Transcript Scientific Computation Using Mathematica

Getting Started
with
Mathematica
by
Hakan Kutucu
What is Mathematica
 Mathematica is a computer program created by
Wolfram Research Inc.(1988)
 Mathematica is a powerful desktop computer
program capable of doing algebraic calculations,
numerical approximations, and computer graphics.
 It is suitable for both numeric and symbolic
work, and it has remarkable word-processing
capabilities as well.
Some Graphical Examples
What is Mathematica
 Mathematica has two parts: the Kernel and the
Front End.
 The Kernel is the main part of the system,
which accepts Mathematica commands, processes
them and sends back results. This is called
evaluating the command.
 The Front End is the part of the system that
handles such things as screen display, printing and
the creation of Mathematica documents.
How does Mathematica work?
Mathematica documents
are called Notebooks. A
Notebook is a bit like a
word-processor
document; you can type
and edit commands, send
them to the Kernel for
evaluation, display the
results and save your
work.
How does Mathematica work?
 You send commands to be evaluated by
holding down the “shift” key and pressing the
“enter” key
 Another easy way to evaluate by pressing the
“enter” key at the right hand side of keyboard.
Functionalities of Mathematica
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Numerical Computation
Equation Solving
Symbolic calculation
Calculus
Graphics
Self-defined functions
List
Pattern Matching
Animation
Mathematica Syntax
 Mathematica commands start with a CAPITAL
letter, e.g. Sin[x]. If the command is really
severalwords in English joined together then each
one starts with a capital but
WithNoSpacesInBetween.
 Case Sensitive: the word Fun is different from
the word fun which is different from the word
fuN.
Mathematica Syntax
 Mathematica uses a lot of brackets and all the
different sorts of them, ( [ { } ] ).
 Arguments of functions are given in square
brackets [ ].
 Curly brackets, called “braces”, {…} are used to
make a list, usually to allow several objects to be
treated as one.
 Expression are cascaded by () pairs.
SPECIAL CONSTANTS
Mathematica protects these names; they
cannot be used as names for other
quantities.
Arithmetic Operations
Operation
Symbol
+
Operation
Name
Add
Sample
-
Substruct
8-3
/
Divide
4/6
* or Space
Multiply
2*4 or 2 4
^
Power
3^4
3+5
Arithmetic Operations
Order of operation is important .
BEDMAS
N (Numerical)
(a) N[expr] displays 6 digits by default
(b) N[expr, n] displays 6 digits when n<16
(c) N[expr, n] does not display more digits than
n originally has.
Some Mathematical Functions
Sqrt[x]
Exp[x]
square root (
)
exponential (ex)
Log[x]
Log[b, x]
natural logarithm lnx
logarithm to base b (logbx)
Sin[x],
Cos[x], Tan[x]
trigonometric functions (with
arguments in radians)
n!
Mod[n, m]
factorial of n
n modulo m
FactorInteger[n]
prime factors of n
Graphics
Mathematica incorporates a wide range of twoand three-dimensional graphics functions. The
simplest is Plot, which generates two-dimensional
Cartesian graphs
Plot[f, {x, xmin, xmax}]
generates a plot of f as a function of x from xmin to
xmax.
Plot[{f1, f2, … }, {x, xmin, xmax}]
plots several functions fi.
Graphics (Example)
Graphics(Options)
Graphics(Options)
For Other Options the command is
Options[Plot]
3 Dimensional Plotting
Plot3D[ f, {x,xmin,xmax}, {y,ymin,ymax}]
will draw the graph of z = f(x,y) over the
rectangle [xmin,xmax]×[ymin,ymax] in the
xy-plane.
3 Dimensional Plotting(Options)
Three Dimensional Plotting(Options)
Defining Functions
 A function may return a symbol, a real
number, a complex matrix, another function....
 Any mathematical object is an acceptable
value for a function.
 A function may be written using mathematical
notation, it may be created using programming
constructs (like for, if/then, etc.), or it could even
be written in another language entirely (Fortran,
C, etc.).
Function & Variable Names
 Mathematica variable names can be long, but
they must not begin with a number,
 Because Mathematica interprets, say,
2dimension as 2 * dimension.
 Names can end with a number, though: x1 is a
useful way of writing in Mathematica a subscripted
variable like x1.
Note also that combinations of letters without
spaces are interpreted as new variables: ax does
not mean a*x.
Defining Functions
A function can be defined by using the following
form:
f[x_] := an expression involving x (no underbar)
Example: f[x_]:=x^2+3x
f[4]
f[a+1]
?f (This shows the definition you made for f. )
Equation solving
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Use Solve to solve an equation with an
exact solution, including a symbolic solution.
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Use NSolve or FindRoot to obtain a
numerical approximation to the solution.
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To use solutions need to use expr/ .x
y.
Matrix Defining
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Entering a matrix:
a={{1,2,3},{4,5,6},{7,8,9}}
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Displays nicely only if told to:
Menus exist but are harder to use than typing
Matrix Operations
Matrix Operations
(Multiplication)
Matrix multiplication
(also called dot or inner
product) is carried out in
Mathematica with the
function Dot, typically
entered with a dot shorthand syntax.
Matrix Operations
(Transpose)
Matrix Operations
(Inverse)
Matrix Operations
Exercises
 x  y  z  2q  6
x  2 y  q   2
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Solve 
 x  y  3z  2q  12
 x  y  4 z  5q  16
1. By using Solve Commabd
2. By using The equation Ax=b