World of Ham Radio 1997/WOHR97_AmSoft_(1997-02-01).iso/formulas/math/amsoft.iii

Text File | 1997-02-01 | 23KB | 603 lines

                     SCIENTIFIC EQUATION SOLVER
                     ==========================


Scientific Equation Solver (SCIEQS) acts as a sophisticated scientific
calculator.  It can perform double precision floating point arithmetic
using an assignment statement syntax similar to BASIC or FORTRAN.
Variables may be assigned values in one statement and then used on the
right-hand side of another statement.

Almost every built-in function found on scientific calculators has
been included: trigonometric, hyperbolic, exponential, logarithmic,
factorial, and square/cube root functions.  Even advanced functions
such as erf(x) and gamma(x) have been included.  You can also define
your own functions with any number of arguments.

There is also an equation solver built into the program.  This feature
allows the program to solve for any variable in a given equation.
Suppose you needed to solve for the variable 'x', where

      (3 * x) + exp(x+1) = 99 .

By entering the command

      !solve  x, [0,100] : (3 * x) + exp(x+1) = 99 ,

the solution 3.483559695247 comes up on your screen in seconds.

SCIEQS also supports numerical integration (also known as quadrature).
This feature allows the program to approximate the values of definite
integrals.  Suppose you needed to evaluate the following integral:

           2
          ⌠
          │  (x^2 * sqrt(x^3 + 1))  dx .
          ⌡
           0

By entering the command

      !quad  x, [0,2] : x^2 * sqrt(x^3 + 1) ,

the value 5.7777777777778 comes up on your screen in seconds.



                        HOW TO START SCIEQS
                        ===================

To begin SCIEQS, type the following command at the MS-DOS command
prompt:
           SCIEQS

The program reads its input from the standard input and writes its
output to the standard output so you may use the data redirection
symbols >, >>, <, and | from the MS-DOS command line.  For example,
the following command reads commands from file POLY.EQS and writes the
results to file RESULTS.TXT:

           SCIEQS  <POLY.EQS  >RESULTS.TXT



                        EXPRESSIONS IN SCIEQS
                        =====================

SCIEQS is an expression oriented language.  Expressions typed by the
user are interpreted, evaluated, and displayed immediately by SCIEQS.
Most commands are of the form

           variable = expression

or simply

           expression

The first form is the assignment statement.  It assigns the expression
on the right side of the equals sign to the variable named on the left
side and also displays the value of the expression.  The second form
simply evaluates the expression and displays its value.

Examples:   2+3      area = pi()*R^2       b = c*d
            x^2      sqrt(2)               (2+3)!

Expressions are composed of numbers, variables, function references,
and arithmetic operators.  Following are the rules for forming
expressions.

Numbers follow a conventional decimal notation, with optional decimal
point.  A power-of-ten scale factor can be included as a suffix.  This
is similar to the number notations used in the computer languages
BASIC and FORTRAN.  The range of numbers is approximately 1E-308 to
1E+308 with approximately 15 significant digits.

Examples:    1.2       +.8         -788       1E6
             4.        1256        -7.3E-5

Variable names and function names must start with a letter and may be
followed by any number of letters, digits, and underscores.  However,
only the first 31 characters of the name are remembered.  SCIEQS is
not case-sensitive so it treats upper and lower case letters as the
same character.  The variable names "Answer" and "answer" refer to the
same variable.

Examples:   SUM       x       days_in_month       xYzzY

An arithmetic expression can be any expression preceded by unary + or
-, two expressions linked by a binary operator (+ - * / % ^), or any
expression followed by unary !.

Arithmetic operators are (from high to low precedence):

                Factorial                     !
                Unary plus and minus          +  -
                Exponentiation                ^
                Mult, Div, and Modulus        *  /  %
                Addition and Subtraction      +  -

Examples:   x+3    -t^2    tan(x)/y    6!    4^2^3 (evaluates to 65536)

Operations at the same precedence level are performed from left to
right except for series of exponentiations, which are performed from
right to left.  The factorial operator appears on the right of the
expression on which it operates.  The order of operations may be
controlled by enclosing expressions in parentheses.

There are two main categories of functions in SCIEQS: built-in and
user-defined functions.  Built-in functions are automatically provided
every time you run SCIEQS.  User-defined functions are defined by you
to handle your specific problem.  To invoke a function, refer to the
function in an expression.

Example:    a = tan(x)/d

Following is a list of the built-in functions.

Mathematical Constants

PI() = 3.141592653589793    ratio of circumference to diameter
E()  = 2.718281828459045    base of natural logarithms


Trigonometric and Inverse Trigonometric Functions in Radians

SIN(x)           Sine of x
COS(x)           Cosine of x
TAN(x)           Tangent of x
ASIN(x)          Arc sine of x
ACOS(x)          Arc cosine of x
ATAN(x)          2-quadrant arc tangent of x
ATAN2(y,x)       4-quadrant arc tangent of y/x
ANGL(x,y)        Angle to point (x,y)
HYPOT(x,y)       Hypotenuse of right triangle


Trigonometric and Inverse Trigonometric Functions in Degrees

SIND(x)          Sine of x
COSD(x)          Cosine of x
TAND(x)          Tangent of x
ASIND(x)         Arc sine of x
ACOSD(x)         Arc cosine of x
ATAND(x)         2-quadrant arc tangent of x
ATAN2D(y,x)      4-quadrant arc tangent of y/x
ANGLD(x,y)       Angle to point (x,y)


Hyperbolic and Inverse Hyperbolic Functions

SINH(x)       Hyperbolic sine of x
COSH(x)       Hyperbolic cosine of x
TANH(x)       Hyperbolic tangent of x
ASINH(x)      Hyperbolic arc sine of x
ACOSH(x)      Hyperbolic arc cosine of x
ATANH(x)      Hyperbolic arc tangent of x


Exponential and Logarithmic Functions

EXP(x)     Number e, 2.718281828459045, raised to the xth power
LN(x)      Natural (base e) logarithm of x
LOG(x)     Common (base 10) logarithm of x
LOG2(x)    Base 2 logarithm of x


Other Mathematical Functions

ABS(x)      Absolute value of x
SQRT(x)     Positive square root of x
CBRT(x)     Cube root of x
ERF(x)      Error function associated with normal distribution curve
ERFC(x)     Complementary error function,  1 - erf(x)
GAMMA(x)    Gamma function, x-1 factorial for integer x>0
RTOD(x)     Convert x in radians to degrees
DTOR(x)     Convert x in degrees to radians
PERM(n,r)   Permutations of n items taken r at a time
COMB(n,r)   Combinations of n items taken r at a time
SGN(x)      Sign (-1, +1 or 0) of x
INT(x)      Integer part of x
ROUND(x)    Nearest integer to x
CEIL(x)     Smallest integer >= x
FLOOR(x)    Largest integer <= x
STEP(x,y)   Returns  1 if x >= y,  0 if x < y



           ASSIGNMENT STATEMENTS AND USER-DEFINED FUNCTIONS
           ================================================

The assignment statement is used to define variables and user-defined
functions.  They have the following formats:

      variable  =  expression

      function(arguments)  =  expression

Once the assignment has been done, the variable or function is
associated with the given expression.  For a user-defined function,
the right-side expression should use the arguments listed in the
function's argument list.  If a variable or function is used in an
expression before it is defined in an assignment statement, an error
message will be displayed.

Examples:
           a = 2
           b = sqrt(a)
           area_circle(radius) = pi() * radius ^ 2
           c = area_circle(a)

Assignment in SCIEQS assigns the expression, not the value of the
expression.  What this means is the value of variables can change if
the variables they depend on change.  For example, after the four
assignments in the above example have been done, the values of the
variables are:

          A = 2
          B = 1.4142135623731
          C = 12.566370614359

But suppose we change the value of variable A by assigning it the
value 3.  Now the values of variables B and C change automatically
because they were defined in terms of variable A.

          A = 3
          B = 1.7320508075689
          C = 28.274333882308

This feature is important when you want to play "What happens if..."
games.  You can build a mathematical model using different formulas
and see what happens when you start changing the values of some of the
variables.


                          COMMANDS IN SCIEQS
                          ==================

Commands in SCIEQS begin with the character "!".  Following is a list
of available commands and what they do.


SOLVE COMMAND
-------------

The solve command uses a numerical approximation method to solve
equations.  It has the form

   !solve  variable ,  [ low, high ]  :  equation

Example: Solve for x where  x^3 - 11*x^2 + 39*x = 27.

   > !solve x, [-100,100]: x^3 - 11*x^2 + 39*x = 27
                   = 0.90375019315913

The variable x now has the value 0.90375019315913 for use in later
calculations.

The values in brackets [ ] represent the low and high bounds of an
interval in which to look for a solution.  The approximation method
first takes a trial value at the midpoint of the specified interval
and iterates until the trial values converge on a solution.  If the
equation to be solved has singularity points, choose the intervals to
avoid the singularities.  SCIEQS may not be able to find a solution if
you include singularity points within the intervals.

If the equation has multiple solutions within the given interval,
SCIEQS will usually find one of them.  To find multiple solutions, you
must narrow the search interval to include only one solution at a
time.  You can narrow down the search intervals by plotting the
equations to find the vicinity of the solutions.


QUAD COMMAND
------------

The quadrature command uses a numerical approximation method to
evaluate definite integrals.  It has the form

   !quad  variable ,  [ low, high ]  :  integrand

The variable indicates the "variable of integration."  The values in
brackets [ ] represent the low and high limits of integration.  The
integrand is the expression to be integrated.  SCIEQS will integrate
the integrand from low to high limits with respect to the specified
variable of integration.
                                           π
                                          ⌠
Example: Evaluate the definite integral   │ sin(x)  dx .
                                          ⌡
                                           0
   > !quad x, [0,pi()] : sin(x)
   estimated error = 4.206476E-016
                   = 2

An estimate of the size of the actual error in the result is first
listed.  If the approximation method encounters problems during the
integration, it may execute much longer than usual in an attempt to
achieve enough accuracy in the result.  If the accuracy can not be
achieved, a warning message will be shown indicating the result may be
unreliable.  The result of the integration is then listed below the
estimated error.  The variable of integration will now have the result
of the integration for use in later calculations.  In the above
example, the variable x will now contain the result.

SCIEQS may not be able to evaluate the integral if there are
singularity points within the limits of integration which cause
floating point overflows.  If a floating point overflow occurs, the
program may terminate unexpectedly.


DERIV COMMAND
-------------

The derivative command uses a numerical approximation method to
evaluate the derivative of a function at a specified point.  It has
the form

   !deriv  variable , point  :  function

The variable indicates the "variable of differentiation."  The
derivative is taken with respect to the variable of differentiation.
Point indicates the value of the variable of differentiation where the
derivative will be evaluated.  It must be a value where the function
is defined.  Function indicates the function to be differentiated.

Example: Evaluate the derivative of f(x) = 3 * x^4  at  x = 7.

   > !deriv x, 7  :  3 * x^4
   estimated error = 0
                   = 4116

An estimate of the size of the actual error in the result is first
listed.  The result of the differentiation is then listed below the
estimated error.  The variable of differentiation will now have the
result of the differentiation for use in later calculations.  In the
above example, the variable x will now contain the result.

SCIEQS may not be able to evaluate the derivative if the function is
undefined near the point of evaluation.  You may restrain the
numerical approximation method by using the following form of the
command.

   !deriv  variable , point , range  :  function

Range specifies the range (point - range) to (point + range).  SCIEQS
will not try to evaluate the function outside of this range.  The
default value of range is ABS(point / 16).  If there are singularity
points within the range (even the default range), the derivative
command will probably fail to evaluate the derivative.  Specifying a
very small value for range may increase the error in the
approximation.


LIST COMMAND
------------

The list command is used to display the values of all the variables
used.  The command has the following forms:

   !list

   !list  filename

The first form displays the variables on the standard output.  The
second form stores the output in the named file.


DEL COMMAND
-----------

The del command is used to delete variables no longer needed.  The
command has the form:

   !del  variable

When the del command is executed, all occurrences of the variable in
formulas and function definitions are replaced by the variable's
current numerical value.  Here is an example:

   a = 2
   b = 5 * a
   !del a
   !list

The list command will not show variable 'a'.  Variable 'b' will have a
value of 10 since the del command replaced the "5 * a" with "5 * 2".


ECHO/NOECHO COMMANDS
--------------------

The echo and noecho commands control the echoing of the input commands
to the output destination.  They have the following forms:

   !echo

   !noecho

The echo command turns on input command echoing while the noecho
command turns it off.  SCIEQS starts out with echoing turned off.  You
may wish to turn on echoing when sending output to a file so the
output file will contain a record of the input as well as the results.


SYS COMMAND
-----------

The sys command allows you to execute MS-DOS commands from within
SCIEQS.  It has the following forms:

   !sys

   !sys  MS-DOS_command

The first form starts the MS-DOS command shell underneath SCIEQS.  You
must type "EXIT" at the MS-DOS command prompt to return to SCIEQS.
The second form executes the given MS-DOS command and automatically
returns to SCIEQS after the command finishes.


EXIT/QUIT COMMANDS
------------------

Both commands terminate SCIEQS execution.  They have the forms

   !exit

   !quit


                      COMMAND FILES IN SCIEQS
                      =======================

SCIEQS supports the inclusion of input from command files.  These
command files are ASCII text files which contain data, numerical
operations, and commands for SCIEQS to interpret.  You may think of
them as "include" files.  To include a file, give the following
command:

   @filename

SCIEQS will open the file named after the "@" character and start
getting its input from that file.  When the end of file is reached,
SCIEQS will resume getting its input from the original source.  You
may specify a directory pathname if the file is not in the current
directory.  You may nest command files within other command files, but
MS-DOS may limit you in the number of files you may open at any one
time.


                        COMMENTS IN SCIEQS
                        ==================

Comments can be included in the input to help document your work.  A
comment begins with a semicolon and continues to end of the line.  You
can put a comment after an expression to explain the operation on that
line.  You can also start a line with a comment in which case the
entire line is a comment.

Examples:   cbrt(x)   ;taking the cube root of variable x
            ; This entire line is a comment.


                       MATH COPROCESSOR USAGE
                       ======================

The SCIEQS program will automatically detect the presence of an 8087,
80287, or 80387 math coprocessor in your computer and use it for
calculations if available.  If a math coprocessor is not available,
the program will still work, but at a slower execution speed.


                      EDITING PREVIOUS COMMANDS
                      =========================

The MS-DOS operating system provides a facility to edit the previous
command given to SCIEQS so that it may be reexecuted.  This is useful
if you had mistyped the last command or if you want to execute several
similar commands in succession.  The following editing keys may be
used to edit the previous command:

   F1   Copies the next character from previous command.
   F2   Copies all characters from previous command up to a specified
          character.  Type specified character after F2 key.
   F3   Copies all remaining characters from previous command.
   F4   Skips over all characters in previous command up to a
          specified character.  Type specified character after F4 key.
   INS  Allows insertion of characters into the command.
   DEL  Skips over one character from previous command.
   <--  Deletes character to left of the cursor on current command.
   ESC  Cancels the current command.

Please refer to your MS-DOS reference manual for more information on
the above editing keys.

Starting with SCIEQS 1.4, you may also recall and edit more than just
the last command by using the program CED.  CED is a command editor
which allows you to use the cursor arrow keys to recall, edit, and
reissue previous commands.  It also works at the MS-DOS command prompt.
CED is available without registration fees on most electronic bulletin
board systems and from user group libraries.  If you can't find it
elsewhere, I'll provide a copy of it when you register SCIEQS.

If you use CED, I suggest you change the default chain character of
caret (^) to control-backslash.  This chain character allows you to
issue more than one command on the same input line.  However, the
CED default value of caret (^) conflicts with the exponentiation
operator used in SCIEQS.  Issue this command at the MS-DOS command
prompt to change the CED chain character:

         CED CHAINCH ^\

The "^\" in the above command is typed by holding down the CTRL key
while pressing the backslash key (\).  You can also type the caret (^)
character followed by the backslash key.  Please refer to the CED
documentation to learn how to have this command issued automatically
when you start your computer.  When you wish to issue more than one
command on the same input line, enter the control-backslash character
between the commands by holding down the CTRL key while pressing the
backslash key (\).


                           UPDATE HISTORY
                           ==============

Version 1.0: Initial release.

Version 1.1: Added feature to evaluate definite integrals.

Version 1.2: Added feature to evaluate derivatives.

Version 1.3: Added command to delete variables.
             Improved comb() and perm() functions.
             Fixed bug in hypot() function.

Version 1.4: Improved error messages with a pointer showing where
               a syntax error occured in the user's input.
             Improved keyboard input so SCIEQS can be used with
               the CED program to recall and edit previous commands.
             Fixed bug where nested evaluations of user-defined
               functions within other user-defined functions did
               not work correctly.


                            DISCLAIMER
                            ==========

The SCIEQS program and associated files are provided "as is" without
any warranty or guarantee, including no warranty of fitness for a
particular purpose.  You assume the entire risk as to the results
generated through your use of this software.


                            DISTRIBUTION
                            ============

The shareware version of SCIEQS is distributed on electronic bulletin
boards and by computer clubs.  You may use the software and make
multiple backup copies.  Versions of SCIEQS designated as "shareware"
may be given to friends and uploaded to electronic bulletin boards
provided no changes have been made to the distribution package.  Do
not distribute copies of the program which do not have the initial
screen explaining the shareware concept.

If you find this program useful, please feel free to give consideration
for my time and effort in developing this tool.  Please send $6.00 to
the address below and ask for SCIEQS registration.  The file ORDER.ME
contains a registration form that can be printed on your printer if
you wish.  In return, you will receive a floppy disk with the latest
version of SCIEQS.  The introductory screen along with its keyboard
entry will NOT appear in the copy of the program you will receive upon
registration.  There will be no advertising in your registered copy.

Please upload the shareware version of this program onto as many local
bulletin boards as possible.  You will be helping me make this tool
available to many more people giving me incentive to make further
improvements to the program.


Send contributions to:  George Yee
                        1847 N. Frances Blvd.
                        Tucson, AZ  85712