Transcript MATHEMATICA - Department of Physics
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