2. 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

3. Chapter 6 “Algorithm Analysis" In this chapter we introduced algorithm analysis and showed 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

4. 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”

5. 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: 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

6. Continued…. Variables defined in MATLAB follow common computer naming conventions: 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 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

7. 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

8. 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 values are displayed, they are stored as double-precision floating- point 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.

9. 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

10. 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

11. 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 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

12. Continued…. In addition to x – y plots, MATLAB offers a variety of plotting 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

13. Continued…. The three-dimensional plotting options in MATLAB include a line 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 three- dimensional 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

14. 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)

15. Continued…. The function name must also be the name of the M-file in which the 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

16. 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

17. 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

18. 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')

19. Continued…. In addition to the interp1 function, MATLAB includes a two- dimensional interpolation function called interp2, a three- dimensional 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

20. Continued….

21. 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 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

22. Continued…. The gradient function uses a forward difference approach to 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

23. Continued….

24. 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