:= Fold x, y x y, 0, a, b, c, d

Out[54]= a b c d

Folding to the left or to the right


In[218]:= Manipulate

Fold F 1, 2 &, x, Take a, b, c, d , step
TreeForm , AspectRatio 1.2, PlotLabel "Fold Left" &,
Fold F 2, 1 &, x, Take a, b, c, d , step
TreeForm , AspectRatio 1.2, PlotLabel "Fold Right" &
GraphicsRow Magnify , 1 &,
step, 0, 4, 1

step

Fold Left Fold Right
F F

Out[218]= F d d F

F c c F

F b b F

x a a x

Power Set

In[56]:= Fold set, element set Append element & set , , Α, Β, Γ

Out[56]= , Α , Β , Γ , Α, Β , Α, Γ , Β, Γ , Α, Β, Γ

One more time

In[57]:= FoldList set, element set Append element & set , , Α, Β, Γ Column

, Α
Out[57]= , Α , Β , Α, Β
, Α , Β , Γ , Α, Β , Α, Γ , Β, Γ , Α, Β, Γ

NestWhileList

x
In[58]:= NestWhileList x , 32, i i 1
2

Out[58]= 32, 16, 8, 4, 2, 1

Pascal Triangle

Pascal triangle can be defined using binomials


In[59]:= Table Binomial n, k , n, 0, 4 , k, 0, n Column , Center &

1
1, 1
Out[59]= 1, 2, 1
1, 3, 3, 1
1, 4, 6, 4, 1

This can defined using the pattern

In[60]:= tuples x_ :
tuples x_, y_, ys___ : x y Join tuples y, ys
pascal h_ : NestWhileList 1 Join tuples Join 1 &, 1 , Length h &
pascal 9 Column , Center &

1
1, 1
1, 2, 1
1, 3, 3, 1
Out[63]= 1, 4, 6, 4, 1
1, 5, 10, 10, 5, 1
1, 6, 15, 20, 15, 6, 1
1, 7, 21, 35, 35, 21, 7, 1
1, 8, 28, 56, 70, 56, 28, 8, 1

And nicely manipulated,

In[64]:= ClearAll hexagon, render ;
2Πk 2Πk
hexagon x_, y_ : Polygon Table Sin x, Cos y , k, 6
6 6
Length l
render l_List : Graphics Hue , hexagon 0, 0 , Text Style , White, Bold, 10 & l
Π
GraphicsRow , ImageSize 50 Length , 30 , Spacings 2, 0 &
Manipulate
Graphics render & pascal h GraphicsColumn , Alignment Center, Spacings 0, 8 &
Magnify , 1.5 &,
h, 1, 5, 1

h

1

1 1
Out[67]=

1 2 1

1 3 3 1

1 4 6 4 1

3 n + 1 Problem


3 n + 1 Problem

n 2 EvenQ n
In[68]:= collatz : n ∂
3n 1 OddQ n
NestWhileList collatz, 200, m m 1

Out[69]= 200, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1

In[70]:= Manipulate
MapIndexed
Text Style 1, Blue, Italic, 45 2 &, NestWhileList collatz, x, m m 1 ,
x, 10, 100, 10

x

Out[70]= 70 35 106 53 160
, , , , ,

80 40 20 10 5 16 8 4 2 1
, , , , , , , , ,

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Example: Binary Search Tree

Haskell

Haskell approach to Algebraic Data Types and Pattern Matching

data Tree a =
Empty | Node (Tree a) a (Tree a) deriving(Show, Eq)
insert :: Ord a => a -> Tree a -> Tree a
insert n Empty = Node Empty n Empty
insert n (Node left x right)
| n < x = Node (insert n left) x right
| otherwise = Node left x (insert n right)
inorder :: Ord a => Tree a -> [a]
inorder Empty = []
inorder (Node left x right) =
inorder left ++ [x] ++ inorder right
bst :: Ord a => [a] -> Tree a
bst [] = Empty
bst (x : xs) = foldr(insert) (insert x Empty) xs

Mathematica

The same algorithm in Mathematica syntax

In[71]:= ClearAll tree, insert, inorder, bst ;
tree nil node _, _, _ ;

insert n_, nil : node nil, n, nil ;
insert n_, node l_, x_, r_ ; n x : node insert n , l , x , r ;
insert n_, node l_, x_, r_ : node l, x, insert n, r ;

inorder nil : ;
inorder node l_, x_, r_ : inorder l Join x Join inorder r ;

bst : nil
bst x_, xs___ : Fold insert 2, 1 & , insert x, nil , xs ;

In[80]:= inorder bst 8, 3, 10, 1, 6, 9, 12, 4, 7, 13, 11

Out[80]= 1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13

Visualizing a Binary Search Tree

In order to be able to visualize a BST, we need to convert the tree of nodes into a {src dst, ...} representation. Using our structure, the
binary search tree of the list {8, 3, 10, 1, 6} is

In[81]:= bst 8, 3, 10, 1, 6

Out[81]= node node node nil, 1, nil , 3, node nil, 6, nil , 8, node nil, 10, nil

We can then define


In[82]:= ClearAll tuple ;
tuple nil :
tuple node nil, x_, nil :
tuple node l : node _, a_, _ , x_, nil : tuple l Join x a
tuple node nil, x_, r : node _, b_, _ : x b Join tuple r
tuple node l : node _, a_, _ , x_, r : node _, b_, _ :
tuple l Join x a, x b Join tuple r

Then

In[88]:= tuple bst 8, 3, 10, 1, 6, 9, 12, 4, 7, 13, 11

Out[88]= 3 1, 3 6, 6 4, 6 7, 8 3, 8 10, 10 9, 10 12, 12 11, 12 13

Plotting the BST

In[89]:= TreePlot tuple bst 8, 3, 10, 1, 0, 4, 6, 5, 9, 12, 2, 7, 13, 11 ,
Automatic, 8, VertexLabeling True, DirectedEdges True,
PlotLabel " 8,3,10,1,0,4,6,5,9,12,2,7,13,11 ", VertexRenderingFunction
ps, v White, EdgeForm Black, Thick , Disk ps, .2 , Black, Text v, ps

8,3,10,1,0,4,6,5,9,12,2,7,13,11

8

3 10

Out[89]=
1 4 9 12

0 2 6 11 13

5 7

We can then visualize a random BST construction step by step


In[214]:= With l RandomInteger 1, 100 , 10 DeleteDuplicates ,
Manipulate

Text Style Framed l , 12, Bold, Black ,
TreePlot
tuple bst Take l, step ,
Automatic, l 1 , VertexLabeling True, DirectedEdges True, VertexRenderingFunction
ps, v White, EdgeForm Black, Thick , Disk ps, .2 , Black, Text v, ps ,
ImageSize 400, 300 , ImagePadding 1
,
Text Style Framed inorder bst Take l, step , 12, Bold, Blue
Column , Center &,
step, 2, Length l , 1

step

91, 57, 43, 99, 92, 50, 11, 48, 89

91

57 99

Out[214]=

43 92

11 50

48

11, 43, 48, 50, 57, 91, 92, 99

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Pattern Matching and Transformation

Matching Cases

Match Cases

In[91]:= Cases a2 , b3 , c4 , d5 , e6 , _2

Out[91]= a2

Match Cases with a predicate

In[92]:= Cases a2 , b3 , c4 , d5 , e6 , _n_ ; EvenQ n

Out[92]= a2 , c4 , e6

Match Cases with a predicate and a transformation rule

In[93]:= Cases a2 , b3 , c4 , d5 , e6 , x_n_ ; OddQ n xn 1

Out[93]= b4 , d6

Pattern Matching and Rules

Swap is simple

In[94]:= a, b . x_, y_ y, x

Out[94]= b, a

Decompose an expression

In[95]:= x Sin Θ y Cos Θ . a_ Sin Α_ b_ Cos Α_ a, b, Α

Out[95]= x, y, Θ

Match rules


In[96]:= ClearAll g ;
g
Μ Α, Μ Β,
Α i, Α j,
Β a, Β b
;
GraphPlot g, VertexLabeling True, AspectRatio 0.2 Magnify , 1.5 &

i b

Out[98]= Α Μ Β

j a

Use delayed rules

In[99]:= find the children of the vertex Β in the Graph g
. Β x_ x , _ & g Flatten

Out[99]= a, b

Example: Palindrome

Generate a sequence of probable palindrome integers

In[100]:= alg196 n_Integer : n IntegerDigits n Reverse FromDigits
NestWhileList alg196, 77, 10 000 000 &

Out[101]= 77, 154, 605, 1111, 2222, 4444, 8888, 17 776, 85 547, 160 105, 661 166, 1 322 332, 3 654 563, 7 309 126, 13 528 163

Recursively test if a sequence is a palindrome

In[102]:= isPalindrome seq_List : seq .
x_ True,
x_, xs___, y_ x y && isPalindrome xs

Test the sequence

In[103]:= Select NestWhileList alg196, 77, 10 000 000 & , isPalindrome IntegerDigits &

Out[103]= 77, 1111, 2222, 4444, 8888, 661 166, 3 654 563

Example: Run Length Encoding

Perform run length encoding on a finite sequence

By Frank Zizza, 1990

Use replace repeated (//.)


In[104]:= runLengthEncoding l_List : Map , 1 &, l .
head___, x_, n_ , x_, m_ , tail___ head, x, n m , tail

In[105]:= runLengthEncoding a, a, a, b, b, c, c, c, c, a, a

Out[105]= a, 3 , b, 2 , c, 4 , a, 2

How does the magic happen?

First define the magical rule

In[106]:= ClearAll rule ;
rule head___, x_, n_ , x_, m_ , tail___ head, x, n m , tail ;

Second, generate a list tuples of the form e1 , 1 , e2 , 1 , ..., en , 1

In[108]:= Map , 1 &, a, a, a, b, b, c, c, c

Out[108]= a, 1 , a, 1 , a, 1 , b, 1 , b, 1 , c, 1 , c, 1 , c, 1

Keep applying the transformation until the input is exhausted

In[109]:= . rule

Out[109]= a, 2 , a, 1 , b, 1 , b, 1 , c, 1 , c, 1 , c, 1

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Example: Mathematica in Bioinformatics

XML in Mathematica
Mathematica supports a large variety of data formats, XML happens to be one of them

In[225]:= xml Import " work presentation Mathematica Conference 2010 code xml graph.xml", "XML"

Out[225]= XMLObject Document ,
XMLElement v, id root , XMLElement v, id a , XMLElement v, id a1, cost 1 , ,
XMLElement v, id a2, cost 2 , , XMLElement v, id a3, cost 3 , ,
XMLElement v, id b , XMLElement v, id b1, cost 4 , , XMLElement v, id b2, cost 5 , ,
XMLElement v, id b3, cost 6 , , XMLElement v, id c , XMLElement v, id c1, cost 7 , ,
XMLElement v, id c2, cost 8 , , XMLElement v, id c3, cost 9 , ,

The XML document represents a graph, each vertex v is represented as an element. Children of an element are connected to the parent, the
hierarchy represents the edges. The following functions use Mathematica XML support to create a garph representation of the XML docu-
ment and plot it

In[226]:= ClearAll root, id, cost, rec ;
root XMLObject _ _, r_, _ : r
id XMLElement _, as_, ___ : "id" . as
cost XMLElement _, as__ , ___ : "cost" . as, _ "0"
rec e : XMLElement "v", _, cs : ___ :
id e id , cost & cs Join rec cs Flatten , 1 &

Recursively walk the element structure and create a Mathematica graph representation

In[231]:= rec root xml

Out[231]= root a, 0 , root b, 0 , root c, 0 , a a1, 1 , a a2, 2 ,
a a3, 3 , b b1, 4 , b b2, 5 , b b3, 6 , c c1, 7 , c c2, 8 , c c3, 9

Plot the extracted graph


In[232]:= rec root xml GraphPlot , VertexLabeling True, EdgeLabeling True & Magnify , 1 &

a1 c3

1 9
a2 c2
2 8
a c
0 0
3 root 7
Out[232]=

a3 c1
0

b
4 6
b1 5 b3

b2

Sequence Alignment

In[118]:= xml Import " work presentation Mathematica Conference 2010 code xml sequenceML.xml", "XML" ;

In[119]:= ClearAll root, sequenceList, sequence, element ;
root XMLObject _ _, r_, _ : r
sequenceList XMLElement "sequenceML", _, seqs_ : sequence seqs
sequence e : XMLElement "sequence", "seqID" id_ , cs_ :

seqID id,
name element "name", cs ,
description element "description", cs ,
aminoAcidSequence element "aminoAcidSequence", cs

element name_, elements_ :
Cases elements, XMLElement n_, , value_ ;n name value First

In[124]:= root xml sequenceList First

Out[124]= seqID gi 58374180 gb AAW72226.1 , name HA,
description Influenza A virus A duck Shandong 093 2004 H5N1 , aminoAcidSequence
MEEIVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFINVPEWSYIVEKANPAND
LCYPGDFNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPYNGKSSFFRNVVWLIKKNSSYPTIKRSYNNTNQEDLLILWGIHHPNDAAE
QTKLYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPEYAYKIVKKGDSAIMKSELEYGNCNTKCQTPMGA
INSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNTPQRERRRKKRGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKAIDGVTNKV
NSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMENERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEFYHKCDNECME
SVKNGTYDYPRYSEEARLNREEISGVKLESMGTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI


In[125]:= first, last root xml sequenceList . x_, ___, y_ x, y

Out[125]= seqID gi 58374180 gb AAW72226.1 , name HA,
description Influenza A virus A duck Shandong 093 2004 H5N1 , aminoAcidSequence
MEEIVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFINVPEWSYIVEKANPA
NDLCYPGDFNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPYNGKSSFFRNVVWLIKKNSSYPTIKRSYNNTNQEDLLILWGIHHPN
DAAEQTKLYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPEYAYKIVKKGDSAIMKSELEYGNCNTKC
QTPMGAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNTPQRERRRKKRGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKA
IDGVTNKVNSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMENERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEF
YHKCDNECMESVKNGTYDYPRYSEEARLNREEISGVKLESMGTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI ,
seqID gi 108671045 gb ABF93441.1 , name hemagglutinin,
description Influenza A virus St Jude H5N1 influenza seed virus 163222 , aminoAcidSequence
MEKIVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFLNVPEWSYIVEKINPA
NDLCYPGNFNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPYQGRSSFFRNVVWLIKKNNAYPTIKRSYNNTNQEDLLVLWGIHHPN
DAAEQTRLYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPENAYKIVKKGDSTIMKSELEYGNCNTKC
QTPIGAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNSPQIETRGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKAIDGV
TNKVNSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMENERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEFYHRC
DNECMESVRNGTYDYPQYSEEARLKREEISGVKLESIGTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI

In[126]:= SequenceAlignment aminoAcidSequence . first, aminoAcidSequence . last

Out[126]= ME, E, K , IVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEF,
I, L , NVPEWSYIVEK, A, I , NPANDLCYPG, D, N , FNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPY,
N, Q , G, K, R , SSFFRNVVWLIKKN, SS, NA , YPTIKRSYNNTNQEDLL, I, V , LWGIHHPNDAAEQT, K, R ,
LYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPE, Y, N , AYKIVKKGDS, A, T ,
IMKSELEYGNCNTKCQTP, M, I , GAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRN, T, S , PQ, R, I , E, RRRKK, T ,
RGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKAIDGVTNKVNSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMEN
ERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEFYH, K, R , CDNECMESV, K, R , NGTYDYP,
R, Q , YSEEARL, N, K , REEISGVKLES, M, I , GTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI

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Example: Mathematica XQuery

XQuery
A fluent XML Query Language from W3C

XQuery FLWOR Expression

An XQuery expression is typically a for, let, where, order by and return construct

for $x in doc("books.xml")/bookstore/book
where $x/price>30
order by $x/title
return $x/title

XML in Mathematica

Mathematica XML Support

Import the XML document

In[127]:= xml Import " work presentation Mathematica Conference 2010 xml books.xml", "XML"

Out[127]= XMLObject Document XMLObject Declaration Version 1.0, Encoding ISO 8859 1 ,
XMLElement bookstore, , XMLElement book, category COOKING ,
XMLElement title, lang en , Everyday Italian , XMLElement author, , Giada De Laurentiis ,
XMLElement year, , 2005 , XMLElement price, , 30.00 ,
XMLElement book, category CHILDREN , XMLElement title, lang en , Harry Potter ,
XMLElement author, , J K. Rowling , XMLElement year, , 2005 , XMLElement price, , 29.99 ,
XMLElement book, category WEB , XMLElement title, lang en , XQuery Kick Start ,
XMLElement author, , James McGovern , XMLElement author, , Per Bothner ,
XMLElement author, , Kurt Cagle , XMLElement author, , James Linn , XMLElement author,
, Vaidyanathan Nagarajan , XMLElement year, , 2003 , XMLElement price, , 49.99 ,
XMLElement book, category WEB , XMLElement title, lang en , Learning XML , XMLElement
author, , Erik T. Ray , XMLElement year, , 2003 , XMLElement price, , 39.95 ,

XQuery like DSL in Mathematica
Define an XQuery like Domain Specific Language (DSL) for XML processing using Mathematica's Functional approach

The tiny language is a set of higher - order functions, each function returns a function that can act on the XML axis


In[128]:= ClearAll doc, where, orderBy, e, att, attribute, data, return ;
doc XMLObject Document _, root_, _ : root;
where Select;
orderBy Sort;
e n_String : es Cases es, XMLElement n , _, _ , ;
att XMLElement _ , rules_, _ , n_String : n . rules Join _ Φ ;
attribute XMLElement _ , rules_, _ , n_String : n n . rules ;
attribute n_String : es attribute , n & es;
attribute n_String, pred_ : el pred att el, n ;
data n_String : es Cases es, XMLElement n , _, d_ d, ;
data n_String, pred_ : es Cases es, XMLElement n , _, d_ ; pred d , ;
return es_, f_ : f es;

Mathematica XQuery DSL in Action

XPath Expressions

All book titles

In[140]:= doc xml e "book" data "title"

Out[140]= Everyday Italian, Harry Potter, XQuery Kick Start, Learning XML

All book authors

In[141]:= doc xml e "book" data "author"

Out[141]= Giada De Laurentiis, J K. Rowling, James McGovern,
Per Bothner, Kurt Cagle, James Linn, Vaidyanathan Nagarajan, Erik T. Ray

The return function

Return a { {author..} title} tuple

In[142]:= doc xml e "book"
return bs
data "author" . a_ a data "title" . t_ t & bs

Out[142]= Giada De Laurentiis Everyday Italian, J K. Rowling Harry Potter,
James McGovern, Per Bothner, Kurt Cagle, James Linn, Vaidyanathan Nagarajan XQuery Kick Start,
Erik T. Ray Learning XML

The where function

All titles with price > 30

where b b data "price", ToExpression 30 &
return data "title"

Out[143]= XQuery Kick Start, Learning XML

All titles in the COOKING category


where b b attribute "category", "COOKING" &
return data "title"

Out[144]= Everyday Italian

All titles in the WEB category with price > 40

where b
b data "price", ToExpression 40 & && b attribute "category", "WEB" &

return data "title"

Out[145]= XQuery Kick Start

The order by function

Order by title in descending order

where b b attribute "category", "WEB" &
orderBy Order 1 data "title" , 2 data "title" 0 &
return data "title"

Out[146]= XQuery Kick Start, Learning XML

All books in WEB category, ordered by title in ascending order, formatted as { title price } list

where b b attribute "category", "WEB" &
orderBy Order 1 data "title" , 2 data "title" 0 &
return bs
data "title" . t_ t data "price" . p_ p & bs

Out[147]= Learning XML 39.95, XQuery Kick Start 49.99

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Example: SQL

Establish a Database Connection

In[148]:= HSQL memory db
Needs "DatabaseLink`"
bookstore OpenSQLConnection ;

Create the Book table

In[150]:= SQLDropTable bookstore, & SQLTableNames bookstore ;
SQLCreateTable bookstore, SQLTable "BOOK" ,
SQLColumn "ID", "DataTypeName" "INTEGER" ,
SQLColumn "TITLE", "DataTypeName" "VARCHAR", "DataLength" 128 ,
SQLColumn "YEAR", "DataTypeName" "VARCHAR", "DataLength" 4 ,
SQLColumn "PRICE", "DataTypeName" "FLOAT"
;
SQLTableNames bookstore

Out[152]= BOOK

Load books from XML

In[153]:= ClearAll books ;
books
doc xml e "book" return
bs

data "title" . t_ t,
data "year" . y_ y,
data "price" . p_ p

& bs

Out[154]= Everyday Italian, 2005, 30.00 , Harry Potter, 2005, 29.99 ,
XQuery Kick Start, 2003, 49.99 , Learning XML, 2003, 39.95

Load books into the Database

In[155]:= MapIndexed
SQLInsert bookstore, "BOOK", "ID", "TITLE", "YEAR", "PRICE" , 2 Join 1 &, books ;

Query the Database

In[156]:= SQLSelect bookstore, "BOOK" TableForm

Out[156]//TableForm=
1 Everyday Italian 2005 30.
2 Harry Potter 2005 29.99
3 XQuery Kick Start 2003 49.99
4 Learning XML 2003 39.95

Close the Database Connection


Close the Database Connection

In[157]:= CloseSQLConnection bookstore

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Example: Geometric Transformation

The RotationTransform Function
Understanding the Function

In[158]:= RotationTransform Θ

Cos Θ Sin Θ 0
Out[158]= TransformationFunction Sin Θ Cos Θ 0
0 0 1

In[159]:= RotationTransform Θ x, y

Out[159]= x Cos Θ y Sin Θ , y Cos Θ x Sin Θ

Creating a Replacement Rule

In[160]:= RotationTransform Θ x, y . a_, b_ x a, y b

Out[160]= x x Cos Θ y Sin Θ , y y Cos Θ x Sin Θ

Testing Our Rule

In[161]:= y x2 . x x Cos Θ y Sin Θ , y y Cos Θ x Sin Θ

2
Out[161]= y Cos Θ x Sin Θ x Cos Θ y Sin Θ

2
In[162]:= y Cos Θ x Sin Θ x Cos Θ y Sin Θ .Θ 90 °

Out[162]= x y2


In[163]:= ContourPlot y x2 , x y2 , x, 2 Π, 2 Π , y, 2 Π, 2 Π ,
Axes True, Frame False, ContourStyle Thick, Red , Thick, Blue

6

4

2

Out[163]=
6 4 2 2 4 6

2

4

6

Creating a Simple Rotate Function

In[164]:= rotate eq_, Θ_ : eq . RotationTransform Θ x, y . a_, b_ x a, y b

In[165]:= rotate y x2 , 90 ° , rotate y Sin x , 30 ° FullSimplify

1
Out[165]= x y2 , x 3 y 2 Sin 3 x y
2


The Rotate Function in Action

In[233]:= Show
ContourPlot
rotate y Sin x , 1 ° Evaluate,
x, 2 Π, 2 Π , y, 2 Π, 2 Π ,
Frame False,
Exclusions rotate y Sin x , 1 ° Evaluate, x2 y2 25 ,
1
ContourStyle Thickness 0.004 , Hue
Π
& Range 0, 360, 10
Magnify , 1 &

Out[233]=

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Example: Tree Chains

Graph Algebraic Data Type

Graph Structure and supporting functions

In[167]:= ClearAll Graph, graph, Vertex, vertex, Leaf, leaf,
Branch, tail, branch, succ, dft, concatMap, chains, toRules, vrf ;

Vertex Vertex d$ : _ ;
vertex d_ : Vertex d ;
value Vertex : d$;

Graph Graph rep$ : _Vertex , _Vertex ... ... ;
graph rep : _Vertex , _Vertex ... ... : Graph rep ;
graph rep : _Vertex _Vertex ... ... :
Graph . x_ y : __ x, y & rep ;
rep Graph : rep$;

Leaf Leaf v$ : Vertex ;
leaf v : Vertex : Leaf v ;
vertex Leaf : v$;

Branch Branch v$ : Vertex, tail$ : __ ;
branch v : Vertex, : leaf v ;
branch v : Vertex, l : ___ : Branch v, l ;
vertex Branch : v$;
tail Branch : tail$;

succ g : Graph, v : Vertex : Cases rep g, u : Vertex, adj_ ; u v adj Flatten;

dft g : Graph, v : Vertex : branch v, dft g, & succ g, v ;

concatMap f_, l : __ : Fold Join, , Flatten f & l ;

chains g : Graph : chains dft g, rep g 1 1 ;
chains l : Leaf : vertex l ;
chains b : Branch : concatMap vertex b, &, chains & tail b Flatten , 1 &;

toRules g : Graph : . x_, y : __ x & y & rep g Flatten;
toRules ch : _Vertex ... : ch . x_ , x_, y_, xs___ x y Join toRules y, xs ;
toRules chs : _Vertex ... ... : toRules & chs;

vrf ps, v White, EdgeForm Black , Disk ps, .3 , Black, Text value v, ps ;

Graph Instance

In[193]:= Dynamic g ;
g graph
vertex Μ vertex Α, Β, Γ ,
vertex Α vertex Α1 , Α2 , Α3 ,
vertex Β vertex Β1 , Β2 , Β3 ,
vertex Γ vertex Γ1 , Γ2 , Γ3 ;


Graph Plot

In[219]:= LayeredGraphPlot g toRules, VertexRenderingFunction vrf Magnify , 1 & Dynamic

Μ

Out[219]= Α Β Γ

Α1 Α2 Α3 Β1 Β2 Β3 Γ1 Γ2 Γ3

Chains and Chains Plot

In[196]:= chains g Dynamic

Out[196]= Vertex Μ , Vertex Α , Vertex Α1 ,
Vertex Μ , Vertex Α , Vertex Α2 , Vertex Μ , Vertex Α , Vertex Α3 ,
Vertex Μ , Vertex Β , Vertex Β1 , Vertex Μ , Vertex Β , Vertex Β2 ,
Vertex Μ , Vertex Β , Vertex Β3 , Vertex Μ , Vertex Γ , Vertex Γ1 ,
Vertex Μ , Vertex Γ , Vertex Γ2 , Vertex Μ , Vertex Γ , Vertex Γ3

In[220]:= GraphPlot , VertexRenderingFunction
vrf Magnify , 1 & & toRules chains g Partition , 3 & TableForm Dynamic

Μ Α Α1 Μ Α Α2

Out[220]= Μ Β Β1 Μ Β Β2

Μ Γ Γ1 Μ Γ Γ2

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Example: Sound

The Sound of Mathematics

In[198]:= n11 11, 2, 7, 9, 11, 2, 7, 5, 11, 2, 7, 4, 2, 2, 2 ;
a11 n11 ;
a12 12 & n11 ;
a13 a12 Join a11 ;
a14 a13 Join Reverse a12 Join n11 ;
s1 Sound SoundNote , .1, "Flute" & a14 ;
n21 2, 3, 7, 2, 3, 7, 2, 3, 9, 9, 7, 3 ;
a21 n21 ;
a22 12 & n21 . xs__ xs, xs ;
a23 a22 Join a21 Join Reverse 12 & a21 Join Reverse a21 ;
s2 Sound SoundNote , .1, "Piano" & a23 ;
EmitSound s1 , s2

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Functional Programming In Mathematica

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