SSG314 Lecture Two Functional Programming

7/31/2019 SSG314 Lecture Two Functional Programming

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Functional ProgrammingMathematica as a Functional Programming Language


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A OOP language has methods, operators and

properties. These are functions. The programs in this paradigm evolves by changes of

the state of objects which are acted upon by theprograms.

Functions in an OOP environment often are used tocreate side effects. Functional Programming takes adifferent approach. The symbolics processor inMathematica supports this way of doing things.

Object-Oriented ProgrammingOOP


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Functional programming is a style of programming that

emphasizes the evaluation of expressions rather than theexecution of commands. Erlang programming language is

described as a functional programming language. Erlang

avoids the use of global variables that can be used in

common by multiple functions since changing such avariable in part of a program may have unexpected effects

in another part.

What is Functional Programming?
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In an earlier definition from the ITU-TS, functionalprogramming is "a method for structuring programsmainly as sequences of possibly nested functionprocedure calls." A function procedure is a relatively

simple program that is called by other programs andderives and returns a value to the program that called it

What is Functional Programming?


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In this section we shall examine the facilities in

Mathematica that support the functional approach toprogramming.

Now a function is a map with a domain and a range.Mathematically, a function has a name, parameters

and a body that is basically a blueprint for creatingthe range out of the domain parameters.

A pure function has no name.

Support for Functional Programming


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It may come as a shock to see that a function doesnot necessarily need a name. If the way to generatethe range from the domain can be specified without aname, we can still have a function. Below arenameless functions with one and two arguments:

Lambda functions

In these examples, we have

specified maps and the way thatthese maps are constructedwithout the superfluity of aname! These are PureFunctions NamelessFunctions


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We can attach a name to a pure function as follows:

The concluding ampersand &signifies the end of the functionbody.

While a function can do without

a name, it cannot be made toproduce a map without its beingendued with a body. The parameters can be specified andnumbered when they are more than one as shown

Attaching a name


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Mathematica calls nameless functions Purefunctions

In this course we shall avoid that nomenclaturebecause we already call functions that have no bodiesin OOP Pure functions. Nameless functions, as wehave seen have bodies. We adopt the commonlanguage agnostic concept of Lambda function.

A Lambda Function can be spacified, as we havealready done without the need for a name.

Attaching a name


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There are four language constructs that are so basic

and will need to master to successfully operate in afunctional environment:

Apply, Map, Fold and Nest.

We will take these in order with simple examples. To

succeed in this course, it will be important to practicewith your own chosen examples and to experimentuntil you have mastered these concepts.

Function Primitives


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Apply resets the Head of anexpression to the new headspecified in the firs argument ofApply.

The Head was changed form Listto f as we can see.

When we change the Head toPlus, the summation wasautomatically applied to the entirelist.

Apply


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There an operator form of Apply as

can be seen in the example here. It works the same way as the

previous page.

Apply can also be used in the infix

way as we could have done for anyother operator:

Apply in Operator form


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Map is a repeated application [Apply used repeatedly]

that can be done in levels. Map can be expressed in the functional or operator

form as follows:

The level of the map in the last example is 2.

Map


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As we saw in apply, there is a Map operator. /@. It is

possible to use both the operator as well as the Infixform to Mapping.

The other two functions: Fold and Nest are based onmapping and applying as we shall see. It is important

to understand the latter in order to fully grasp the fullmeaning of the former.

Map Operator


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FoldList


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In the last example, a Lambda function is folded as

shown. You can see the power of these namelessfunctions and the intricate computations performedon the fly that would have required a number ofiterations.

FoldList


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Once the idea of FoldList is fully understood,

remember that often we are interested only in thefinal fold result. In the case of the penultimateexample, we may be interested only in the last sum.The Fold[] function gives that as in:

Fold


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Nest and NestList combo work in a similar way as theabove:

Nest, NestList


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Nest supplies the ultimate result

in a NestList with the sameparameters.

Nest[] as well as NestList[]repeatedly applies the Head to a

single argument while FoldApply[](s) to a list of arguments.

Nest


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We first define a more elaborate factorial functionthat admits non-integer values. A plot of this functionis shown and the DownValues show the search route

UpValues & DownValues


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With UpValues, we can redefine Mathematica

functions in new contexts. See the cookbook for a detailed example.

Upvalues


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A quick comparison of the attributes of the Plus and dividefunctions help to see the meaning of some

Orderless and Flat are absent from Divide. The first showthat Plus is commutative in its arguments while Divide isnot. Further, Plus (f[f[x,y],z]) can be flattened out (f[x,y,z])

while Divide cannot. Listable shows that function can be supplied with a List

and it will be treaded over such a list: for example,Log[{1,3,6}] will create the list {Log[1],Log[3},Log[6]}

Attributes


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NumericFunction shows that when numeric

arguments are passed to the function, Mathematicacan assume the function itself is Numeric.

Protected shows that the values supplied areprotected from modification.

Others


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Sometimes you may want Mathematica to hold an

argument in the function without evaluation. This is anadvanced usage may not be obvious now. HoldFirst Mathematica defines a function Hold which prevents

its argument from being evaluated. The attribute HoldFirstallows you to give this feature to the first argument of afunction. All remaining arguments will behave normally.

HoldRest This is the opposite of HoldFirst; the first argumentis evaluated normally, but all remaining arguments are kept inunevaluated form.

HoldAll All arguments of the function are kept unevaluated.This is equivalent to using both HoldFirst and HoldRest.

Holding Arguments


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SSG314 Lecture Two Functional Programming

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