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M A T H P L O T Mathematical Function Plotting Program Phillip H. Sherrod A "shareware" program. MATHPLOT allows you to specify complicated 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 may be generated 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. MATHPLOT -- Mathematical Function Plotter Page 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. Printed copies of plots may be produced on HP LaserJet printers. MATHPLOT -- Mathematical Function Plotter Page 2 INSTALLING MATHPLOT The MATHPLOT system consists of the following files: MATHPLOT.EXE -- The executable program. MATHPLOT.HLP -- Text file accessed by HELP command. MATHPLOT.FON -- Font file used for title line and axis labels. MATHPLOT.LJF -- Font file for HP LaserJet printer. MATHPLOT.FUN -- Example functions MATHPLOT.DOC -- Documentation file. 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.HLP, 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 help and 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 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. MATHPLOT prints a greater-than sign (">") to prompt you for a command. Begin by specifying a simple function to be plotted. I suggest you enter the command: MATHPLOT -- Mathematical Function Plotter Page 3 Y=SIN(X) You may use either upper or lower case letters when you type a command. Next, command MATHPLOT to evaluate this function and plot the results by entering the command: PLOT You should see a sine wave displayed on your screen. Press Enter to return to command mode. Now let's make the function slightly more complicated by entering the command: Y=SIN(X)+SIN(3*X)/3 Enter the PLOT command again to see this function plotted. 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 Now use the PLOT command to display the result. 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 use the PLOT command to display them. You should see a simple sine wave (the Y2 function) superimposed on the previous function (the Y function). Use the EXIT command to exit MATHPLOT and return to DOS. 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 -- Mathematical Function Plotter Page 4 MATHPLOT.FUN: MATHPLOT MATHPLOT.FUN MATHPLOT -- Mathematical Function Plotter Page 5 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. 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). 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. 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... MATHPLOT -- Mathematical Function Plotter Page 6 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 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. 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. MATHPLOT -- Mathematical Function Plotter Page 7 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. 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. MATHPLOT -- Mathematical Function Plotter Page 8 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. 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)). MATHPLOT -- Mathematical Function Plotter Page 9 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. MATHPLOT -- Mathematical Function Plotter Page 10 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: 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 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)) 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. 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 MATHPLOT -- Mathematical Function Plotter Page 11 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. 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 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. 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) MATHPLOT -- Mathematical Function Plotter Page 12 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. MATHPLOT -- Mathematical Function Plotter Page 13 MATHPLOT COMMANDS The commands described in this section are used to control MATHPLOT. When MATHPLOT is waiting for a command, it displays a greater-than sign (">") as a prompt. 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. When MATHPLOT detects a minus sign as the last character of a line, it deletes the minus sign and requests additional command input using the prompt "->". 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 Startup 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 startup command file may contain commands to set preferred default values, or it 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. 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 commands to set default values. For example, if you normally like to have grid lines turned on, create a file named MATHPLOT -- Mathematical Function Plotter Page 14 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. 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. ALPHABETICAL LIST OF COMMANDS AXES {ON | OFF} (default=ON) -- Specifies whether to display axes without labels. Use the LABELS command to display axes with labels. CALCULATE cexpr -- Evaluate the specified expression, which must contain only numbers, operators, and built in functions, and print the result. CAXES cexpr (default=3) -- Specify the color to be used for the axis lines and labels. 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. MATHPLOT -- Mathematical Function Plotter Page 15 CGRID cexpr (default=8) -- Specify the color to be used for grid lines. CLEAR -- Clear the screen. 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, 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. CTITLE cexpr (default=7) -- Specify the color to use for the title line. DATA filename -- Causes X and Y data values to be read from an external file rather than being computed as 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". DISPLAY string -- Display the string on the console. This command may be placed in command file to cause information to be printed while the command file is being processed. 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". See also the description of the SAVE command. MATHPLOT -- Mathematical Function Plotter Page 16 DOMAIN cexpr1,cexpr2 (default=0,2*pi) -- Define the domain of the independent variable over which the functions are evaluated and plotted. The lower end of the domain (cexpr1) must be less than the upper end (cexpr2). This command may be abbreviated to "DOM". EXIT -- Stop MATHPLOT and return to DOS. GRID {ON | OFF} (default=OFF) -- Specify whether grid lines are to be displayed. HELP -- Display several pages of help text. LABEL {ON | OFF} (default=ON) -- Specify whether axes with labels are to be displayed. LIST {ON | OFF} (default=OFF) -- Specify whether commands executed from an external file by use of the DO command are to be listed as they are executed. 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'. 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. 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. MATHPLOT -- Mathematical Function Plotter Page 17 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. 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. 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. This command may be abbreviated to the letter 'P'. 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). RESET -- Reset all parameters and options to their initial values and remove all function definitions. 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". TABULATE -- Evaluate the functions and print the values in tabular form rather than plotting them. MATHPLOT -- Mathematical Function Plotter Page 18 TITLE string -- Define a title line to be displayed at the top of the plot. WAXES cexpr (default=3) -- Specify the width (in dots) of the axis lines for printed output. This parameter does not affect the screen display. 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. WGRID cexpr (default=1) -- Specify the width (in dots) of the grid lines. This only affects hard copy output. MATHPLOT -- Mathematical Function Plotter Page 19 ADVANCED APPLICATIONS 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. 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 MATHPLOT -- Mathematical Function Plotter Page 20 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 MATHPLOT -- Mathematical Function Plotter Page 21 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 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 "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 MATHPLOT -- Mathematical Function Plotter Page 22 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. MATHPLOT -- Mathematical Function Plotter Page 23 USE AND DISTRIBUTION OF MATHPLOT MATHPLOT is a "shareware" product. You are welcome to make copies of this program and pass them on to friends or post this program on bulletin boards. However, if you find MATHPLOT to be useful and/or entertaining you are expected to send to the author the registration form printed on the next page with $20 to help cover the development and support of MATHPLOT. In return, you will receive the most recent version of the program and a bound copy of the manual. Add $5 if Mathplot is being shipped out of the United States. See also the special offer with Nonlin that follows. You are welcome to write to the author at: Phillip H. Sherrod 4410 Gerald Place Nashville, TN 37205-3806 Both the MATHPLOT program and documentation are copyright (c) 1991-1992 by Phillip H. Sherrod. You are not authorized to modify the program. "MATHPLOT" is a trademark. Disclaimer MATHPLOT is provided "as is" without warranty of any kind, either expressed or implied. This program may contain "bugs" and inaccuracies, and its results should not be assumed to be correct unless they are verified by independent means. The author assumes no responsibility for the use of MATHPLOT and will not be responsible for any damage resulting from its use. MATHPLOT -- Mathematical Function Plotter Page 24 N O N L I N Nonlinear Regression Analysis Program -- Special Offer 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 $36. 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. If you do any data analysis, or would just like to learn about the incredibly useful regression analysis technique, you need Nonlin! MATHPLOT -- Mathematical Function Plotter Page 25 ===================================================================== Software Order Form ===================================================================== NAME ______________________________________________________ ADDRESS ___________________________________________________ CITY _______________________ STATE _______ ZIP ___________ TELEPHONE _________________________________________________ COMPUSERVE ACCOUNT (optional) _____________________________ MATHPLOT VERSION (on title screen) ________________________ 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 ($20). ___ I wish to register Mathplot and order Nonlin ($36). Add $5 to any amount shown above if the software is being shipped out of the United States. 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 at: Phillip H. Sherrod 4410 Gerald Place Nashville, TN 37205-3806