Art of Symbolic Computing - I using MATHEMATICA

(A Fully Integrated Environment for Technical Computing) Software and Applications

R.C. Verma

Physics Department Punjabi University Patiala – 147 002

Refresher Course in Physics Panjab University Sept. 2010

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Education (Some Indian Statistics)

Total population below 14 is around 250 million.

Maximum number of ‘out of school’ children in world.

Nowhere Children:- Around 90 million children are neither enrolled in school nor accounted for as labour. Only 60% of children reach grade V.

Many of those ‘completing’ primary school cannot read or write.

Every child is curious, creative, intelligent, innocent and beautiful. Why and where he loose these qualities. Somewhere society (present normal setup of mind, current values and education modes) is responsible for this state of affairs.

Hope:- Efforts under UGC, DST, NMEICT of MHRD etc.

http://www.sakshat.ac.in/

NMEICT (MHRD) www.sakshat.ac.in

• Around 30 programs: 1. e-content development in all the branches of knowledge at UG and PG levels, 2. Development of Pedagogical methods of teaching and research in e-learning, 3. Teachers empowerment through training in Computer Applications to various subjects in Universities and Colleges, 4. Development of Computer Simulation packages in various subjects of science and humanities, 5. Introduction of one compulsory paper/project for on-line training at PG level • More details on these and other activities of the mission can be obtained from the following file web-link: •

http://www.sakshat.ac.in/PDF/Missiondocument.pdf

MATHEMATICA – AN INTRODUCTION Part I: Basics and Arithmetics

Input and Output Number Representation & Arithmetic Computations Built-in Constants & Functions Complex Arithmetic Operations

Introduction:-

Computer-based symbolic computation has a very significant role in computer world. Still it is a very under-used, probably because of a reluctance to break from tradition. Today symbolic manipulation has become a powerful tool, which boosts productivity It runs on most popular workstation OS, including Microsoft Windows, Apple Macintosh OS, Linux, and other Unix-based systems. Mathematica today is used in all the branches of knowledge: • Physical science, • Biological science, • Social science, • Commerce, • Education, • Enginering, • Computer science, etc. Over

100

specialized commercial

packages

and over

200 books

for Mathematica and its applications

0. Phases of the computer algebra

:-

First Generation:

Reduce has packages suited to the needs of the high energy physicist. Macsyma is slow and lacks a friendly interface.

Second Generation:

Maple has strong algebraic and graphics capability.

Third Generation:

Mathematica is a very user-friendly interface and superb graphic abilities. A standard tool for research, development and production of software.

MatLab & MathCad are also available with similar facilities.

1. Who Created Mathematica?

S

tephen

Wolfram

is the creator of Mathematica. His early work: mainly in high-energy physics, quantum field theory , and cosmology . In 1979, he began the construction the computer algebra system. He began the development of Mathematica in 1986 . 1st version released on June 23, 1988, hailed as a major advance in computing world. In 1991, its Window version 2 was developed by him and his team.

Before 2004, versions 3, 4 and 5, now 6 introduced with new capabilities.

Parallel Processing

and

Grid-Mathematica

have also been launched.

http://www.wolfram.com/

2. What can Mathematica do for you?

STANDARD ARITHMETIC Operations with Integer, Rational, Real, and Complex nos with LARGE precision RICH in Built-in Constants & functions; Special functions, Creating New functions ALGEBRA Factorization, Simplification, Reduction, Solution of Simultaneous equations List manipulation: Vector, Matrix & Tensor operations; Sums & Products GRAPHICS 2D, 3D, Contour, Density, Parametric plots, Listplot; Sound and Animation CALCULUS Limits, Differentiation and integration, Taylor Series, Solving Differential Eqns.

NUMERICAL ANALYSIS: Root Finding, Numerical Integration, Solving Differential Eqns., Curve Fitting, etc.

PROGRAMMING In-built Interpreted Programming Language, also with Compilation facility FORMATTED OUTPUT: TeX, C and Fortran output,

3. Structure of Mathematica

Modular software system: KERNEL : main engine of the system containing all the functions (over 1000) WMATHEXE.EXP 3,979,305 (for Window Math2.2 version) FRONT-END: serves as the channel on which a user communicates with the kernel. STANDARD PACKAGES: special topics, like vector analysis, statistics, algebra, to graphics, etc. MATHSOURCE: collection of packages and notebooks created by Mathematica users.

Feyndia, Math-Tensor, Graphics, and other Addons

WebMathematica

4. Ten Commandments

1. Mathematica distinguishes between

uppercase

and

lowercase

letters.

2. Mathematica commands, built-in functions, & constants start with a

capital

letter.

3. Use

lowercase

letters for defining

variables

or functions. 4. Arguments of all

commands

& functions are enclosed in

square brackets

[ ].

5. Use

curly brackets

{ } for lists of items, and range of

parameters

of the function.

6. Parenthesis

( ) is reserved for indicating the

grouping

of terms.

7. Double brackets

[ [ ] ] are used for indexing the

components

of an object. 8. All functions, for

numerical

calculations, start with a capital

N

. 9. Giving

Remarks

. Statements starting with

(*

and ends with

*)

are not executed.

10. When Mathematica detects a syntax

error

, it prints a

message

.

5. How to Start Mathematica and Execute its Commands?

Double

click

the Mathematica icon, or Math.Exe file When Mathematica starts, it shows you a blank

notebook

. Enter Mathematica commands into the notebook, and then press

Shift-Enter

keys to

process

the input given. Pressing the

Enter

key generates

a new line

.

For getting Help on Mathematica Commands,

use

Help

or type double question marks

??

before the command name. For instance ??

Integrate

Math 6.0 - Help

• Documentation Center • Index of Functions • 5 minutes with Mathematica • Demonstration Projects

•

Special Features (given in Help)

Core Language

The uniquely powerful symbolic language that is the foundation for

Mathematica

Mathematics and Algorithms

The world's largest integrated web of mathematical capabilities and algorithms

Data Handling & Data Sources

Powerful primitives and sources for large volumes of data in hundreds of formats

Visualization and Graphics

Symbolic graphics and unparalleled function and data visualization

Systems Interfaces & Deployment

Unique customizability and connectivity powered by symbolic programming & parallel programming

Dynamic Interactivity

Capabilities that define a new kind of dynamic interactive computing

Notebooks and Documents

Program-constructible symbolic documents with uniquely flexible formatting.

6. Input and Output Labels In[1] : =

An input label appears at the beginning of every Input cell. These are numbered according to the order of evaluation. The result of the command is displayed in an Output cell,

Out[n]=

which is labeled according to the input label. Mathematica formats the material in Output cells in mathematical notation. e.g.

type 5.3+2.9 and press Shift-Enter keys Mathematica shows: In[1]:= 5.3 + 2.9 Out[1] = 8.2

7. Number Representation & Arithmetic Computations

Mathematica can distinguish between

integer, rational, real, complex numbers .

Following symbols are used to denote them:

Addition + Subtraction Multiplication * Division / Exponentiation ^ Parenthesis ( )

can be used to change order of operations.

Note: - Multiplication can be represented by a space.

Caution:

x y means x * y but xy is treated as a new

variable

xy .

8. MIXED MODE OPERATIONS:

When two constants or variables of the same type are combined through any of these fundamental operations, the result would be of the same type as that of the operands. Mathematica gives out of integer operation as integer, In[2] := 7^37 Out[2] = 18562115921017574302453163671207 Mathematica allows the mixed mode operations, e.g, where real number and integer are used. Result in such a case is given as real number.

In[3] := 7.0^37 Out[3] = 1.85621 10 31

9. Variable Assignments

A single equal sign is used to assign values directly to a variable. In[4]:= weight= 0.542

Out[4]= 0.542

In[5]:= counter = 1 Out[5]= 1 Arithmetic expression may also be assigned to a variable, In[6]:= y = x*x Out[6] = 2 x

10. Writing Results:

Results obtained in a program are generally written using the following statement: Print[ variable ] Messages can be printed on screen by enclosing them in double quote (

“

) sign: In[8]:= Print[“You are welcome!”] Out[8] = You are welcome!

11. Numerical Resolution

Mathematica can produce its approximate form, when N[ expression ] command is used.

In[9]:= N[ 3^85 ] Out[9]= 40 3.59175 10 Here, N stands for numerical, and the command can be given as expression // N. e.g.

In[9]:= 3^85//N To get the result up to a desired number of decimal place, say n digit precision, type N[ expression, n ] In[10]:= N[3^85, 10] Out[10]= 3.591754555 10 40

12. Built-in Constants

Pi (ratio of circumference to diameter, Pi = 3.141592

…), In[11]:= N[ Pi, 100] Out[11]=3.141592653589793238462643383279502884197169399375\ 105820974944592307816406286208998628034825342117068 E (the base e of the natural logarithm, e = 2.71828

….), In[12]:= N[ E ] Out[12]= 2.71828

Degree (Pi/180: degree to radian conversion factor) In[13]:= Sin[60 Degree ] Out[13]= Sin[60 Degree] In[14]:= N[Sin[60 Degree]] Out[14]= 0.866025

Mathematica is also familiar with iota I

i

  1 and Infinity .



Built-in Functions in Mathematica 13. Numerical Functions

Abs[x] gives the

absolute

value of x. In[21]:= Abs[ -5 ] Out[21]= 5 Sqrt[x] returns the

square root

In[22]:= N[ Sqrt[15], 50] (with positive sign) Out[22] = 3.8729833462074168851792653997823996108329217052916

Round[x] produces the

closest integer

In[26]:= Round[3.67] to x.

Out[26]= 4 Sign [x] yields

sign

In[27]:= Sign[-4] as –1, 0, 1 for x > 0, x = 0, x < 0 respectively.

Out[27]= -1

Factorial: n!

yields the In[23]:= 6!

factorial

of integer n. Out[23]= 720 In[24]:= Factorial[6] Out[24]= 720 n!!

yields

double factorial

n(n-2)(n 4)…..

In[25]:= 10!!

Out[25]= 3840 Prime[k] gives kth

prime number.

In[28]:= Prime[5] Out[28]= 11 FactorInteger[ number] gives

prime factors

In[29]:= FactorInteger[20645081] Out[29] = {{1753, 1}, {11777, 1}} of the given number Chopping a result Chop[ expression ] Replaces all approximate real numbers with magnitude less than 10 -10 by zero.

14. Logarithmic and Exponential Functions

Log[x] gives

logarithm

of x to base

e

. In[30]:= N[ Log [4 Pi], 40] Out[30]= 2.5310242469692907929778915942694118477983

Log[b, x] yields

logarithm

of x to base

b

. In[31]:= Log[3, 50.0] Out[31]= 3.56088

Exp[x] gives

exponential

of x. In[32]:= Exp[-2.5] Out[33]= 0.082085

15. Trigonometric Functions

Sin[x] sine of x x must be in radians.

Cos[x] cosine of x -do Tan[x] Cot[x] Sec[x] tangent of x -do cotangent of x -do – secant of x -do – Csc[x] cosecant of x -do – In[33]:= Cos[Pi/4] Out[33]= 1 ------ Sqrt[2] In[34]:= Tan[3Pi/4] Out[34]= -1

15.1 Inverse Trigonometric Functions

ArcSin[x] inverse of sine ArcCos[x] inverse of cosine ArcTan[x] inverse of tangent ArcCot[x] inverse of cotangent ArcSec[x] inverse of secant ArcCsc[x] inverse of cosecant In[35]:= ArcTan[-1] Out[35]= -Pi -- 4

15.25 Hyperbolic Functions

Sinh[x] hyperbolic-sine of x Cosh[x] hyperbolic-cosine of x Tanh[x] hyperbolic- tangent of x Coth[x] hyperbolic-cotangent of x Sech[x] hyperbolic-secant of x Csch[x] hyperbolic-cosectant of x In[36]:= Sinh[2.0] Out[36]= 3.62686

15.3 Inverse Hyperbolic Functions

ArcSinh[x] inverse-hyperbolic-sine of x ArcCosh[x] inverse-hyperbolic-cosine of x ArcTanh[x] inverse-hyperbolic- tangent of x ArcCoth[x] inverse-hyperbolic-cotangent of x ArcSech[x] inverse-hyperbolic-secant of x ArcCsch[x] inverse-hyperbolic-cosectant of x In[37]:= ArcCosh[2.0] Out[37]= 1.31696

16. Functions involving two or more arguments

MOD[m, n] command yields the

remainder

on division of m by n. In[38]:= Mod[237, 13] Out[38]= 3 Max[x1,x2, x3, ….] gives

largest number

In[39]:= Max[ 2.3, 1.2, 6.7, 3.8, 0.1] Out[39]= 6.7

Min[x1,x2, x3, ….] gives

smallest number

In[40]:= Min[2.3, 1.2, 6.7, 3.8, 0.1] Out[40]= 0.1

17. Complex Arithmetic Operations

Complex numbers can be entered in the form

x + y * l

, ln[41]:= z = 3.5 + 6.8 I Out[41]= 3.5 + 6.8 I Re[z] yields

real part,

ln[42]:= Re[z] Out[42]= 3.5

x, of complex number z = x + i y Im[z] yields

imaginary part

, y, of complex number z = x + i y. Abs[z] yields

absolute value,

ln[43]:= Abs[z] Out[43]= 7.64788 |z| = V(x^2 +y^2), of complex number z = x + i y Arg[z] or  yields

argument

, Tan-1(y/x) of a complex number z = x + i y, for z = |z|ei  ln[44]:= Arg[z] Out[44]= 1.09545

l Conjugate[z] produces

complex conjugate

, z* = x – i y, for z = x + i y.

ln[45]:= Conjugate[z] Out[45]= 3.5 - 6.8 I

One can performs basic arithmetic operations,

addition, subtraction, multiplication, division, exponentiation

on the complex numbers.

ln[46]:= (2+3I)(4-I) Out[46]= 11 + 10 I In[47]:= (1.5 - 0.2 I)^8 Out[47] = 13.4344 - 23.9958 I

End of Part I