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M A T H P L O T Mathematical Function Plotting Program A "shareware" program Phillip H. Sherrod Member, Association of Shareware Professionals (ASP) Mathplot allows you to specify mathematical functions using ordinary algebraic expressions and immediately plot them. Four types of functions may be specified: cartesian (Y=f(X)); parametric cartesian (Y=f(T) and X=f(T)); polar (Radius=f(Angle)); and parametric polar (Radius=f(T) and Angle=f(T)). Up to four functions may be plotted simultaneously. Scaling is automatic. Options are available to control axis display and labeling as well as grid lines. Hard copy output can be generated to HP LaserJet printers as well as screen display. Mathplot is an ideal tool for engineers, scientists, math and science teachers, and anyone else who needs to quickly visualize mathematical functions. Table of Contents 1. Using Mathplot . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Installing Mathplot . . . . . . . . . . . . . . . . . . . . 1 1.3 Getting Started . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Demonstration Command File . . . . . . . . . . . . . . . . 4 1.5 Command Editor . . . . . . . . . . . . . . . . . . . . . . 4 1.5.1 Function Keys . . . . . . . . . . . . . . . . . . . . . 4 1.5.2 Control Characters . . . . . . . . . . . . . . . . . . 4 1.6 Function Specification . . . . . . . . . . . . . . . . . . 4 1.6.1 Arithmetic Operators . . . . . . . . . . . . . . . . . 5 1.6.2 Numeric Constants . . . . . . . . . . . . . . . . . . . 5 1.6.3 Symbolic Constants . . . . . . . . . . . . . . . . . . 5 1.6.4 Parameter values . . . . . . . . . . . . . . . . . . . 5 1.7 Built in Functions . . . . . . . . . . . . . . . . . . . . 6 1.8 Function Types . . . . . . . . . . . . . . . . . . . . . . 8 1.8.1 Cartesian Functions . . . . . . . . . . . . . . . . . . 8 1.8.2 Parametric Cartesian Functions . . . . . . . . . . . . 9 1.8.3 Polar Functions . . . . . . . . . . . . . . . . . . . . 9 1.8.4 Parametric Polar Functions . . . . . . . . . . . . . . 9 1.9 Plotting Simultaneous Functions . . . . . . . . . . . . . . 10 2. Mathplot Commands . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Start up Command File . . . . . . . . . . . . . . . . . . . 11 2.2 Initialization file . . . . . . . . . . . . . . . . . . . . 11 2.3 Command Arguments . . . . . . . . . . . . . . . . . . . . . 12 2.4 Alphabetical List of Commands . . . . . . . . . . . . . . . 12 2.4.1 AXES . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.2 CALCULATE . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.3 CAXES . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.4 CFn . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.5 CGRID . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.6 COLORBORDER . . . . . . . . . . . . . . . . . . . . . . 13 2.4.7 COLOREDIT . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.8 COLORHEAD . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.9 COLORHELP . . . . . . . . . . . . . . . . . . . . . . . 13 2.4.10 COLORHELPHEAD . . . . . . . . . . . . . . . . . . . . 13 2.4.11 COLORMESSAGE . . . . . . . . . . . . . . . . . . . . . 13 2.4.12 COMMONSCALE . . . . . . . . . . . . . . . . . . . . . 13 2.4.13 CTITLE . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.14 DATA . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.15 DISPLAY . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.16 DO . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.17 DOMAIN . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.18 EXIT . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.19 GRID . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.20 HELP . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.21 LABEL . . . . . . . . . . . . . . . . . . . . . . . . 15 i Contents ii 2.4.22 LIST . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.23 NUMPOINTS . . . . . . . . . . . . . . . . . . . . . . 15 2.4.24 ORIGIN . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.25 Pn . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.26 PAUSE . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.27 PDEVICE . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.28 PRESOLUTION . . . . . . . . . . . . . . . . . . . . . 16 2.4.29 WIDTH . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.30 PLOT . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.31 PRINT . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.32 RESET . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.33 SAVE . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.34 TITLE . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.35 WAXES . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.36 WFn . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.37 WGRID . . . . . . . . . . . . . . . . . . . . . . . . 17 3. Advanced Applications . . . . . . . . . . . . . . . . . . . . 18 3.1 Root Finding . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Multiple Domains . . . . . . . . . . . . . . . . . . . . . 18 3.3 Interesting Functions . . . . . . . . . . . . . . . . . . . 19 3.4 "Artistic" Plots . . . . . . . . . . . . . . . . . . . . . 20 4. Use and Distribution of Mathplot . . . . . . . . . . . . . . . 21 4.1 Copyright Notice . . . . . . . . . . . . . . . . . . . . . 22 4.2 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . 22 5. Other Software . . . . . . . . . . . . . . . . . . . . . . . . 23 5.1 Nonlin . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 TSX-32 Operating System . . . . . . . . . . . . . . . . . . 24 6. Software Order Form . . . . . . . . . . . . . . . . . . . . . 25 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Chapter 1 Using Mathplot 1.1 Introduction Mathplot is a program for IBM-PC computers which allows interactive entry and plotting of mathematical functions. Some of the features of Mathplot are listed below: . Direct entry of complicated mathematical functions using normal algebraic expressions with embedded operators and functions. . An assortment of built-in functions including trigonometric, square root, log, Gamma, Bessel, elliptic integrals, normal probability distribution, etc. . The ability to plot four types of functions: cartesian (Y=f(X)); parametric cartesian (Y=f(T) and X=f(T)); polar (Radius=f(Angle)); and parametric polar (Radius=f(T) and Angle=f(T)). . The ability to simultaneously plot up to four functions. . Automatic scaling and axis labeling. . The ability to save function specifications in command files which then can be easily executed. . The ability to accept X-Y data points from an external file. . A command editor allows easy recall and editing of commands. . Printed copies of plots may be produced on HP LaserJet printers. 1.2 Installing Mathplot The Mathplot system consists of the following files: 1 Chapter 1. Using Mathplot 2 MATHPLOT.EXE -- The executable program. MATHPLOT.FON -- Font file used for title and axis labels. MATHPLOT.LJF -- Font file for HP LaserJet printer. MATHPLOT.FUN -- Example functions MATHPLOT.DOC -- Documentation file. REGISTER.DOC -- Software registration form. To install Mathplot, copy the files into the directory of your choice. If you do not plan to generated hard copy output for a LaserJet printer, you may delete the MATHPLOT.LJF file. If the MATHPLOT.FON and MATHPLOT.LJF files are not in your current directory, you must place a command of the following form in your AUTOEXEC.BAT file to tell Mathplot where to look for its font files: SET MATHPLOT=directory Where "directory" is the name of the device and directory where the files are located. For example, if the files are located in a directory named MATHPLOT on the C disk, the following command could be used: SET MATHPLOT=C:\MATHPLOT 1.3 Getting Started Mathplot is exceptionally easy to use. Although there are a fair number of commands to control options, you only need to know a couple of commands to begin using it. Start the program by entering the command: MATHPLOT Mathplot will display a title screen; press Enter to proceed to command mode. The main Mathplot screen is divided into three sections. The top window explains the function keys. The bottom window is used to display status and error messages. The middle window is used by the command editor to display commands that you enter. Several of the common Mathplot operations can be performed either by pressing a function key or by typing a command. These commands are described below: Function key Command Action ------------ ------- ----------------------------------- ESC EXIT Exit from Mathplot to DOS F1 PLOT Plot the current functions F2 Print current functions on LaserJet F3 HELP Display help menu F4 RESET Erase functions and reset Mathplot F5 Run an example Chapter 1. Using Mathplot 3 The F5 key executes built-in example plots. Each time you press F5 a different example is displayed. I suggest you use this feature to display a few plots in order to get a feeling for how Mathplot works and what types of commands are used to produce plots. Press any key to continue with execution after you finish looking at a plot. Note that after each example is completed the commands that produced it are displayed on your screen. By examining a few sets of example commands you should be able to get a pretty good idea about how to use Mathplot. Once you have used F5 to display some plots, it is time to try entering your own commands. Begin by specifying a simple function to be plotted. I suggest you enter the command: Y=SIN(X) You may use either upper or lower case letters when you type a command. Press Enter when you are finished typing the line. Next, press F1 to command Mathplot to evaluate this function and plot the results. You should see a sine wave displayed on your screen. Press Enter to return to the command editor. Now let's make the function slightly more complicated by entering the command: Y=SIN(X)+SIN(3*X)/3 This replaces the previous function definition. Press F1 to see this function plotted. If you make a mistake when typing a command, use the arrow and editing keys such as Backspace and Delete to correct your mistake, then press Enter to reexecute the command. So far, the function has been evaluated over the interval (0,2*pi), which is the initial default domain. Use the following command to change the domain to (-2*pi,2*pi): DOMAIN -2*PI,2*PI Also, use the following command to turn on grid line display: GRID ON Press F1 to display the plot. As a final exercise, use the following command to define a second function to be displayed simultaneously with the first one: Y2=SIN(X) And, again press F1 to display them. You should see a simple sine wave (the Y2 function) superimposed on the previous function (the Y function). Press ESC to exit Mathplot and return to DOS. Chapter 1. Using Mathplot 4 1.4 Demonstration Command File The Mathplot distribution includes a command file named MATHPLOT.FUN containing a number of interesting example plots. You can execute this command file by using the Mathplot "DO" command, or by specifying the command file on the invocation line. The following DOS command starts the Mathplot program and instructs it to execute the commands stored in the file MATHPLOT.FUN: MATHPLOT MATHPLOT.FUN 1.5 Command Editor The middle window of the main Mathplot screen is used by the command editor to accept and display commands you type. In addition to typing commands, you can also use the arrow keys and other editing functions to recall, edit, and reexecute commands. If you enter a command that has an error, a message will appear in the bottom window. You can use the up-arrow key to move the cursor up to the previous line, edit it, and then reexecute it by pressing Enter. Each time you press Enter, the line in which the cursor is positioned is executed. Using the arrow keys to move up and down does not execute commands. 1.5.1 Function Keys The following function keys can be used with the command editor: Delete -- Delete the character under the cursor. Home -- Move to the left end of the line. End -- Move to the right end of the line. Page up -- Move up to the previous page of commands. Page down -- Move down to the next page of commands. Insert -- Switch between insert and over type mode. 1.5.2 Control Characters The following control characters can be used with the command editor: Enter -- Execute the command line the cursor is on. Backspace -- Delete the character to the left of the cursor. Control-U -- Delete characters to the left of the cursor. Control-D -- Delete all characters in the current line. 1.6 Function Specification Much of the power of Mathplot comes from its ability to evaluate complicated functions entered in ordinary algebraic form. This section explains the arithmetic operators and built in functions that are used to define a function. Chapter 1. Using Mathplot 5 1.6.1 Arithmetic Operators The following arithmetic operators may be used in expressions: + addition - subtraction or unary minus * multiplication / division ** or ^ exponentiation Exponentiation has the highest precedence, followed by multiplication and division, and then addition and subtraction. Parentheses may be used to group terms. As a convenience, Mathplot allows you to omit the multiplication operator between a numeric constant and a following variable or function. For example, the expressions "2pi", and "2 pi" are equivalent to "2*pi". Similarly, "5X" is equivalent to "5*X". However, if you specify a number before the letter "E", it will be taken as the exponential form of a number (see below) rather than the number times the constant E (base of natural logarithms). 1.6.2 Numeric Constants Numeric constants may be written in their natural form (1, 0, 1.5, .0003, etc.) or in exponential form, n.nnnEppp, where n.nnn is the base value and ppp is the power of ten by which the base is multiplied. For example, the number 1.5E4 is equivalent to 15000. All numbers are treated as "floating point" values, regardless of whether a decimal point is specified or not. 1.6.3 Symbolic Constants There are two numeric constants that may be specified using symbolic names. The symbolic name "PI" is equivalent to the value of pi, 3.14159... Similarly, the symbolic constant "E" is equivalent to the base of natural logarithms, 2.7182818... 1.6.4 Parameter values Mathplot allows you to use parameters named P0, P1, ..., P9 in function definitions. You can then assign values to the parameters and replot the function without having to retype the function to change a value. To assign a value to a parameter, type the parameter name as a command followed by the value to be assigned. For example, the following command assigns the value 0.67 to the parameter P0: P0 0.67 Note that parameters are like constants during the evaluation and plotting of a function (i.e., their values do not change). But you can assign new values to parameters between plot generations. For example, consider the following commands which define a function Chapter 1. Using Mathplot 6 and then plot it twice with two different values of the P0 parameter: Y = EXP(-P0*X) * COS(X) P0 0.6 PLOT P0 0.8 PLOT This function represents the response of a damped oscillator with the P0 coefficient controlling the damping factor. By changing the value of the P0 parameter you can examine different forms of the function without having to retype the function. 1.7 Built in Functions The following functions are built into Mathplot and may be used in expressions: ABS(x) -- Absolute value of x. ACOS(x) -- Arc cosine of x. Angles are measured in radians. ASIN(x) -- Arc sine of x. Angles are measured in radians. ATAN(x) -- Arc tangent of x. Angles are measured in radians. BETAI(x,a,b) -- Incomplete beta function: Ix(a,b). The incomplete beta function can be used to compute a variety of statistical functions. For example, the probability of Student's t with `df' degrees of freedom can be computed with BETAI(df/(df+t^2),.5*df,.5). The probability of the F statistic with df1 and df2 degrees of freedom can be computed with 2*BETAI(df2/(df2+df1*f),.5*df2,.5*df1). COS(x) -- Cosine of x. Angles are measured in radians. COSH(x) -- Hyperbolic cosine of x. COT(x) -- Cotangent of x. (COT(x) = 1/TAN(x)). CSC(X) -- Cosecant of x. (CSC(x) = 1/SIN(x)). DEG(x) -- Converts an angle, x, measured in radians to the equivalent number of degrees. EI1(alpha,phi) -- Elliptic integral of the first kind. Computes the integral from 0 to phi radians of the function d.phi/sqrt(1-k**2*sin(phi)**2), where k = sin(alpha). alpha and phi must be in the range 0 to pi/2. Chapter 1. Using Mathplot 7 EI2(alpha,phi) -- Elliptic integral of the second kind. Computes the integral from 0 to phi radians of the function sqrt(1-k**2*sin(phi)**2)*d.phi, where k = sin(alpha). alpha and phi must be in the range 0 to pi/2. EIC1(alpha) -- Complete elliptic integral of the first kind. Computes the integral from 0 to pi/2 radians of the function d.phi/sqrt(1-k**2*sin(phi)**2), where k = sin(alpha). alpha must be in the range 0 to (less than) pi/2. EIC2(alpha) -- Complete elliptic integral of the second kind. Computes the integral from 0 to pi/2 radians of the function sqrt(1-k**2*sin(phi)**2)*d.phi, where k = sin(alpha). alpha must be in the range 0 to pi/2. ERF(x) -- Standard error function of x. EXP(x) -- e (base of natural logarithms) raised to the x power. FAC(x) -- x factorial (x!). Note, the FAC function is computed using the GAMMA function (FAC(x)=GAMMA(x+1)) so non-integer argument values may be computed. GAMMA(x) -- Gamma function. Note, GAMMA(x+1) = x! (x factorial). GAMMAI(x) -- Reciprocal of GAMMA function (GAMMAI(x) = 1/GAMMA(x)). GAMMALN(x) -- Log (base e) of the GAMMA function. HAV(x) -- Haversine of x. (HAV(x) = (1-COS(x))/2). J0(x) -- Bessel function of the first kind, order zero. J1(x) -- Bessel function of the first kind, order one. JN(n,x) -- Bessel function of the first kind, order n. LOG(x) -- Natural logarithm of x. LOG10(x) -- Base 10 logarithm of x. MAX(x1,x2) -- Maximum value of x1 or x2. MIN(x1,x2) -- Minimum value of x1 or x2. NORMAL(x) -- Normal probability distribution of x. X is in units of standard deviations from the mean. See also the description of the NPD function. NORMAL(x) = NPD(x,0,1). NPD(x,mean,std) -- Normal probability distribution of x with specified mean and standard deviation. X is in units of standard deviations from the mean. Chapter 1. Using Mathplot 8 PAREA(x) -- Area under the normal probability distribution curve from -infinity to x. (i.e., integral from -infinity to x of NORMAL(x)). PULSE(a,x,b) -- Pulse function. If the value of x is less than a or greater than b, the value of the function is 0. If x is greater than or equal to a and less than or equal to b, the value of the function is 1. In other words, it is 1 for the domain (a,b) and zero elsewhere. If you need a function that is zero in the domain (a,b) and 1 elsewhere, use the expression (1-PULSE(a,x,b)). RAD(x) -- Converts an angle measured in degrees to the equivalent number of radians. SEC(x) -- Secant of x. (SEC(x) = 1/COS(x)). SIN(x) -- Sine of x. Angles are measured in radians. SINH(x) -- Hyperbolic sine of x. SQRT(x) -- Square root of x. STEP(a,x) -- Step function. If x is less than a, the value of the function is 0. If x is greater than or equal to a, the value of the function is 1. If you need a function which is 1 up to a certain value and then 0 beyond that value, use the expression STEP(x,a). T(n,x) -- Chebyshev polynomial of order n. TAN(x) -- Tangent of x. Angles are measured in radians. TANH(x) -- Hyperbolic tangent of x. Y0(x) -- Bessel function of the second kind, order zero. Y1(x) -- Bessel function of the second kind, order one. YN(n,x) -- Bessel function of the second kind, order n. 1.8 Function Types Mathplot allows four different types of functions to be plotted: cartesian, parametric cartesian, polar, and parametric polar. Each of the function types is described below: 1.8.1 Cartesian Functions A cartesian function has the form Y=f(X), where X is the independent variable that is plotted along the horizontal axis and Y is the dependent variable plotted on the vertical axis. The value specified with the DOMAIN command controls the interval of X Chapter 1. Using Mathplot 9 values over which the function is evaluated. Some examples of this type of function are listed below: Y=SIN(X) Y=2*X**2-3*X+5 Y=SIN(X)/EXP(X) Y=1/SQRT(2*(X+1)) 1.8.2 Parametric Cartesian Functions Parametric functions use a third variable, which we will call T, as the independent variable. Both X and Y are dependent variables defined as functions of T. The value of the T variable is not directly displayed on the plot but is used only in the functions that define the values of the X and Y variables that are plotted. For example, the following commands define X and Y as functions of T: X=COS(T) Y=SIN(T) When parametric functions are defined, the DOMAIN statement specifies the interval over which the T variable is computed. If the two functions of X and Y specified above are plotted over the domain (0,2*PI), the result is a circle with radius 1 centered around the origin. 1.8.3 Polar Functions A polar function specifies the distance (or radius) of a point from the origin as a function of the angle swept around the origin. The angle (A) is the independent variable and the radius (R) is the dependent variable. The DOMAIN statement specifies the interval of values that the angle is to be evaluated over. The angle begins on the positive X axis; positive angles are measured counterclockwise around the origin. Angles are measured in radians; the RAD and DEG functions can be used to convert between degrees and radians. The simplest polar function is R=1 which specifies that for all angles the radius is 1. If this function is plotted over the domain (0,2*pi) the result is a circle of radius 1 centered around the origin. Another simple polar function is R=A which defines a spiral with the radius increasing with the angle. 1.8.4 Parametric Polar Functions A parametric polar function defines the radius (R) and angle (A) as functions of the parametric variable, T. The DOMAIN statement defines the interval of T values over which the function is Chapter 1. Using Mathplot 10 evaluated. The following is an example of a parametric polar function definition: R=COS(T) A=SIN(T) Over the domain (0,2*pi), this produces a figure-eight pattern. 1.9 Plotting Simultaneous Functions Up to four functions may be plotted at the same time. When defining multiple functions, add a single-digit suffix to the dependent variable name for the function. For example, the following commands define four functions to be plotted simultaneously: Y1=SIN(X) Y2=COS(X) Y3=SIN(2*X) Y4=COS(2*X) If no number is specified, the default value is 1. Thus, the following two function definitions are equivalent: Y=2*X**2 Y1=2*X**2 You can remove a function definition by defining another function for the same dependent variable or by specifying the dependent variable without a function. For example, the following command removes the definition for Y2: Y2 When a function of a type (cartesian, polar, etc.) different from the current type is defined, all of the currently defined functions are removed. Chapter 2 Mathplot Commands The commands described in this section are used to control Mathplot. When Mathplot is waiting for a command, the cursor is positioned in the middle window of the screen. If you need to type a command that is longer than a single line, you may continue a command by typing a minus sign as the last character on the line to be continued. Note that a minus sign as the last character on a line is interpreted by Mathplot as a continuation marker. It is NOT used as a minus sign in the expression. Commands may be abbreviated to the first three letters of their keyword. Commands may be typed using upper or lower case letters. You may place comments on commands by preceding the comment by an exclamation point. For example, the following commands have comments: ! Define square wave function Y=SIN(X)+SIN(3*X)/3 ! Approximate square wave DOMAIN 0,2*PI ! One cycle of the function 2.1 Start up Command File The DOS command to start Mathplot is: MATHPLOT [filename] where "filename" is an optional parameter specifying the name of a file containing commands to be executed by Mathplot as soon as it is started. This start up command file may contain one or more complete sets of statements to define functions and plot them. The default file extension is ".FUN". See also the descriptions of the DO and SAVE commands for additional information about command files. 2.2 Initialization file Each time Mathplot is started it attempts to open a file named MATHPLOT.INI. If this file exists, all commands in it are executed. You may create such a file to contain initialization 11 Chapter 2. Mathplot Commands 12 commands to set default values. For example, if you normally like to have grid lines turned on, create a file named MATHPLOT.INI containing the command "GRID ON". Mathplot looks for this file first in the current directory and if it does not find it there it checks to see if a MATHPLOT environment variable has been defined to specify a directory. The initialization file is executed before any file specified on the command that starts Mathplot. 2.3 Command Arguments Many of the commands accept numeric arguments. In the descriptions below, the notation "cexpr" is shown wherever a numeric argument may be specified. These numeric arguments may consist of numbers, symbolic constants, and expressions using operators and built in functions. For example, the following is a valid DOMAIN command: DOMAIN -2*PI,PI*SQRT(2) The argument expressions may NOT contain variables such as X, Y, R, A, or T, or parameters such as P0 or P1. Arguments shown in brackets are optional. Braces around a set of options indicate that you must choose one of the options in the set. 2.4 Alphabetical List of Commands 2.4.1 AXES AXES {ON | OFF} (default=ON) -- Specifies whether to display axes. Use the LABELS command to control whether axis labels are displayed. 2.4.2 CALCULATE CALCULATE cexpr -- Evaluate the specified expression, which must contain only numbers, operators, and built in functions, and display the result in the bottom window. 2.4.3 CAXES CAXES cexpr (default=3) -- Specify the color to be used for the axis lines and labels. 2.4.4 CFn CFn cexpr (default=2,1,4,3) -- Define the color to be used when plotting function 'n'. For example, the command "CF1 3" specifies that color 3 is to be used when plotting function 1. Chapter 2. Mathplot Commands 13 2.4.5 CGRID CGRID cexpr (default=8) -- Specify the color to be used for grid lines. 2.4.6 COLORBORDER COLORBORDER color1 [ON color2] (default=WHITE) -- Specify the color of the border lines. The 'color1' and 'color2' parameters must be one of the following keywords: WHITE, BLACK, BLUE, GREEN, RED, CYAN, MAGENTA, BROWN. The 'color1' parameter specifies the foreground color (i.e., the color of the characters). If the "ON color2" phrase is used 'color2' is the background color. If the "ON color2" phrase is omitted, "ON BLACK" is used by default. For example, to get white characters on a blue background, specify: COLORBORDER WHITE ON BLUE This command (and the other COLORxxx) commands can be placed in the MATHPLOT.INI initialization file to cause your selected colors to be initialized automatically each time Mathplot is started. 2.4.7 COLOREDIT COLOREDIT color1 [ON color2] (default=WHITE) -- Specify the color of the center portion of the screen where commands are typed and displayed. 2.4.8 COLORHEAD COLORHEAD color1 [ON color2] (default=CYAN) -- Specify the color of the text in the heading portion of the screen (i.e., "ESC=Exit F1=Plot", etc.). 2.4.9 COLORHELP COLORHELP color1 [ON color2] (default=WHITE) -- Specify the color of the help screen text. 2.4.10 COLORHELPHEAD COLORHELPHEAD color1 [ON color2] (default=MAGENTA) -- Specify the color of the heading of the help display screen. 2.4.11 COLORMESSAGE COLORMESSAGE color1 [ON color2] (default=GREEN) -- Specify the color of information and error messages displayed in the lower portion of the screen. 2.4.12 COMMONSCALE COMMONSCALE {ON | OFF} (default=OFF) -- If this option is turned on, the X and Y directions are forced to have the same scale. If this option is off, the X and Y ranges are scaled independently, Chapter 2. Mathplot Commands 14 which may cause figures to be distorted - for example, a circle may display as an ellipse. However, turning this option on may result in one of the dimensions using only a small portion of the screen if the X and Y ranges are very different. This option is always on for polar function plots. 2.4.13 CTITLE CTITLE cexpr (default=7) -- Specify the color to use for the title line. 2.4.14 DATA DATA filename -- Causes X and Y data values to be read from an external file rather than being computed from function definitions. After the data has been read, use the PLOT command to display it. Any function definition following this command will clear the external data values and revert to function mode. Each X,Y data pair must be specified as a separate line in the file with a space separating the X value from the Y value. The default extension for the file is ".DAT". 2.4.15 DISPLAY DISPLAY string -- Display the string in the bottom window of the screen. The string is displayed for 3 seconds. This command may be placed in command file to cause information to be printed while the command file is being processed. 2.4.16 DO DO filename -- Execute the Mathplot commands stored in an external file. This allows you to specify a complete plot request, including functions and options, and then execute it without having to retype the commands. An external file may itself contain a DO command and this nesting may be performed to a depth of 10 files. The default file extension is ".FUN". Use the LIST command to control whether the commands in the file are displayed in the edit window. See also the description of the SAVE command. 2.4.17 DOMAIN DOMAIN cexpr1,cexpr2 (default=0,2*pi) -- Define the domain of the independent variable over which the functions are evaluated and plotted. This command may be abbreviated to "DOM". 2.4.18 EXIT EXIT -- Stop Mathplot and return to DOS. Pressing ESC will also cause Mathplot to exit. Chapter 2. Mathplot Commands 15 2.4.19 GRID GRID {ON | OFF} (default=OFF) -- Specify whether grid lines are to be displayed. 2.4.20 HELP HELP -- Display the help menu. You can also do this by pressing F3. 2.4.21 LABEL LABEL {ON | OFF} (default=ON) -- Specify whether axis labels are to be displayed. 2.4.22 LIST LIST {ON | OFF} (default=OFF) -- Specify whether commands executed from an external file by subsequent use of the DO command are to be listed as they are executed. 2.4.23 NUMPOINTS NUMPOINTS cexpr (default=100) -- Specify the number of points at which the functions are to be evaluated over the domain. Specifying a larger number of points results in a smoother plot but increases the computation time. The maximum number of points that may be computed is 2000. This command may be abbreviated to the single letter 'N'. 2.4.24 ORIGIN ORIGIN {ON | OFF} (default=OFF) -- Specify whether the origin (0,0) is to be forced to be included in the plot. If this option is off and the range of values does not span 0, then the origin may not be displayed. Turning this option on allows the range of the function relative to the origin to be observed, but may result in a small scale factor if the range is far from the origin. 2.4.25 Pn Pn cexpr -- Specify a value for parameter Pn where 'n' is in the range 0 to 9. Mathplot allows you to use up to ten parameter values in function specifications. The parameters are named P0, P1, ..., P9. You can assign a new value to a parameter and replot the function without having to retype the function. 2.4.26 PAUSE PAUSE [cexpr] (default=0) -- This command may be placed in a command file to cause execution to pause until a key is pressed or the specified number of seconds elapse. If cexpr is omitted or has the value 0, execution pauses for an indefinite time until a key is pressed. Chapter 2. Mathplot Commands 16 2.4.27 PDEVICE PDEVICE device (default=PRN) -- Allows you to specify the device or file to which printer output is written when printing is turned on by use of the PRINT ON command. The default device is "PRN" but you may specify another device such as LPT2 or COM. You may also direct output to a disk file that you can printer later using the DOS PRINT or COPY commands. 2.4.28 PRESOLUTION PRESOLUTION value (default=150) -- Specifies whether plots sent to HP LaserJet printers should use 150 or 300 dot-per-inch resolution. The value parameter must be 150 or 300. The default value is 150 causes the plots to use most of the horizontal width of an 8.5x11 inch page. These plots are suitable for direct transfer to overhead transparencies. Specifying 300 for the resolution produces smaller plots that are suitable for inclusion in printed documents. 2.4.29 WIDTH WIDTH value (optional) -- Specify the width, in inches, of printed plots. Due to memory space considerations, the maximum width is limited to about 7.9 inches for 150 DPI resolution and 4.5 inches for 300 DPI resolution. If you have limited memory space, you may have to reduce the width to be able to produce printed plots. This statement is ignored unless you request that a plot be printed. 2.4.30 PLOT PLOT [cexpr] (default=0) -- Evaluate the currently defined functions and plot them. If an optional value (cexpr) is specified, the function is displayed for the specified number of seconds or until a key is pressed. If the optional value is omitted or is zero, the plot is displayed for an indefinite time until a key is pressed. The use of the optional value is most useful when the PLOT command occurs in a command file producing a "slide show" of functions. You can also plot the function(s) by pressing F1. 2.4.31 PRINT PRINT {ON | OFF} (default=OFF) -- Turns printer output on or off. When turned on, any plot displayed on the screen is also written to the printer (or file as directed by the PDEVICE command). To produce a printed copy of the currently defined function(s) press F2. 2.4.32 RESET RESET -- Reset all parameters and options to their initial values and remove all function definitions. You can also perform this function by pressing F4. Chapter 2. Mathplot Commands 17 2.4.33 SAVE SAVE filename -- Write the current function definitions and the values of all options and parameters to the specified file. The DO command can then be used at a later time to redisplay the function. The default file extension is ".FUN". 2.4.34 TITLE TITLE string -- Define a title line to be displayed at the top of the plot. 2.4.35 WAXES WAXES cexpr (default=3) -- Specify the width (in dots) of the axis lines for printed output. This parameter does not affect the screen display. 2.4.36 WFn WFn cexpr (default=3) -- Specify the width (in dots) of the line used to draw the plot of function 'n'. This only affects hard copy output. 2.4.37 WGRID WGRID cexpr (default=1) -- Specify the width (in dots) of the grid lines. This only affects hard copy output. Chapter 3 Advanced Applications 3.1 Root Finding Mathplot can be used to find the roots of equations. To do this, turn on axis labeling (LABEL ON) and plot the function over the domain in which a root occurs. Observe the approximate X value at which the function crosses the X axis and then reset the domain to closely span the crossing point. Replot the function and obtain a new X estimate. After several iterations you will be able to determine the root to several significant digits. For example, use this technique to find the root of the function: Y = X^4 - EXP(X) in the domain (2,9.5). For a course in Algebra I, Mathplot can be used to demonstrate the graphical method of solving two simultaneous linear equations by locating the point of intersection. For an Algebra II course, Mathplot can be used to locate minimum and maximum points for quadratic and cubic equations. 3.2 Multiple Domains Mathplot only allows specification of a single continuous domain of values. However, by using the multiple function plotting capability it is sometimes possible to simulate multiple domains. For example, the function X*Y=5 defines a hyperbola with branches in both the positive X-Y domain (upper right quadrant) and the negative X-Y domain (lower left quadrant). The following set of commands defines two parametric functions to draw each of the branches: X1=T Y1=5/T X2=-T Y2=-5/T DOMAIN 0.5,10 LABEL ON COMMONSCALE ON PLOT 18 Chapter 3. Advanced Applications 19 3.3 Interesting Functions The following specifications plot interesting and attractive functions: TITLE Four Leaved Rose CTITLE 4 R=2*SIN(2*A) CF1 3 DOMAIN 0,2*PI NUMPOINTS 150 AXES ON CAXES 1 PLOT TITLE Prolate Cycloid CTITLE 4 Y=1-2*COS(T) X=T-2*SIN(T) CF1 2 DOMAIN -4*PI,4*PI NUMPOINTS 400 CAXES 1 COMMONSCALE ON PLOT TITLE Hypocycloid of four cusps (Asteroid) CTITLE 4 Y=SIN(T)^3 X=COS(T)^3 CF1 2 DOMAIN 0,2*PI NUMPOINTS 100 AXES ON CAXES 1 COMMONSCALE ON PLOT TITLE Bifolium CTITLE 4 Y=2*SIN(T)^2*COS(T)^2 X=2*SIN(T)*COS(T)^3 CF1 3 DOMAIN 0,2*PI NUMPOINTS 100 AXES ON CAXES 1 COMMONSCALE ON PLOT Chapter 3. Advanced Applications 20 TITLE Epicycloid CTITLE 4 Y=5*SIN(T)-SIN(5*T) X=5*COS(T)-COS(5*T) CF1 3 DOMAIN 0,2*PI NUMPOINTS 150 AXES ON CAXES 1 COMMONSCALE ON PLOT 3.4 "Artistic" Plots When Mathplot plots a function, it evaluates the function at the number of points specified by the last NUMPOINTS command and then connects the points together using straight lines. By specifying very large domain values for polar or parametric functions, the lines connecting the points span large distances and produce interesting and "artistic" plots. It is best to turn off axis display (AXES OFF, LABELS OFF) and turn on common scaling (COMMONSCALE ON) for these plots. Here are some examples to try: Y=SIN(T)^3 X=COS(T)^3 CF1 2 DOMAIN 0,4000 NUMPOINTS 500 R=2*SIN(2*A) CF1 3 DOMAIN 0,2000 NUMPOINTS 500 Y=5*SIN(T)-SIN(5*T) X=5*COS(T)-COS(5*T) CF1 3 DOMAIN 0,6000 NUMPOINTS 400 Parametric and polar functions seem to produce the best artistic plots. By varying the NUMPOINTS and DOMAIN values you can usually produce many different plots from the same functions. Chapter 4 Use and Distribution of Mathplot There are two versions of the Mathplot program: shareware and registered. You are welcome to make copies of the shareware version of Mathplot and pass them on to friends or post this program on bulletin boards or distribute it via disk catalog services, CD ROMS, or other means provided the entire Nonlin distribution is included in its original, unmodified form. A distribution fee may be charged for the cost of the diskette, shipping and handling. Vendors are encouraged to contact the author to get the most recent version of Nonlin. As a shareware product, you are granted a no-cost, trial period of 30 days during which you may evaluate Mathplot. If you find Mathplot to be useful, educational, and/or entertaining, and continue to use it beyond the 30 day trial period, you are required to compensate the author by sending the registration form printed on page 25 (and in REGISTER.DOC) with the appropriate registration fee to help cover the development and support of Mathplot. In return for registering, you will be authorized to continue using Mathplot beyond the trial period and you will receive the registered version of the program, a bound, typeset manual, and three months of support via telephone, mail, or e-mail. Your registration fee will be refunded if you encounter a serious bug that cannot be corrected. See also the special offer for Nonlin on page 23. This program is produced by a member of the Association of Shareware Professionals (ASP). ASP wants to make sure that the shareware principle works for you. If you are unable to resolve a shareware-related problem with an ASP member by contacting the member directly, ASP may be able to help. The ASP Ombudsman can help you resolve a dispute or problem with an ASP member, but does not provide technical support for members' products. Please write to the ASP Ombudsman at 545 Grover Road, Muskegon, MI 49442 or send a CompuServe message via CompuServe Mail to ASP Ombudsman 7007,3536. 21 Chapter 4. Use and Distribution of Mathplot 22 You are welcome to contact the author: Phillip H. Sherrod 4410 Gerald Place Nashville, TN 37205-3806 USA 615-292-2881 (evenings) CompuServe: 76166,2640 Internet: 76166.2640@compuserve.com 4.1 Copyright Notice Both the Mathplot program and documentation are copyright (c) 1991-1994 by Phillip H. Sherrod. You are not authorized to modify the program. "Mathplot" is a trademark. 4.2 Disclaimer This software and documentation are provided on an "as is" basis. This program may contain "bugs" and inaccuracies, and its results should not be assumed to be correct unless they are verified by independent means. Phillip H. Sherrod disclaims all warranties relating to this software, whether expressed or implied, including but not limited to any implied warranties of merchantability or fitness for a particular purpose. Neither Phillip H. Sherrod nor anyone else who has been involved in the creation, production, or delivery of this software shall be liable for any indirect, consequential, or incidental damages arising out of the use or inability to use such software, even if Phillip H. Sherrod has been advised of the possibility of such damages or claims. The person using the software bears all risk as to the quality and performance of the software. This agreement shall be governed by the laws of the State of Tennessee and shall inure to the benefit of Phillip H. Sherrod and any successors, administrators, heirs and assigns. Any action or proceeding brought by either party against the other arising out of or related to this agreement shall be brought only in a state or federal court of competent jurisdiction located in Davidson County, Tennessee. The parties hereby consent to in personam jurisdiction of said courts. Chapter 5 Other Software 5.1 Nonlin If you like Mathplot, you should check out Nonlin -- the nonlinear regression analysis program by the same author. And, if you register your use of Mathplot and order Nonlin at the same time, you can get both for the special price of $60. What is regression analysis? Regression analysis is a mathematical technique for determining the best values of parameters to fit an equation to a set of data points. For example, you might want to develop an equation of the form price = p0 + p1*age + p2*miles to predict the price of a used car based on its age and the number of miles driven. With Nonlin you can collect data from car ads and then perform the analysis using the following set of commands: VARIABLES PRICE,AGE,MILES PARAMETERS P0,P1,P2 FUNCTION PRICE = P0 + P1*AGE + P2*MILES DATA Nonlin will analyze the data and determine the best values of the parameters P0, P1, and P2 to fit the data values. Ordinary linear regression programs can only determine parameter values for linear (straight line) equations. Nonlin, on the other hand, can handle multivariate, linear, polynomial, and general nonlinear equations. For example, using Nonlin you can easily determine the best values for the parameters Offset, Amplitude, and Frequency for an equation of the form: Y = Offset + Amplitude * sin(Frequency * X) Nonlin uses the same expression evaluator as Mathplot so you can model complicated equations using the full set of operators and library functions available in Mathplot. Nonlin comes with a 45 page manual that explains regression analysis and gives many examples. Nonlin is in use at many universities and research labs around the world. 23 Chapter 5. Other Software 24 5.2 TSX-32 Operating System If you have a need for a multi-user, multi-tasking operating system, you should look into TSX-32. TSX-32 is a full-featured, high performance, multi-user operating system for the 386 and 486 that provides both 32-bit and 16-bit program support. With facilities such as multitasking and multisessions, networking, virtual memory, X-Windows, background batch queues, data caching, file access control, real-time, and dial-in support, TSX-32 provides a solid environment for a wide range of applications. A two user, shareware version of TSX-32 called TSX-Lite is also available. TSX-32 is not a limited, 16-bit, multi-DOS add-on. Rather, it is a complete 32-bit operating system which makes full use of the hardware's potential, including protected mode execution, virtual memory, and demand paging. TSX-32 sites range from small systems with 2-3 terminals to large installations with more than 64 terminals on a single 386. In addition to supporting most popular 16-bit DOS programs, TSX-32 also provides a 32-bit "flat" address space with both Phar Lap and DPMI compatible modes of execution. Since the DOS file structure is standard for TSX-32, you can directly read and write DOS disks. And, you can run DOS part of the time and TSX-32 the rest of the time on the same computer. TSX-32 allows each user to control up to 10 sessions. Programs can also "fork" subtasks for multi-threaded applications. The patented Adaptive Scheduling Algorithm provides consistently good response time under varying conditions. The TSX-32 network option provides industry standard TCP/IP networking through Ethernet and serial lines. Programs can access files on remote machines as easily as on their own machine. The SET HOST command allows a user on one machine to log onto another computer in the network. FTP, Telnet, and NFS are available for interoperability with other systems. System requirements: 386 or 486 system, 4MB memory, 12MB of free disk space (Stacker and DoubleSpace are not supported). TSX-32 is the best and most powerful operating system available for the 386 and 486. For additional information contact: S&H Computer Systems, Inc. 1027 17th Avenue South Nashville, TN 37212 USA 615-327-3670 (voice) 615-321-5929 (fax) CompuServe: 71333,27 Internet: 71333.27@compuserve.com ================================================================= Software Order Form ================================================================= Name ______________________________________________________ Address ___________________________________________________ City _______________________ State _______ Zip ___________ Country ____________________ Telephone ___________________ Internet address (optional) _______________________________ Mathplot version you currently have _______________________ Bulletin board where you found Mathplot ___________________ Comments __________________________________________________ Check the box below which indicates your order type: ___ I wish to register Mathplot ($20). ___ I wish to order Nonlin ($45). ___ I wish to register Mathplot and order Nonlin ($60). Add $5 to any amount shown above if the software is being shipped out of the United States. I cannot accept checks from non-US banks. Visa, MasterCard and American Express credit card charges are accepted but a check, money order, or cash is preferred. If you wish to use a credit card specify the billing name, address, card number, and expiration date. In return for registering, you will receive the registered version of the program, a laser-printed, bound copy of the manual, and three months of telephone or CompuServe support. Your registration fee will be refunded if you find a serious bug that cannot be corrected. Distribution disk choice (check one): 3.50" HD (1.4 MB) ______ 5.25" HD (1.2 MB) ______ 5.25" DD (360 KB) ______ Send this form with the amount indicated to the author: Phillip H. Sherrod 4410 Gerald Place Nashville, TN 37205-3806 USA 615-292-2881 (evenings) CompuServe: 76166,2640 Internet: 76166.2640@compuserve.com Index 26 Abbreviating commands, 11 DEG function, 6 ABS function, 6 Device for output, 16 ACOS function, 6 Disclaimer, 22 Arc cosine, 6 DISPLAY command, 14 Arc sine, 6 DO command, 14 Arc tangent, 6 DOMAIN command, 14 Artistic plots, 20 Domains ASIN function, 6 multiple, 18 ASP, 21 Edit area color, 13 Asteroid, 19 EI1 function, 6 ATAN function, 6 EI2 function, 7 AXES command, 12 EIC1 function, 7 Bessel function, 7, 8 EIC2 function, 7 Beta function, 6 Elliptic integral, 6, 7 BETAI function, 6 Epicycloid, 20 Bifolium, 19 ERF function, 7 Border color, 13 Error color, 13 Build-in functions, 6 EXIT command, 14 Calculate command, 12 EXP function, 7 Cartesian functions, 8 Exponential, 7 CAXES command, 12 Exponentiation operator, 5 CFn command, 12 External command file, 14 CGRID command, 13 External data file, 14 Chebyshev polynomial, 8 FAC function, 7 Color Factorial, 7 of axes, 12 Four leaved rose, 19 of border, 13 GAMMA function, 7 of edit area, 13 GAMMALN function, 7 of errors, 13 GAMMI function, 7 of function, 12 Grid color, 13 of grid, 13 GRID command, 15 of heading, 13 HAV function, 7 of help, 13 Haversine, 7 of messages, 13 Heading color, 13 of title, 14 Help color, 13 COLORBORDER command, 13 HELP command, 15 COLOREDIT command, 13 Hyperbolic cosine, 6 COLORHEAD command, 13 Hyperbolic sine, 8 COLORHELP command, 13 Hyperbolic tangent, 8 COLORHELPHEAD command, 13 Hypocycloid, 19 COLORMESSAGE command, 13 Initialization file, 11 Comments, 11 Interesting functions, 19 COMMONSCALE command, 13 Inverse gamma, 7 Continuing lines, 11 J0 function, 7 Copyright notice, 22 J1 function, 7 COS function, 6 JN function, 7 Cosecant function, 6 LABEL command, 15 COSH function, 6 LIST command, 15 COT function, 6 LOG function, 7 Cotangent function, 6 Log gamma, 7 CSC function, 6 LOG10 function, 7 CTITLE command, 14 Long lines, 11 Curve fitting, 23 MATHPLOT.INI, 11 Cycloid, 19 MAX function, 7 DATA command, 14 Message color, 13 Index 27 MIN function, 7 WGRID command, 17 Multiple domains, 18 WIDTH command, 16 Nonlin, 23 Y0 function, 8 Nonlinear regression, 23 Y1 function, 8 NORMAL function, 7 YN function, 8 NPD function, 7 Numeric constants, 5 NUMPOINTS command, 15 Operator precedence, 5 Order form, 25 ORIGIN command, 15 Parameter values, 5, 15 Parametric functions, 9 PAREA function, 7 PAUSE command, 15 PDEVICE command, 16 PI constant, 5 PLOT command, 16 Pn command, 15 Polar functions, 9 Precedence of operators, 5 PRESOLUTION command, 16 PRINT command, 16 Printer resolution, 16 Probability distribution, 7 Prolate cycloid, 19 PULSE function, 8 RAD function, 8 REGISTER.DOC, 21 Registration form, 25 Regression analysis, 23 RESET command, 16 Resolution of printer, 16 Root finding, 18 Rose figure, 19 SAVE command, 17 Scaling, 13 SEC function, 8 Secant, 8 SIN function, 8 SINH function, 8 SQRT function, 8 Standard error function, 7 Start up file, 11 STEP function, 8 Symbolic constants., 5 T function, 8 TAN function, 8 TANH function, 8 Title color, 14 TITLE command, 17 Trademark notice, 22 TSX-32, 24 WAXES command, 17 WFn command, 17