Transcript Embedded Communications in Wireless Sensor Network

Chapter 7:
MuPAD Objects III
Series, Table, Array, Matrix, Poly
MATLAB
for Scientist and Engineers
Using Symbolic Toolbox
You are going to
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See that MuPAD handles objects
Get to know MuPAD series, tables, arrays,
matrices, and polynomial types
Use these objects for various purposes
2
Series Expansion
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Taylor Series
Look at the differences!
Coefficient of a particular order
3
Series Expansion (cont.)
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Laurent Series
4
Exercise #1
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Besides the arithmetical operators, some
other system functions such as diff or int work
directly for series. Compare the result of
taylor(diff(1/(1-x), x), x) and the
derivative of taylor(1/(1-x), x).
Mathematically, both series are identical. Can
you explain the difference in MuPAD?
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Exercise #2
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The function
tends to zero
for large x, i.e.
Show that the approximation
is valid
for large values of x. Find better asymptotic
approximations of f.
6
Tables
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Association of arbitrary indices with arbitrary
values
Index
Value
Explicit Table Generation
Implicit Table Generation
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Tables – Accessing Elements
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Use [ ] or _index( )
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Check with contains( )
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Tables - Operation
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Getting the elements
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Does this table have it on LHS or RHS?
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Tables – Replacing and Removing
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Current
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Deleting
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Replacing
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Tables – select and split
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Select rows which have 'a'.
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Split table: has a, does not have a, unknown
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Exercise #3
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Generate a table T with the following entries:
Ford 1815, Reagan 4711, Bush 1234, Clinton 5678.
Look up Ford’s number. How can you find out
whose number is 5678?
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Exercise #4

Generate the table T(1 = 1, 2 = 2, …, n = n) and
the list [1, 2, …, n] of length n = 100000. Add a
new entry to the table and to the list.
How long does this take?
Hint: the call time((a:= b)) returns the execution
time for an assignment.
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Arrays
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Multidimensional Data Structure
Integer Indices, Fixed Size
2D Array
3D Array
Starting index starts from 2!
Values
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Arrays (cont.)
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Values Only
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Accessing Elements
Dimension Info: 0-th Operand
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All Elements
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Exercise
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Generate a so-called Hilbert matrix H of
dimension 20 × 20 with entries
Hij = 1/(i + j − 1).
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Vectors and Matrices
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Generating matrices
Domain type. Default is Any Object.
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Matrix Constructor
Dimension
Values
Value generating function
Sub-matrix range
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Matrix Operation
Inverse Matrix
Matrix Concatenation
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Matrix – System Functions
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System Functions for Matrices
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Matrix – System Functions (cont.)
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System Functions for Matrices
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Exercise
Hilbert Matrix
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Matrix – Special Methods
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Matrix Algebra
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The Libraries linalg and numeric
Determinant
Eigenvalues
Transpose
Cholesky Decomposition
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Exercise

Consider the following matrices:
Let BT be the transpose of B. Compute the
inverse of 2A + B BT .
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Polynomials
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Polynomials - Functions
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System Functions for Polynomials
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Polynomials – Functions (cont.)
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System Functions for Polynomials
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Polynomial - Manipulations
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Evaluations
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Integration and Differentiation
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Exercise
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Consider the polynomials p = x7 − x4 + x3 − 1
and q = x3 − 1.
Compute p − q2.
Does q divide p?
Factor p and q.
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Key Takeaways
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Now, you are able to
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approximate an expression using Taylor and
Laurent series,
use MuPAD tables for key-value association
applications,
deal with fixed-size multi-dimensional data using
MuPAD arrays,
apply various linear algebraic functions on
MuPAD matrices,
and to manipulate MuPAD polynomials.
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Summary

Explain the following expressions
taylor(sin(x),x=0,5)
has(T,b)
T[a]:= b
lhs(T)
select(T,has,x)
rhs(T)
contains(T,a)
op(T)
split(T,has,x)
A := array([[1,2],[x,y]])
M := matrix([[1,2],[x,y]])
delete T[a]
transpose(M)
op(A,0)
matrix::tr(M)
M[1,1..2]
linalg::det(M)
poly(a*x+b*x^2,[x],Dom::Integer
mapcoeffs(poly,_plus,2)
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