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Information about “MATHPLOT”
MATHPLOT is a set of graphics-aided programs designed to help
scientists and engineers analyze data. Routines are included for
least-squares fitting polynomials to sets of data points, solving
simultaneous linear equations, numerically integrating a curve defined
by a set of data points, calculating simple statistical parameters for
data sets, and for calculating the coefficients of a Fourier series for
a periodic function described by a set of data points.
Programs are also included for calculating and plotting Fourier spectra
and Bessel functions, for plotting data sets and for plotting and
comparing user-specified functions with data sets.
FILE1863.TXT
Disk No: 1863
Disk Title: Mathplot
PC-SIG Version: S1
Program Title: MATHPLOT
Author Version: 2.2
Author Registration: $25.00
Special Requirements: Graphics Adapter (Herc, CGA, EGA, VGA) Graphics Pr
MATHPLOT is a set of graphics aided programs designed to help
scientists and engineers analyze data. Routines are included for
least-squares fitting polynomials to sets of data points, solving
simultaneous linear equations, numerically integrating a curve defined
by a set of data points, calculating simple statistical parameters for
data sets, and for calculating the coefficients of a Fourier series
for a periodic function described by a set of data points.
Programs are also included for calculating and plotting Fourier spectra
and Bessel functions, for plotting data sets and for plotting and
comparing user-specified functions with data sets. Screen graphics
capabilities are used by most of the programs, and the host computer
must be equipped with a graphics card and monitor (CGA, EGA, VGA, MCGA
or Hercules). MATHPLOT detects which graphics hardware is present and
provides the appropriate drivers.
PC-SIG
1030D East Duane Avenue
Sunnyvale Ca. 94086
(408) 730-9291
(c) Copyright 1989 PC-SIG, Inc.
GO.TXT
╔═════════════════════════════════════════════════════════════════════════╗
║ <<<< Disk #1863 MATHPLOT >>>> ║
╠═════════════════════════════════════════════════════════════════════════╣
║ To start program, type: MATHPLOT ║
║ ║
║ To print documentation, type: COPY MANUAL.TXT PRN ║
╚═════════════════════════════════════════════════════════════════════════╝
INVOICE.TXT
INVOICE
Date: ________________
Remit to: From:
J. H. McCrary ___________________________
1708 Pomona Drive ___________________________
Las Cruces, N. M. 88001 ___________________________
Ph. 505-522-9327 or 522-0805 ___________________________
Qty Unit price Total
___ MATHPLOT $25.00 ______
MANUAL.TXT
- - M A T H P L O T - -
Version 2.2
graphics aided mathematics
for data analysis
J. H. McCrary
1708 Pomona Drive
Las Cruces, N. M. 88001
505-522-0805 or 522-9327
May 1989
(C) J. H. McCrary (1989)
mathplot
Table of Contents
page
Introduction 1
Files on distribution disk 1
Starting MATHPLOT 2
Main menu 3
Hardcopies of graphics screens 3
Least squares 4
Simultaneous equations 8
Numerical integration 10
Matrix inversion 12
Fourier series 14
Bessel functions 17
Statistics 18
Data plots 21
Function plots 23
Data files 26
___________________________________________________
MATHPLOT requires an IBM-PC/XT/AT (or compatible)
computer with at least 256 kbytes of RAM, PC-DOS
2.0 or higher, an IBM CGA, EGA, VGA or MCGA
graphics card (or compatible), or a Hercules Gra-
phics Card (or compatible) with appropriate moni-
tor. For printed output (text and graphics) an
Epson MX, FX or LQ series (or compatible) printer
is required.
___________________________________________________
i
mathplot
- - M A T H P L O T - -
Introduction:
MATHPLOT is a set of graphics aided programs desig-
ned to help scientists and engineers analyze data. Rou-
tines are included for least-squares fitting polynomials
to sets of data points, solving simultaneous linear eq-
uations, numerically integrating a curve defined by a
set of data points, calculating simple statistical para-
meters for data sets, and for calculating the coeffi-
cients of a Fourier series for a periodic function de-
scribed by a set of data points. Programs are also in-
cluded for calculating and plotting Fourier spectra and
Bessel functions, for plotting data sets and for plot-
ting and comparing user-specified functions with data
sets. Screen graphics capabilities are used by most of
the programs, and the host computer must be equipped
with a graphics card and monitor (CGA, EGA, VGA, MCGA or
Hercules). Mathplot detects which graphics hardware is
present and provides the appropriate drivers. The dis-
tribution disk contains the following files:
Filename Function
mathplot.exe main system file
poly2.mpd sample data
poly3.mpd "
poly4.mpd "
fplsinx.mpd "
foursqr.frd "
fouramp.frd "
simeq2.sim "
simeq4.sim "
simeq6.sim "
matrix2.inv "
matrix3.inv "
manual.txt owners manual
MATHPLOT.EXE is compatible with computers with or
without a numeric co-processor. If a numeric co-proces-
sor is present in the host computer, it is detected and
utilized by MATHPLOT in its math intensive operations.
The system file requires IBM-PC or MS DOS version 2.0 or
higher.
The hardware requirements to utilize all of the
features of MATHPLOT are as follows:
1
mathplot
* IBM PC (or compatible) with one disk drive.
* Color or monochrome graphics capability.
* Epson MX or FX printer (or compatible).
* numeric math co-processor (optional).
* 256 Kbytes of RAM.
Starting MATHPLOT:
The differences between MATHPLOT Versions 2.2 and
2.1 are the following: (1) the number of data points the
programs can handle has been increased from 100 to 200,
(2) logarithms (base e and base 10) can now be taken for
data point coordinate values in the Least Squares pro-
gram, (3) the ESCape key can be used from any prompt in
any program to return to the previous menu.
A working disk should be prepared by copying the
appropriate MATHPLOT files to a formatted floppy disk or
to a directory on a fixed disk. Such a disk should not
be write protected since data may be written to it
during operation. The following procedure is suggested.
Insert the MATHPLOT system disk in drive A. Copy the
MATHPLOT.EXE file to the working disk in Drive B or to
the working directory on the hard disk. Copy any or all
of the sample data files and the owner's manual text
file as desired. The MATHPLOT working disk is now ready
for use.
With the working files in the default drive type
MATHPLOT followed by ENTER. The introductory screen is
displayed until a key is pressed after which the Main
Menu (next page) appears.
Throughout all of the MATHPLOT programs, various
menus and data input prompts are presented. When re-
sponding to a menu or to a "Yes or No" prompt, it is
necessary only to press a single key without pressing
the ENTER key. When entering data or other mathematical
parameters (order of polynomial, number of equations,
etc.), the ENTER key must be pressed after entering the
desired number. At any prompt in any of the programs,
the ESC key may be used to return to the previous menu.
In most of the programs an opportunity is given to have
a quick look or "preview" of the data on the graphics
screen immediately after it has been entered and checked.
2
mathplot
main menu 8:34 am
- - M A T H P L O T - -
Select from the following options:
1 - Least squares fitted polynomial:
2 - Solution of simultaneous equations:
3 - Numerical integration:
4 - Matrix inversion:
5 - Fourier series:
6 - Bessel functions:
7 - Statistics:
8 - Data plot:
9 - Function plot:
0 - Data files:
ESC - Exit to DOS:
Enter selection:
The mathematical functions will be discussed in
separate sections to follow. To leave MATHPLOT and re-
turn to DOS simply press the ESCape key. Selection of
option 0 causes the MATHPLOT data files on the disk in
the default drive (or directory) to be listed on the
screen. These data files will also be discussed in the
following sections. In all of the MATHPLOT programs, the
prompt "press any key" is used at pauses in program flow.
If ESCape is pressed at this prompt, the program does
not continue and control is returned to the menu for
that program.
Hardcopies of Graphics Screens:
Hardcopies of MATHPLOT graphics screens can be
printed by Epson (or compatible printers). In order to
obtain hardcopies of graphics screens it is necessary
that programs not supplied with MATHPLOT be run before
MATHPLOT is started. The procedure for making hardcopies
be described below.
3
mathplot
A. Hercules Monochrome Graphics Card
The utilities program entitled HGC.COM sup-
plied with the Hercules card must be run before
MATHPLOT is started. The HGC.COM file must be of a
recent version (6/17/85 or later) and must contain
the Hercules PRINT routine. To run this program,
insert the Hercules disk in the default drive, turn
the printer ON, and type HGC PRINT <enter>. MATH-
PLOT may then be started, and from a graphics
screen a hardcopy may be obtained by pressing SHIFT
PRTSC followed by 0 (zero). The screen dump process
takes about two minutes after which MATHPLOT opera-
tions may be continued. Screen dumps from Hercules
clone cards generally are not possible. The user is
referred to the literature accompanying that card.
B. IBM CGA Graphics Card
The PC-DOS program named GRAPHICS.COM must be
run before MATHPLOT is started. To run this pro-
gram, insert a DOS disk in the default drive, turn
the printer ON, and type GRAPHICS <enter>. MATHPLOT
may then be started, and from a graphics screen a
hardcopy may be obtained by pressing SHIFT PRTSC.
The screen dump process takes about two minutes
after which MATHPLOT operations may be continued.
It should be pointed out that difficulties with the
screen dump process may be encountered if memory resi-
dent programs such as Sidekick or Prokey are present.
If the computer locks up during or just after printing a
graphics screen, the problem may be due to the presence
of one of these utilities. The computer should be re-
booted with a DOS system disk which does not load memory
resident programs.
least squares
1. Using the Least Squares program:
Pressing 1 from the Main Menu causes MATHPLOT to
enter the Least Squares program. This program fits a
polynomial of any order between 0 and 9 to a set of data
points. Up to 200 data points may be used. For the poly-
nomial to be meaningful, its order should be much less
than the number of data points. The first screen after
the Main Menu gives a brief description of the program.
Press any key to continue to the input phase, and the
following Least Squares Input Menu will be displayed.
4
least squares
least squares 8:48 am
Select from the following options;
1 - Retrieve data from disk file.
2 - Enter new data.
3 - Return to previous menu.
Enter selection:
The Input Menu is self explanatory, but some com-
ments are in order. Data may be retrieved from a file in
drive or directory, new data may be entered, or control
may be returned to the previous menu by pressing 1, 2,
or 3 respectively.
If 1 is selected from the Least Squares Input Menu,
all files in the default drive whose extension is .MPD
(MathPlot Data) will be listed on the screen, and the
user will be asked to type the name of the file to be
used. If no files are present on the default drive disk,
the message "No .MPD files:" will appear. Data files
from any disk or directory may be used by typing the ap-
propriate drive letter, path and filename. The filename
may be entered in upper or lower case letters with or
without the extension (.MPD). If the filename is entered
incorrectly, the message "file not found - try again"
will appear until the filename is correctly entered or
ESC is pressed. If none of the listed files is to be
used, pressing ENTER or ESC from the "Enter file name:"
prompt returns control to the Input Menu. The distribu-
tion disk contains three sample data files, POLY2.MPD,
POLY3.MPD, and POLY5.MPD which can be used to demon-
strate the Least Squares program.
After the .MPD data file has been read, the X and Y
coordinates of all the data points are displayed sequen-
tially, and the following data check menu is displayed.
Press C (change data), G (graphic preview), S (save)
ENTER to continue
If C (or c) is pressed, the next prompts will be "Press
ENTER to register number, ESC to exit" and "i = ? ".
The user enters the value of the index for the data po-
int to be changed. Prompts are then given for new X and
Y coordinate values. This process continues until the
user presses the ENTER or ESC key at the "i = ?" prompt.
The data may be viewed on the graphics screen by pres-
sing G from the data check menu. Pressing S provides an
opportunity to save the data to a disk file with the
5
least squares
.MPD extension. To continue with the program, press
ENTER from the data check menu.
If 2 is selected from the Least Squares Input Menu,
the user is prompted for the number of data points to be
entered. The number must be an integer between 3 and
200. The user is then prompted for X and Y coordinates
of each point until all data have been entered. The
data points may be entered in any order since they are
later sorted in increasing values of X. After the data
have been entered, the data check procedure described
above will give the user an opportunity to make any ne-
cessary changes. New or changed data may be previewed
on the graphics screen and saved to a .MPD disk file.
The user is next prompted for, "Order of fitted
polynomial, m = ? ". A positive integer between 0 and
9 must be entered. After a short time the least squares
fitted polynomial coefficients are displayed. Pressing
any key will display the Least Squares Menu as follows:
least squares 9:13 am
Select from the following options:
1 - Display results on graphics screen.
2 - Calculate projected values from polynomial.
3 - Print copy of polynomial and data.
4 - Calculate another polynomial from these data.
5 - More calculations with new data.
6 - Modify data set (interchange X,Y; log Y).
7 - Check/change data.
8 - Exit to main menu.
Enter selection:
Option 1 from the Least Squares Menu displays the
data points and the fitted polynomial on the graphics
screen. The first prompt after selecting option 1 is:
"Specify axis labels and ranges ? Y or N". Answering
"N", "n" or ENTER causes a graphics display with auto-
matically selected axis labels and ranges. This range
includes all of the data points plus an additional 10%
of the range spanned by these points. Answering the
prompt with "Y" or "y" results in additional prompts for
axis labels and for minimum and maximum values of X and
Y to be plotted. Axis label strings may include as many
as 18 characters. Additional characters are truncated.
6
least squares
The graphics screen shows the data points as X's and the
calculated polynomial as a dotted curve. Vertical and
horizontal axes pass through the origin of the coordin-
ate system. To exit the graphics mode, any key may be
pressed, and control will return to the Least Squares
Menu.
Option 2 from the Least Squares Menu permits the
calculation of values of Y from input values of X using
the least squares fitted polynomial. Prompts are given
for values of X, and the calculated values of Y are
displayed. All such projected values are stored in
memory and are printed out when option 3 is selected
from this menu. Pressing ESCape at the "X = ? " prompt
returns control to the Least Squares Menu.
Option 3 prompts the user to enter a print-out
header. This may be any combination of as many as 70
characters. It is intended to permit the user to iden-
tify the set of calculations being printed. This is fol-
lowed by a prompt to turn on the printer. The polynomial
coefficients, all input data with calculated values of
Y for each input value of X, and all projected values
of Y along with their corresponding values of X are then
printed. Control is automatically returned to the Least
Squares Menu after printing has stopped.
Option 4 results in a prompt for a value for the
order of a new polynomial to be fitted to the same data.
Any value from 0 through 9 may be selected (consistent
with the number of data points). Results from the new
calculations are presented as described above.
Option 5 from the Least Squares Menu returns pro-
gram control to the Least Squares Input Menu, from which
new data may be entered and new polynomials calculated.
Option 6 causes the display of a menu which permits
the user to interchange the X,Y values for all points in
the data set or to replace data set Y values with base e
or base 10 logarithms of the Y values. Non-positive Y
values are replaced with zero. The interchange feature
also permits logarithms to be taken of data set X values.
Option 7 permits the checking and changing of cur-
rent data prior to calculating new results. Changed data
may also be saved to a disk file with a .MPD extension.
Option 8 returns control to the MATHPLOT Main Menu.
7
simeq
2. Using the Simultaneous Equations program:
Selecting option 2 from the Main Menu leads into
the Simeq program and displays its introductory screen.
Simeq solves a system of up to 12 simultaneous linear
equations through the use of the matrix algebra equation
A(i,j)*X(i) = B(j). Program input consists of the mat-
rices A(i,j) and B(j). The coefficient matrix A(i,j)
must be square and the column matrix B(j) must be of the
same order. Recall that in matrix notation the first
subscript is the row number and the second is the column
number. The kth row in the coefficient matrix is made up
of the coefficients in the kth equation
A(k,1)*X(1) + A(k,2)*X(2) + ... + A(k,n)*X(n) = B(k)
If the determinant of A(i,j) is zero, its inverse does
not exist and the problem cannot be solved.
Press any key and the Simeq Input Menu is displayed.
This menu is identical with the Least Squares Input Menu
described in Section 1. The options are treated slightly
differently however due to the format of the Simeq data.
If 1 is selected from the Simeq Input Menu, all
files in the default drive whose extension is .SIM will
be listed on the screen and the user will be asked to
type the name of the file to be used. If no files are
present on the default drive disk, the message "No .SIM
files:" will appear. Data files from any drive or direc-
tory may be used by typing the drive letter, path and
filename. The filename may be typed in upper or lower
case letters with or without the extension (.SIM).
If the filename is entered incorrectly, the message
"file not found - try again" will appear until the file-
name is correctly entered or ESC is pressed. If none of
the listed files is needed, pressing ENTER or ESC from
the "Enter file name:" prompt returns control to the In-
put Menu. The distribution disk contains three sample
files, SIMEQ2.SIM, SIMEQ4.SIM, and SIMEQ6.SIM which can
be used for demonstrating the program.
After the .SIM data file has been read, the ele-
ments of the A(i,j) matrix will be displayed, and the
following data check menu is displayed.
press C (change data), ENTER to continue
If C (or c) is pressed, the next prompt will be "Press
ENTER to register number, ESC to exit" followed by "i =
? ". Enter the row number of the matrix element to be
changed followed by ENTER. The column number of the ele-
ment to be changed is entered after the "j = ? " prompt.
8
simeq
The correct value of the matrix element A(i,j) is then
entered. This process continues until the user presses
ENTER or ESC at the "i = ? " prompt. Press ENTER again
to continue. The B(j) matrix is then displayed and may
be changed similarly. Pressing S from the B matrix data
check menu provides an opportunity to save the data to a
disk file.
If 2 is selected from the Simeq Input Menu, the
user is prompted for the number of linear equations to
be solved. The number must be an integer not larger
than 12. The user is then prompted for each element of
the A(i,j) matrix until all of the elements have been
entered. The elements of the B(j) matrix are then en-
tered in a similar fashion. After the data have been
entered, the data check procedure described above will
give the user an opportunity to make necessary changes
in either matrix and to save the data to a .SIM disk
file.
After a short period of time, the solution matrix
X(i) is displayed. Pressing any key will result in the
display of the Simeq Menu as follows.
simeq 9:31 am
Select from the following options:
1 - Print copy of problem and answer.
2 - More calculations with new data.
3 - Check/change data and recalculate.
4 - Exit to main menu.
Enter selection:
Option 1 prompts the user to enter a header line
and to then turn on the printer. The program than prints
the input matrices and the solution matrix and returns
control to the Simeq Menu after printing has ended.
Option 2 from the Simeq Menu returns program con-
trol to the Simeq Input Menu. New data may be entered,
checked and saved and new sets of equations solved.
Option 3 permits the checking and changing of cur-
rent data prior to re-calculating new results. Changed
data may also be saved to a disk file.
Option 4 returns control to the MATHPLOT Main Menu.
9
integrate
3. Using the Numerical Integration program:
Selecting 3 from the Main Menu causes MATHPLOT to
enter the Integrate program and display its introductory
screen. Integrate computes the integral of a function
which is defined by up to 200 unequally spaced discrete
function values between the minimum and maximum values
of the independent variable, X. The method of cubic
splines is used, and a sort routine precludes the need
for entering the data points in any particular order.
After entering and checking the data, they may be saved
to a disk file and viewed on the graphics screen by re-
sponding appropriately to the prompts.
Depress any key and the Integrate Input Menu is
displayed. This menu is identical with the Least Squares
Input Menu described in Section 1 and the options are
treated basically the same.
If 1 is selected from the Integrate Input Menu, all
disk files in the default drive whose extension is .MPD
will be listed on the screen and the user will be asked
to type the name of the file to be used. If no files are
present on the default drive disk, the message "No .MPD
files:" will appear. A file from any drive or directory
may be used by typing the drive letter, path and file-
name. The filename may be typed in upper or lower case
letters with or without the extension (.MPD). If the
filename is entered incorrectly, the message "file not
found - try again" will appear until the filename is
correctly entered or ESC is pressed. If none of the lis-
ted files is needed, pressing ENTER or ESC from the "En-
ter file name:" prompt returns control to the Input Menu.
The distribution disk contains three sample files, POLY-
2.MPD, POLY3.MPD, and POLY5.MPD which can be used for
demonstrating the Integrate program.
After the .MPD data file has been read, the X and Y
coordinates of all the data points representing the
function to be integrated will be displayed sequential-
ly, and the data check menu will appear. This menu is
identical to the one shown and described on page 5 of
this manual. The user may change data coordinate values,
view a plot of the data on the graphics screen, save da-
ta to a disk file or continue with the program from this
menu.
If 2 is selected from the Integrate Input Menu, the
user is prompted for the number of data points to be en-
tered. The number must be an integer between 3 and 200.
The user is then prompted for X and Y coordinates of
each point until all data have been entered. The data
10
integrate
points may be entered in any order since they are later
sorted in increasing values of X. After the data have
been entered, the data check procedure described above
will give the user an opportunity to make any necessary
changes and to save the data to a .MPD disk file.
After a short time, the integral of Y(x) between
the lower and upper limits of X is displayed on the
screen. Pressing any key will result in the display of
the Integrate Menu as follows.
integrate 9:34 am
Select from the following options:
1 - Print copy of problem and answer.
2 - More calculations with new data.
3 - Check/change data and recalculate.
4 - Exit to main menu.
Enter selection:
Option 1 prompts the user to enter a header line
and to turn on the printer. The program than causes the
input data and the results of the numerical integration
calculation to be printed. Control is automatically re-
turned to the Integrate Menu after printing has ended.
Option 2 from the Integrate Menu returns program
control to the Integrate Input Menu. New data may be
entered, checked and saved and new integrals computed.
Option 3 permits the checking and changing of cur-
rent data prior to recalculating new results. Changed
data may also be saved to a disk file.
Option 4 returns control to the MATHPLOT Main Menu.
11
invert matrix
4. Using the Matrix Inversion program:
Selecting 4 from the Main Menu results in the dis-
play of the introductory screen for the Matrix Inversion
program. This program inverts a square, non-singular ma-
trix whose order may be as high as 12 by using a Gauss-
Jordan reduction method with partial pivoting. The square
matrix A(i,j) which is to be inverted must be input with
the usual matrix notation (i is the row number and j the
column number). If the determinant of A(i,j) is zero,
its inverse does not exist.
Press any key and the Matrix Inversion Input Menu
is displayed. This menu is identical with the Least
Squares Input Menu described in Section 1. The options
are treated slightly differently however due to the
matrix format of the input data.
If 1 is selected from the Matrix Inversion Input
Menu, all files in the default drive whose extension is
.INV will be listed on the screen and the user will be
asked to type the name of the file to be used. If no
files are present on the default drive disk, the message
"No .INV files:" will appear. As with the other programs
in MATHPLOT, data files in any drive or directory may be
used by entering the appropriate drive and path. The
filename may be typed in upper or lower case letters
with or without the extension (.INV). If the filename is
entered incorrectly, the message "file not found - try
again" will appear until the filename is correctly
entered or ESC is pressed. If none of the listed files
is needed, pressing ENTER or ESC from the "Enter file
name: " prompt returns control to the Input Menu. The
distribution disk contains two sample files, MATRIX2.INV
and MATRIX3.INV which can be used for demonstrating the
Matrix Inversion program.
After the .INV data file has been read, the ele-
ments of the A(i,j) matrix will be displayed, and the
following data check menu is displayed.
press C (change data), S (save), ENTER to continue
If C (or c) is pressed, the next prompts will be "Press
ENTER to register number, ESC to exit" and "i = ? ". En-
ter the row number of the matrix element to be changed
followed by ENTER. The column number of the element to
be changed is entered after the "j = ? " prompt. The
correct value of the matrix element A(i,j) is then en-
tered. This process continues until the user presses the
ENTER or ESC key at the "i = ?" prompt. Pressing S from
the data check menu provides an opportunity to save the
data to a disk file. Pressing ENTER causes the program
to continue.
12
invert
If 2 is selected from the Matrix Inversion Input
Menu, the user is prompted for the order of the matrix
to be inverted. The number must be an integer not lar-
ger than 12. The user is then prompted for each element
of the input matrix until all of the elements have been
entered. The data check procedure described above will
then give the user an opportunity to make any necessary
changes and to save the data to a disk file.
After a short time, the inverted matrix is dis-
played, or if the input matrix A(i,j) is singular the
calculation will not proceed and an error message
"A(i,j) is singular." will appear. Pressing any key will
result in the display of the Matrix Inversion Menu as
follows.
invert matrix 9:52 am
Select from the following options:
1 - Print copy of problem and answer.
2 - More calculations with new data.
3 - Check/change data and recalculate.
4 - Exit to main menu.
Enter selection:
Option 1 prompts the user to enter a header line
and to turn on the printer. The program than causes the
input and the inverted matrices to be printed. Control
is automatically returned to the Matrix Inversion Menu
after printing has ended.
Option 2 from the Matrix Inversion Menu returns
program control to the Matrix Inversion Input Menu. New
data may be entered, checked and saved to .INV files and
new matrices inverted.
Option 3 permits the checking and changing of current
data prior to recalculating new results. Changed data
may also be saved to a disk file.
Option 4 returns control to the MATHPLOT Main Menu.
13
fourier
5. Using the Fourier Series program:
Pressing 5 from the Main Menu causes MATHPLOT to
enter the Fourier program. This program calculates co-
efficients for a trigonometric Fourier series for an em-
pirical periodic function that is input in the form of
discrete function values distributed at equal intervals
over one period of the function. 2 M + 1 such coeffi-
cients are calculated, where M is the input order of the
series. M must be an integer between 1 and 10. The
number of values of the function, N, must be an odd in-
teger between 5 and 199. The coefficients are calcula-
ted by numerically integrating the product of the
input function and the appropriate trigonometric func-
tion over one period of the independent variable. Accu-
racy in the values of the calculated coefficients, of
course, increases with N. The first screen after the
Main Menu displays a brief description of Fourier. Any
key may be pressed to continue to the input phase of the
program with the display of the Fourier Input Menu.
This menu is also identical with the other input menus
with the exception of the heading in the upper left hand
corner of the screen. The "Enter new data." option
however is treated slightly differently.
If 1 is selected from the Fourier Input Menu, all
default drive files whose extension is .FRD (Fourier
data) will be listed on the screen and the user will
be asked to type the name of the file to be used. The
filename may be typed in upper or lower case letters
with or without the extension (.FRD). Files may be
called from any drive or directory by using the appro-
priate path. If no files are present on the default
drive disk, the message "No .FRD files:" will appear.
If the filename is entered incorrectly, the message
"file not found - try again" will appear. If none of
the listed files is to be used, pressing ENTER or ESC
from the "Enter file name: " prompt returns control to
the Input Menu. The distribution disk contains two files,
FOURSQR.FRD and FOURAMP.FRD which can be used to demon-
strate the Fourier program.
After the .FRD data file has been read, the N
equally spaced values of the periodic function and the
corresponding values of the independent variable will be
displayed sequentially, and the data check menu will ap-
pear. This menu is identical to the one shown and de-
scribed on page 5 of this manual. The user may change
data coordinate values, view a plot of the data on the
graphics screen, save data to a disk file or continue
with the program from this menu.
14
fourier
If 2 is selected from the Fourier Input Menu, the
user is prompted for the number of data points to be en-
tered. The number must be an ODD integer between 5 and
199. The user is then prompted for the minimum and max-
imum values of the independent variable, X. These must
correspond to the end points of a single period in the
periodic function. The program then calculates the lo-
cation of the N equally spaced points and prompts for
the corresponding function values at each location. Af-
ter the data have been entered, the data check procedure
described above will give the user an opportunity to
make necessary changes. New or changed data may be saved
to a .FRD disk file for future use and previewed on the
graphics screen.
The user is next prompted for "Order of the series,
M = ? ". A positive integer no larger than 10 must be
entered. After a period of time, the series coeffi-
cients will be displayed. The computation time varies
rather dramatically with both M and N. The time to
calculate a 10th order series from 51 data points is
about 40 seconds on an ordinary PC with no coprocessor
and about 1 or 2 seconds on a PC with an 8087. Pressing
any key will result in the display of the Fourier Menu
as follows.
fourier 10:12 am
Select from the following options:
1 - Display periodic function on graphics screen.
2 - Display Fourier spectrum on graphics screen.
3 - Print copy of problem and results.
4 - Calculate another series from these data.
5 - More calculations with new data.
6 - Check/change data.
7 - Exit to main menu.
Enter selection:
Option 1 from the Fourier Menu displays the data
points and the calculated series on the graphics screen.
The first prompt after selecting option 1 asks: "Speci-
fy axis labels and ranges ? Y or N". Answering "N",
"n" or ENTER causes a graphics display with automatical-
ly selected axis labels and ranges. This range includes
all of the data points plus an additional 10% of the
range spanned by these points. Answering the prompt with
"Y" or "y" results in additional prompts for axis la-
bels and for minimum and maximum values of X and Y to be
15
fourier
plotted. Axis label strings may include as many as 18
characters. Additional characters are truncated. The
graphics screen shows the data points as X's and the
calculated Fourier series function as a dotted curve.
Vertical and horizontal axes pass through the origin of
the coordinate system. To exit the graphics mode, any
key may be pressed to return control to the Fourier Menu.
Option 2 produces a plot of the Fourier spectrum on
the graphics screen. The square root of the sum of the
squares of the calculated A and B coefficients are plot-
ted versus the frequency. A more detailed spectrum may
be obtained by specifying a larger value for M, the or-
der of the series (see p. 14). To exit the graphics
mode, press any key.
Option 3 prompts the user to enter a header line as
described in the Least Squares section above and to turn
on the printer. The Fourier series coefficients and all
input data are then printed. Control is automatically
returned to the Fourier Menu after printing has stopped.
Option 4 results in a prompt for a value for the
order of a new Fourier series to be calculated from the
same data. Any value from 1 through 10 may be selected.
Results from the new calculations are presented as des-
cribed above.
Option 5 from the Fourier Menu returns program
control to the Fourier Input Menu, whence new data may
be entered and new series calculated.
Option 6 permits the checking and changing of cur-
rent data prior to calculating new results. Changed data
may also be saved to a disk file.
Option 7 returns control to the MATHPLOT Main Menu.
16
bessel
6. Using the Bessel program:
Selecting option 6 from the Main Menu causes MATH-
PLOT to enter the Bessel program. This program calcu-
lates and plots Bessel functions of the first or second
kind. Orders for J(n,X) may range from -15 to +15 and
for Y(n,X) from 0 to 15. Any positive value may be en-
tered for the argument X. Both functions are calculated
from truncated series for values of the argument less
than 20. For larger values of X an asymptotic formula is
used. For X < 20, the error in both functions is less
than 1E-6. For X > 20, the accuracy is high so long as X
is much greater than the order. The first screen after
the Main Menu displays a brief description of the Bes-
sel program. Any key may be pressed to continue to the
Bessel Menu shown below.
bessel 10:38 am
Select from the following options:
1 - Calculate values of J(n,X).
2 - Plot J(n,X) vs. X.
3 - Calculate values of Y(n,X).
4 - Plot Y(n,X) vs. X.
5 - Print calculated values.
6 - Return to main menu.
Enter selection.
If 1 is selected from the Bessel Menu, the program
prompts for the desired order of the Bessel function of
the first kind. Any number between (and including) -15
and +15 may be entered for the order. The user is next
prompted for the argument of the function. Any POSITIVE
number may be entered. The value of the Bessel func-
tion for the entered order and argument is then dis-
played. Such calculations may continue until the user
presses ESC at either prompt, whence control is returned
to the Bessel Menu.
If 2 is selected from the Bessel Menu, the program
prompts for the order of the Bessel function of the
first kind to be plotted. Any number between -15 and +15
may be entered. The program then prompts (in turn) for
the minimum and maximum values of the range of the ar-
gument, X, to be plotted. Any positive values may be
entered. X max must, of course, be greater than X min.
The function plotted over the specified range will then
be displayed. Any key may be pressed to exit the graph-
17
bessel
ics screen and return to the Bessel Menu.
Options 3 and 4 result in calculations or plots for
Bessel functions of the second kind. The only difference
from the procedures described above is that orders for
Y(n,X) are restricted to the range 0 through +15.
Option 5 prompts the user to enter a header line
and to turn on the printer. All functions calculated in
the preceding session are then printed, and control is
returned to the Bessel Menu.
Option 6 returns control to the MATHPLOT Main Menu.
statistics
7. Using the Statistics program:
Selecting 7 from the Main Menu causes MATHPLOT to
enter the Statistics program and to display its intro-
ductory screen. This program prompts for the number of
variables (1 or 2) and the number of observations (3 to
200). It calculates the mean and the standard deviation
for each variable. If two variables are used, the cor-
relation coefficient and linear regression coefficients
are also calculated. From calculated linear regression
coefficients, the user can find projected values of ei-
ther variable from entered values of the other. In the
graphics mode, Statistics plots single variable values
versus their index number, or if two variables are used,
the second variable, Y is plotted against the first. In
both cases the mean of the variable(s) is indicated by
appropriate axis lines. Statistics works with data
files of the .MPD type and will produce hardcopy of data
and results on the printer.
After entering the number of variables, the Statis-
tics Input Menu is displayed. This menu is identical
with the Least Squares Input Menu described in Section 1
and the options are treated basically the same. If one
variable is being studied, however, prompts will be
given only for X values.
If 1 is selected from the Statistics Input Menu,
all default drive .MPD files will be listed on the
screen and the user will be asked to type the name of
the file to be used. A drive letter, path and filename
18
statistics
may be typed in upper or lower case letters with or
without the extension (.MPD). If no files are present
in the specified drive, the message "No .MPD files:"
will appear. If the filename is entered incorrectly, the
message "file not found - try again" will appear. If
none of the listed files is needed, pressing ENTER or
ESC from the "Enter file name: " prompt returns control
to the Input Menu. The distribution disk contains three
sample files, POLY2.MPD, POLY3.MPD, and POLY5.MPD which
can be used for demonstrating the Statistics program.
After the .MPD data file has been read, the X and Y
values of the observations will be displayed sequential-
ly, and the data check menu will appear. This menu is
identical to the one shown and described on page 5 of
this manual. The user may change observation values,
view a plot of the data on the graphics screen, save the
data to a disk file or continue with the program from
this menu.
If 2 is selected from the Statistics Input Menu,
the user is prompted for the number of observations to
be entered. The number must be an integer between 3
and 200. The user is then prompted for values of each
observation until all data have been entered after which
the data check procedure described above will give the
user an opportunity to make necessary changes and to
save the data to a .MPD disk file.
After a short time, the statistical results are
displayed, showing the number of variables, the number
of observations, the sum, mean value and standard devia-
tion from the mean for each variable. If two variables
are being studied, the correlation factor is also dis-
played. Pressing any key will result in the display of
the Statistics Menu as follows.
statistics 10:51 am
Select from the following options:
1 - Display data on graphics screen.
2 - Linear regression calculations.
3 - Print copy of data and results.
4 - More calculations with new data.
5 - Interchange X and Y variables.
6 - Check/change data and recalculate.
7 - Exit to main menu.
Enter selection:
19
statistics
Option 1 from the Statistics Menu produces a graph-
ical display of the data. If a single variable is being
used, its values are plotted vertically versus the order
in which they were entered. A horizontal axis passes
through the mean of their values. The plotted area is
not user selectable. If observations of two variables
were entered, the Y values are plotted (vertically) ver-
sus the X values for each observation. The first prompt
after selecting option 1 asks: "Specify axis labels and
ranges ? Y or N". Answering "N", "n" or ENTER causes a
graphics display with automatically selected axis labels
and ranges. This range includes all of the data points
plus an additional 10% of the range spanned by these
points. Answering the prompt with "Y" or "y" results in
additional prompts for axis labels and for minimum and
maximum values of X and Y to be plotted. Axis label
strings may include as many as 18 characters. Additional
characters are truncated. A horizontal axis passes
through the mean value of Y and a vertical axis through
the mean value of X.
Selecting Option 2 causes Statistics to enter the
linear regression calculation routine, in which case it
is assumed that the two-variable statistical data appro-
ximate a straight line. If single variable data were
entered an error message appears, otherwise the program
prompts the user "Calculate Y(x) or X(y) ? Enter Y or
X". Typing Y (or y) causes the program to prompt "X =
?". From the entered value of X the program calculates
the corresponding value of Y from the linear regression
coefficients. In a similar manner, calculations of X
may be made from entered values of Y. This process may
be continued until ESC is pressed at the "X = ? " or "Y
= ? " prompt.
Option 3 prompts the user to turn on the printer.
The program than causes the input data, the results of
the statistical calculations and the results of all lin-
ear regression calculations to be printed. Control is
automatically returned to the Statistics Menu after
printing has ended.
Option 4 from the Statistics Menu returns program
control to the Statistics Input Menu. New data may be
entered, checked and saved and new calculations made.
Option 5 interchanges the X variable value with the
corresponding Y value for each observation. The new co-
ordinates are sorted according to increasing values of X
before being displayed or utilized.
20
statistics
Option 6 permits the checking and changing of current
data prior to recalculating new results. Whenever a
change is made in the data, an opportunity is given to
write the changed data to disk.
Option 7 returns control to the MATHPLOT Main Menu.
data plot
8. Using the Data Plot program:
Pressing 8 from the Main Menu causes MATHPLOT to
enter the Data Plot program. This program plots up to
200 data points on the graphics screen in a user speci-
fied region of the X,Y plane.
Press any key and the Data Plot Input Menu is dis-
played. This menu is identical with the Least Squares
Menu described in Section 1, and the options are treated
the same.
If 1 is selected from the Data Plot Input Menu, all
disk files whose extension is .MPD will be listed on
the screen and the user will be asked to type the name
of the file to be used. If no files are present on the
default drive disk, the message "No .MPD files:" will
appear. The drive letter, path and filename may be typed
in upper or lower case letters with or without the ex-
tension (.MPD). If the filename is entered incorrectly,
the message "file not found - try again" will appear.
If none of the listed files is to be used, pressing EN-
TER or ESC from the "Enter file name: " prompt returns
control to the Input Menu. Any of the .MPD files on the
distribution disk can be used to demonstrate the Data
Plot program.
After the .MPD data file has been read, the X and Y
coordinates of all the data points will be displayed se-
quentially, and the data check menu will appear. This
menu is identical to the one shown and described on page
5 of this manual. The user may change coordinate values,
view a plot of the data on the graphics screen, save the
data to a disk file or continue with the program from
this menu.
If 2 is selected from the Data Plot Input Menu, the
user is prompted for the number of data points to be en-
tered. The number must be an integer between 3 and 200.
The user is then prompted for X and Y coordinates of
21
data plot
each point until all data have been entered. The data
check procedure described above will give the user an
opportunity to make changes. New or changed data may be
saved to a .MPD disk file for future use.
Pressing 3 transfers control back to the Main Menu.
The user is then prompted: "Specify axis labels and
ranges ? Y or N". Answering "N", "n" or ENTER causes
a graphics display with automatically selected axis
labels and ranges. This range includes all of the data
points plus an additional 10% of the range spanned by
these points. Answering the prompt with "Y" or "y" re-
sults in additional prompts for axis labels and for
minimum and maximum values of X and Y to be plotted.
Axis label strings may include as many as 18 characters.
Additional characters are truncated. The graphics plot
is then displayed. To exit the graphics mode, any key
may be pressed, and control will return to the Main Menu.
There is no interaction between the Data Plot prog-
ram and the printer other than the graphics screen dump.
The *.MPD files can, however, be printed from the Integ-
rate program.
function plot
9. Using the Function Plot program:
Pressing 9 in the Main Menu causes MATHPLOT to be-
gin the Function Plot program. In this program the user
may enter any mathematical function of an independent
variable X, which can be written from combinations of
the supported functions and operators described below.
The program evaluates this expression at a large number
of points in a user specified range and plots a continu-
ous curve depicting the entered function on the graphics
screen. A set of data points may also be plotted on the
same graphics screen. With Function Plot the user may
thus make graphical comparisons of data sets with mathe-
matical functions and obtain analytical representations
of that data.
The following functions are supported:
sin, cos, tan, asin, acos, atan, sinh, cosh,
tanh, ln, log, exp, abs, sqrt, sqr, fact, pi
22
function plot
The following operators are also supported:
+ - * / and ^
The independent variable in the expression must be
represented by X or x. Arguments of the supported func-
tions must be enclosed by parentheses, brackets or bra-
ces. The multiplication operator (*) may be omitted be-
tween a number and a parentheses, between two parenthe-
ses and between a number and a function name. Numbers
may be entered in ordinary notation (123.456) or in ex-
ponential notation (-1.23456E-7). Decimal fractions may
be entered without a leading zero. Arguments of trigo-
nometric functions must be in radians. Following is an
example of an acceptable function.
F(x) = -1.23E-1 {[sin(x)]^0.33} + pi/6
Operator precedence is as follows:
1. factorialization (fact)
2. minus sign as a prefix negation operator
3. multiplication
4. division
5. exponentiation (^)
6. addition
7. subtraction
Any key may be pressed to leave the introductory
screen of Function Plot. The user is then prompted to
"Enter the desired function", "F(x) = ". Using the
information above, the user may then complete the equa-
tion to be studied. Pressing ENTER then displays the
Function Plot Menu as follows:
function plot 12:34 pm
Select from the following options:
1 - Plot the function and the current data set.
2 - Calculate values of F(x).
3 - Enter a new function.
4 - Enter a new data set.
5 - Check/change data.
6 - Print results.
7 - Exit to main menu.
Enter selection:
23
function plot
If no data set has been entered, option 1 in the
Function Plot Menu leads to prompts for "X axis label
= ? " and "Y axis label = ? ". Any string of 18 or
fewer characters may be entered, or pressing ENTER from
the prompt(s) will result in the use of the default
prompts "X AXIS" and "FUNCTION". Additional prompts ap-
pear for plotting ranges. These must be answered with
consistent values (xmax > xmin, etc.). After a short
period, dependent upon the length of the entered func-
tion and hardware configuration, a plot of the function
will appear. Pressing any key removes the graphics dis-
play which is followed by the prompt "Plot a new func-
tion ? Y or N ". Answering positively permits the in-
troduction of a new function. Pressing "N", "n" or
ENTER or ESC from both returns control to the Function
Plot Menu.
If a data set has been entered, option 1 causes the
prompt "Specify axis labels and ranges ? Y or N " to
appear. A positive response leads to the axis label and
plotting range sequence; a negative response results in
the use of default values for all these variables. De-
fault ranges include a 10% border around the area en-
closing all of the data points. On the graphics screen
the user selected function is plotted as a dotted, con-
tinuous curve. The data points are shown as X's. Press-
ing any key from the graphics screen returns the prompts
for more plots and new functions as described above. The
user may continue to enter new functions and compare
them with the data on the graphics screen until a satis-
factory analytical representation of the data has been
developed. The current function appears on all of the
screens in the Function Plot program. All of the .MPD
sample data files on the distribution disk may be used
in demonstrating the Function Plot program.
Option 2 prompts the user to enter a value for the
independent variable x. The Function Plot program then
calculates the entered function for this value of x. All
such function evaluations are retained in memory for la-
ter printing.
Option 3 causes the current function to be display-
ed and prompts for the entry of a new function.
Option 4 in the Function Plot Menu displays the In-
put Menu described in previous programs. This menu per-
mits data to be entered from the keyboard or to be read
from disk files exactly as described in the Least
Squares program instructions (see pages 4 through 7).
Option 5 permits the current data set to be checked
and changes to be made. The modified data set may then
be saved and plotted.
24
function plot
Option 6 prompts the user to enter a header line
and to turn on the printer. The current function, the
data set and all of the function values calculated in
Option 3 (above) are then printed. In addition to the
data set coordinates, values calculated from the current
function for the X coordinate of each data point are al-
so printed.
Option 7 returns control to the MATHPLOT Main Menu.
If both a function and a data set are present in
the program, the standard deviation between the data set
Y values and the function Y values is calculated. This
value is presented as (Sigma = ______ ) on the Function
Plot Menu between and to the right of menu items 3 and 4.
Sigma is calculated as follows:
Sigma = SQRT((Y(func)-Y(data))^2/(n-1))
This value of sigma is also printed by Option 6 (above).
files
10. Data files:
Pressing 0 from the Main Menu causes MATHPLOT to
display the prompt "Enter d:\path\ ". Pressing ENTER
from this prompt results in a display of all of the data
files in the default drive. If a drive and path are en-
tered, then the MATHPLOT data files in that drive/path
are displayed. Four types of data files are used by the
MATHPLOT programs. These types are distinguished by
their extension: .MPD, .SIM, .INV and .FRD.
Least Squares, Integrate, Statistics, Data Plot and
Function Plot use .MPD files. These files are confi-
gured to contain pairs of real coordinates for as many
as 200 data points. All of these programs can be used to
read, display, change and write .MPD files. Hardcopies
of .MPD files can be obtained from the Least Squares,
Integrate, Statistics and Function Plot programs. The
Data Plot program also uses .MPD files but does not in-
_____________________
Acknowledgement: Portions of the Function Plot program
utilize material from "Super Tools Library", published
by Paragon Courseware, 4954 Sun Valley Road, Del Mar,
California 92014.
25
files
teract with the printer except through the graphics
screen dump. Data files of the .MPD type can be printed
out conveniently and quickly from the Integrate program.
The Simeq and Matrix Inversion programs use .SIM
and .INV files respectively. The files are configured
to contain 1 and 2 dimension arrays of real numbers
whose order may be as high as 12. The .SIM files con-
tain 1 two-dimension array A(i,j) and 1 one-dimension
array B(j), whereas the .INV files contain only the two
dimension array of the elements for the matrix to be
inverted. Both programs permit the user to read, dis-
play, change, write to disk and print out their respec-
tive files.
The Fourier Series program uses .FRD files. These
files are configured like the .MPD files. However, each
file must contain an odd number of coordinate pairs
evenly spaced along the X axis.
All data files are written to disk in ASCII format.
They may be viewed and edited or written with a word
processor or displayed directly from DOS with the "TYPE"
command. These files were written with a text editor in
the non-document mode. This format may be used in wri-
ting other data files. It must be noted that fractional
numbers in data files must have a zero inserted before
the decimal point. Leading decimal points in real num-
bers cannot be read by the program.
26
Directory of PC-SIG Library Disk #1863
Volume in drive A has no label
Directory of A:\
BIGPOLY2 MPD 1947 5-01-89 6:30a
FILE1863 TXT 2295 12-22-89 2:52p
FOURAMP FRD 768 8-05-88 6:30a
FOURSQR FRD 512 8-05-88 6:30a
FPLSINX MPD 261 4-29-89 6:30a
GO BAT 38 1-01-80 1:37a
GO TXT 540 1-01-80 2:16a
INVOICE TXT 565 6-09-89 6:30a
MANUAL TXT 77059 5-09-89 6:30a
MATHPLOT EXE 133120 6-19-89 6:30a
MATRIX2 INV 128 8-05-88 6:30a
MATRIX3 INV 128 8-05-88 6:30a
POLY2 MPD 128 8-05-88 6:30a
POLY3 MPD 130 8-05-88 6:30a
POLY5 MPD 213 8-05-88 6:30a
SIMEQ2 SIM 128 8-05-88 6:30a
SIMEQ4 SIM 128 8-05-88 6:30a
SIMEQ6 SIM 109 8-05-88 6:30a
18 file(s) 218197 bytes
92160 bytes free