Transcript Lecture32.pptx

Chapter 4 "Inheritance"
 Inheritance is a powerful feature that allows the reuse of code.
 However, be sure that functions applied to objects created at
run time through the new operator are bound at run time. This
feature is known as dynamic binding, and the use of virtual
functions is required to ensure that run-time decisions are
made.
 Reread this chapter as often as necessary to ensure that you
understand the distinctions among
 non-virtual functions
 virtual functions
 pure virtual functions
Chapter 6 “Algorithm Analysis"
 In this chapter we introduced algorithm analysis and showed
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that algorithmic decisions generally influence the running time
of a program much more than programming tricks do
We also showed the huge difference between the running times
for quadratic and linear programs and illustrated that cubic
algorithms are, for the most part, unsatisfactory
We examined an algorithm that could be viewed as the basis
for our first data structure
The binary search efficiently supports static operations thereby
providing a logarithmic worst-case search
Later in the text we examine dynamic data structures that
efficiently support updates
Data Visualization with MATLAB
 Chapter 2 “MATLAB Environment”
 Chapter 3 “Built-in MATLAB Functions”
 Chapter 5 “Plotting”
 Chapter 6 “User-Defined Functions”
 Chapter 13 “Numerical Techniques”
Chapter 2 "MATLAB
Environment"
 In this chapter, we introduced the basic MATLAB structure
 The MATLAB environment includes multiple windows, four of
which are open in the default view:
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Command window
Command history window
Workspace window
Current folder window
 In addition, the
 Document window
 Graphics window
 Edit window
open as needed during a MATLAB session
Continued….
 Variables defined in MATLAB follow common computer naming
conventions:
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Names must start with a letter.
Letters, numbers, and the underscore are the only characters allowed.
Names are case sensitive.
Names may be of any length, although only the first 63 characters are
used by MATLAB
 Some keywords are reserved by MATLAB and cannot be used as
variable names
 MATLAB allows the user to reassign function names as variable names,
although doing so is not good practice.
 The basic computational unit in MATLAB is the matrix. Matrices
may be
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Scalars ( 1 x 1 matrix)
Vectors ( 1 x n or n x 1 matrix, either a row or a column)
Two-dimensional arrays ( m x n or n x m )
Multidimensional arrays
Continued….
 Matrices often store numeric information, although they can store other kinds of
information as well
 Data can be entered into a matrix manually or can be retrieved from stored data files
 When entered manually, a matrix is enclosed in square brackets, elements in a row
are separated by either commas or spaces, and a new row is indicated by a
semicolon:
a = [1 2 3 4; 5 6 7 8]
 Evenly spaced matrices can be generated with the colon operator. Thus, the
command
b = 0:2:10
creates a matrix starting at 0, ending at 10, and with an increment of 2
 The linspace and logspace functions can be used to generate a matrix of specified
length from given starting and ending values, spaced either linearly or
logarithmically
 The help function or the MATLAB Help menu can be used to determine the
appropriate syntax for these and other functions
Continued….
 MATLAB follows the standard algebraic order of operations
 MATLAB supports both standard (decimal) and scientific notation
 It also supports a number of different display options. No matter how
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values are displayed, they are stored as double-precision floatingpoint numbers
MATLAB variables can be saved or imported from either .MAT or
.DAT files
The .MAT format is proprietary to MATLAB and is used because it
stores data more efficiently than other file formats
The .DAT format employs the standard ASCII format and is used
when data created in MATLAB will be shared with other programs
Collections of MATLAB commands can be saved in script M-files.
This is the best way to save the list of commands used to solve a
problem so that they can be reused at a later time
Cell mode allows the programmer to group M-file code into sections
and to run each section individually. It is especially convenient when
one M-file is used to solve multiple problems.
Chapter 3 “Built-in MATLAB
Functions"
 General mathematical functions, such as
 exponential functions
 logarithmic functions
 roots
 Rounding functions
 Functions used in discrete mathematics, such as
 factoring functions
 prime-number functions
 Trigonometric functions, including
 standard trigonometric functions
 inverse trigonometric functions
 hyperbolic trigonometric functions
 trigonometric functions that use degrees instead of radians
Continued….
 Data analysis functions, such as
 maxima and minima
 averages (mean and median)
 sums and products
 sorting
 standard deviation and variance
 Random-number generation for both
 uniform distributions
 Gaussian (normal) distributions
 Functions used with complex numbers
Chapter 5 “Plotting"
 The most commonly used graph in engineering is the x – y plot
 This two-dimensional plot can be used to graph data or to visualize
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mathematical functions
No matter what a graph represents, it should always include a title
and x - and y -axis labels
Axis labels should be descriptive and should include units, such as
ft/s or kJ/kg MATLAB includes extensive options for controlling the
appearance of your plots
The user can specify the color, line style, and marker style for each
line on a graph
A grid can be added to the graph, and the axis range can be adjusted
Text boxes and a legend can be employed to describe the graph
The subplot function is used to divide the plot window into an mxn
grid
Inside each of these subwindows, any of the MATLAB plots can be
created and modified
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 In addition to x – y plots, MATLAB offers a variety of plotting
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options, including polar plots, pie charts, bar graphs,
histograms, and x – y graphs with two y –axes
The scaling on x – y plots can be modified to produce
logarithmic plots on either or both x – and y- axes
Engineers often use logarithmic scaling to represent data as a
straight line
The function fplot allows the user to plot a function without
defining a vector of x - and y –values
MATLAB automatically chooses the appropriate number of
points and spacing to produce a smooth graph
Additional function-plotting capability is available in the
symbolic toolbox
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 The three-dimensional plotting options in MATLAB include a line
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plot, a number of surface plots, and contour plots
Most of the options available in two dimensional plotting also apply
to these three-dimensional plots
The meshgrid function is especially useful in creating threedimensional surface plots
Interactive tools allow the user to modify existing plots
These tools are available from the figure menu bar
Plots can also be created with the interactive plotting option from the
workspace window
The interactive environment is a rich resource
Figures created in MATLAB can be saved in a variety of ways, either
to be edited later or to be inserted into other documents
MATLAB offers both proprietary file formats that minimize the
storage space required to store figures and standard fi le formats
suitable to import into other applications
Chapter 6 “User Defined
Functions"
 MATLAB contains a wide variety of built-in functions
 However, you will often find it useful to create your own
MATLAB functions
 The most common type of user-defined MATLAB function is
the function M-file, which must start with a function-definition
line that contains
 the word function
 a variable that defines the function output
 a function name
 a variable used for the input argument
 For example,
function output = my_function(x)
Continued….
 The function name must also be the name of the M-file in which the
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function is stored
Function names follow the standard MATLAB naming rules
Like the built-in functions, user-defined functions can accept
multiple inputs and can return multiple results
Comments immediately following the function-definition line can be
accessed from the command window with the help command
Variables defined within a function are local to that function. They
are not stored in the workspace and cannot be accessed from the
command window
Global variables can be defined with the global command used in
both the command window and a MATLAB function. Good
programming style suggests that define global variables with capital
letters. In general, however, it is not wise to use global variables
Continued….
 Groups of user-defined functions, called “toolboxes,” may be stored
in a common directory and accessed by modifying the MATLAB®
search path. This is accomplished interactively with the path tool,
either from the menu bar, as in
File -> Set Path
or from the command line, with pathtool
 MATLAB provides access to numerous toolboxes developed at The
MathWorks or by the user community
 Another type of function is the anonymous function, which is
defined in a MATLAB session or in a script M-file and exists only
during that session
 Anonymous functions are especially useful for very simple
mathematical expressions or as input to the more complicated
function functions
Chapter 13 “Numerical
Techniques"
 Tables of data are useful for summarizing technical information
 However, if a value is needed that is not included in the table,
must approximate that value by using some sort of
interpolation technique
 MATLAB includes such a technique, called interp1 . This
function requires three inputs: a set of x -values, a
corresponding set of y -values, and a set of x -values for which
you would like to estimate y -values
 The function defaults to a linear interpolation technique, which
assumes that approximate these intermediate y -values as a
linear function of x that is,
y = f(x) = ax + b
Continued….
 A different linear function is found for each set of two
data points, ensuring that the line approximating the data
always passes through the tabulated points
 The interp1 function can also model the data by using
higher-order approximations, the most common of which
is the cubic spline
 The approximation technique is specified as a character
string in a fourth optional field of the interp1 function
 If it’s not specified, the function defaults to linear
interpolation. An example of the syntax is
new_y = interp1(tabulated_x, tabulated_y, new_x, 'spline')
Continued….
 In addition to the interp1 function, MATLAB includes a two-
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dimensional interpolation function called interp2 , a threedimensional interpolation function called interp3 , and a
multidimensional interpolation function called interpn
Curve-fitting routines are similar to interpolation techniques
However, instead of connecting data points, they look for an
equation that models the data as accurately as possible
Once you have an equation, you can calculate the
corresponding values of y
The curve that is modeled does not necessarily pass through the
measured data points
MATLAB’s curve-fitting function is called polyfit and models
the data as a polynomial by means of a least-squares regression
technique
Continued….
 The function returns the coefficients of the polynomial equation of
the form
 𝑦 = 𝑎1 𝑥 𝑛 + 𝑎2 𝑥 𝑛−1 + 𝑎3 𝑥 𝑛−2 + ⋯ … … … … + 𝑎𝑛 𝑥 + 𝑎𝑛+1
 These coefficients can be used to create the appropriate expression in
MATLAB, or they can be used as the input to the polyval function to
calculate values of y at any value of x
 For example: the following statements find the coefficients of a
second-order polynomial to fi t the input x–y data and then calculate
new values of y , using the polynomial determined in the first
statement:
 coef = polyfit(x,y,2)
 y_first_order_fit = polyval(coef,x)
 These two lines of code could be shortened to one line by nesting
functions:
 y_first_order_fit = polyval(polyfit(x,y,1),x)
Continued….
 MATLAB also includes an interactive curve-fitting capability that allows
the user to model data not only with polynomials, but with more
complicated mathematical functions
 The basic curve-fitting tools can be accessed from the Tools menu in the
figure window
 More extensive tools are available in the curve-fitting toolbox, which is
accessed by typing
 cftool
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in the command window.
Numerical techniques are used widely in engineering to approximate both
derivatives and integrals
Derivatives and integrals can also be found with the symbolic toolbox
The MATLAB diff function finds the difference between values in adjacent
elements of a vector
By using the diff function with vectors of x - and y -values, we can
approximate the derivative with the command
 slope = diff(y)./diff(x)
 The more closely spaced the x and y data are, the closer will be the
approximation of the derivative
Continued….
 The gradient function uses a forward difference approach to
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approximate the derivative at the first point in an array
It uses a backward difference approach for the final value in the
array, and a central difference approach for the remainder of
the points
In general, the central difference approach gives a more
accurate approximation of the derivative than either of the
other two techniques
Integration of ordered pairs of data is accomplished using the
trapezoidal rule, with the trapz function
This approach can also be used with functions, by creating a set
of ordered pairs based on a set of x values and the
corresponding y values
Continued….
Integration of functions is accomplished more directly with one of two
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quadrature functions: quad or quadl
 These functions require the user to input both a function and its limits of
integration
 The function can be represented as a character string, such as
 'x.^2-1‘
as an anonymous function, such as
 my_function = @(x) x.^2-1
or as an M-fi le function, such as
 function output = my_m_file(x)
 output = x.^2-1;
 Any of the three techniques for defining the function can be used as input,
along with the integration limits—for example,
 quad('x.^2-1',1,2)
 Both quad and quadl attempt to return an answer accurate to within 1 ×
10−6
 The quad and quadl functions differ only in the technique they use to
estimate the integral
 The quad function uses an adaptive Simpson quadrature technique, and the
quadl function uses an adaptive Lobatto quadrature technique
Continued….
 MATLAB includes a series of solver functions for first-order
ordinary differential equations and systems of equations
 All of the solver functions use the common format
 [t,y] = odesolver(function_handle,[initial_time, final_time],
[initial_cond_array])
 A good first try is usually the ode45 solver function, which
uses a Runge–Kutta technique
 Other solver functions have been formulated for stiff
differential equations and implicit formulations
 The ode solver functions require that the user know the initial
conditions for the problem
 If, instead, boundary conditions are known at other than the
starting conditions, the bvp4 function should be used