USING MATHOBJECTS

To use MathObjects in your own problems, you need to load the "MathObjects.pl" macro file:

loadMacros("Parser.pl");

which defines the commands you need to interact with MathObjects. Once you have done that, you can call the MathObjects functions to create formulas for you. The main call is Formula(), which takes a string and returns a parsed version of the string. For example:

$f = Formula("x^2 + 3x + 1");

will set $f to a reference to the parsed version of the formula.

Working With Formulas

A formula has a number of methods that you can call. These include:

$f->eval(x=>5)

Evaluate the formula when x is 5. If $f has more variables than that, then you must provide additional values, as in $f->eval(x=>3,y=>1/2);

$f->reduce

Tries to remove redundent items from your formula. For example, Formula("1x+0") returns "x". Reduce tries to factor out negatives and do some other adjustments as well. (There still needs to be more work done on this. What it does is correct, but not always smart, and there need to be many more situations covered.) All the reduction rules can be individually enabled or disabled using the Context()->reduction->set() method, but the documentation for the various rules is not yet ready.

$f->substitute(x=>5)

Replace x by the value 5 throughout (you may want to reduce the result afterword, as this is not done automatically). Note that you can replace a variable by another formula, if you wish. To make this easier, substitute will apply Formula() to any string values automatically. E.g., Formula("x-1")->substitute(x=>"y")

returns "y-1" as a formula.

$f->string

returns a string representation of the formula (should be equivalent to the original, though not necessarily equal to it).

$f->TeX

returns a LaTeX representation of the formula. You can use this in BEGIN_TEXT...END_TEXT blocks as follows:

BEGIN_TEXT
Suppose \(f(x) = \{$f->TeX}\). ...
END_TEXT

$f->perl

returns a representation of the formula that could be evaluated by perl's eval() function.

$f->perlFunction

returns a perl code block that can be called to evaluate the function. For example:

$f = Formula('x^2 + 3')->perlFunction;
$y = &$f(5);

will assign the value 28 to $y. You can also pass a function name to perlFunction to get a named function to call:

Formula('x^2 + 3')->perlFunction('f');
$y = f(5);

If the formula involves more than one variable, then the paramaters should be given in alphabetical order.

Formula('x^2 + y')->perlFunction('f');
$z = f(5,3);  # $z is 28.

Alternatively, you can tell the order for the parameters:

Formula('x^2 + y')->perlFunction('f',['y','x']);
$z = f(5,3); $ now $z is 14.

Combining Formulas

There is a second way to create formulas. Once you have a formula, you can create additional formulas simply by using perls' built-in operations and functions, which have been overloaded to handle formulas. For example,

$x = Formula('x');
$f = 3*x**2 + 2*$x - 1;

makes $f be a formula, and is equivalent to having done

$f = Formula("3x^2 + 2x - 1");

This can be very convenient, but also has some pitfalls. First, you need to include '*' for multiplication, since perl doesn't do implied multiplication, and you must remember to use '**' not '^'. (If you use '^' on a formula, the parser will remind you to use '**'.) Second, the precedences of the operators in perl are fixed, and so changes you make to the precedence table for the parser are not reflected in formulas produced in this way. (The reason '^' is not overloaded to do exponentiation is that the precedence of '^' is wrong for that in perl, and can't be changed.) As long as you leave the default precedences, however, things should work as you expect.

Note that the standard functions, like sin, cos, etc, are overloaded to generate appropriate formulas when their values are formulas. For example,

$x = Formula('x');
$f = cos(3*$x + 1);

produces the same result as $f = Formula("cos(3x+1)"); and you can then go on to output its TeX form, etc.

Special Syntax

This parser has support for some things that are missing from the current one, like absolute values. You can say |1+x| rather than abs(1+x) (though both are allowed), and even |1 - |x|| works.

Also, you can use sin^2(x) (or even sin^2 x) to get (sin(x))^2.

Finally, you can use sin^-1(x) to get arcsin(x).

There is an experimental set of operator precedences that make it possible to write sin 2x + 3 and get sin(2x) + 3. See examples/7-precedence.pg for some details.

The Formula Types

The parser understands a wide range of data types, including real and complex numbers, points, vectors, matrices, arbitrary lists, intervals, unions of intervals, and predefined words. Each has a syntax for use within formulas, as described below:

numbers        the usual form:  153, 233.5, -2.456E-3, etc.

complex        a + b i  where a and b are numbers:  1+i, -5i, 6-7i, etc.

infinitites    the words 'infinity' or '-infinity' (or several
               equivalents).

point          (a,b,c)  where a, b and c are real or complex numbers.
               any number of coordinates are allowed.  Eg, (1,2), 
               (1,0,0,0), (-1,2,-3).  Points are promoted to vectors 
               automatically, when necessary.

vector         <a,b,c>  or  a i + b j + c k  (when used in vector context).
               As with points, vectors can have any number of 
               coordinates.  For example, <1,0,0>, <-1,3>, <x,1-x>, etc.

matrix         [[a11,...,a1n],...[am1,...amn]], i.e., use [..] around 
               each row, and around the matrix itself.  The elements 
               are separated by commas (not spaces).  e.g,
                    [[1,2],[3,4]]     (a 2x2 matrix)
                    [1,2]             (a 1x2 matrix, really a vector)
                    [[1],[2]]         (a 2x1 matrix, ie. column vector)
               Points and vectors are promoted to matrices when 
               appropriate.  Vectors are converted to column vectors 
               when needed for matrix-vector multiplication.  Matrices 
               can be 3-dimensional or higher by repeated nesting of 
               matrices.  (In this way, a 2-dimensional matrix is really 
               thought of as a vector of vectors, and n-dimensional 
               ones as vectors of (n-1)-dimensional ones.)

list            (a,b,c)  where a,b,c are arbitrary elements.
               For example, (1+i, -3, <1,2,3>, Infinity).
               The empty list () is allowed, and the parentheses are 
               optional if there is only one list.  (This makes it 
               possible to make list-based answer checkers that
               really know where the separations occur.)

interval        (a,b), (a,b], [a,b), [a,b], or [a,a]  where a and b are
               numbers or appropriate forms of infinity.
               For example, (-INF,3], [4,4], [2,INF), (-INF,INF).

union           represented by 'U'.  For example  [-1,0) U (0,1].

string          special predefined strings like NONE and DNE.

These forms are what are used in the strings passed to Formula(). If you want to create versions of these in perl, there are several ways to do it. One way is to use the Compute() command, which takes a string parses it and then evaluates the result (it is equivalent to Formula(...)->eval). If the formula produces a vector, the result will be a Vector constant that you can use in perl formulas by hand.

For example:

$v = Compute("<1,1,0> >< <-1,4,-2>");

would compute the dot product of the two vectors and assign the resulting vector object to $v.

Another way to generate constants of the various types is to use the following routines. If their inputs are constant, they produce a constant of the appropriate type. If an input is a formula, they produce corresponding formula objects.

Real(a)            create a real number with "fuzzy"
                   comparisons (so that 1.0000001 == Real(1) is true).

Complex(a,b)       create a complex number a + b i

Infinity           creates the +infinity object
-(Infinity)        creates -infinity

Point(x1,...xn) or Point([x1,...,xn])      produces (x1,...,xn)

Vector(x1,...,xn) or Vector([x1,...,xn])   produces <x1,...,xn>

Matrix([a11,...,a1m],...,[am1,...,amn]) or
Matrix([[a11,...,a1m],...,[am1,...,amn]])  produces an n x m matrix

List(a,...,b)   produces a list with the given elements

Interval('(',a,b,']')   produces (a,b], (the other endpoints work as
                        expected.  Use 'INF' and '-INF' for infinities.)

Union(I1,...,In)  takes the union of the n intervals. (where I1 to In 
                  are intervals.)

String(word)  Produces a string object for the given word (if it
              is a known word).  This is mostly to be able to
      call the ->cmp and ->TeX methods.

For example,

$a = random(-5,5,1)
$V = Vector($a,1-$a,$a**2+1);

produces a vector with some random coordinates.

Objects of these types also have TeX, string and perl methods, so you can use:

Vector(1,2,3)->TeX

to produce a TeX version of the vector, just as you can with formulas.

There are several "constant" functions that generate common constant values. These include pi, i, j, k and Infininty. you can use these in perl expressions as though they were their actual values:

$z = $a + $b * i;
$v = $a*i + $b*j + $c*k;
$I = Infinity;

Note that because of a peculiarity of perl, you need to use -(pi) or - pi (with a space) rather than -pi, and similary for the other functions. Without this, you will get an error message about an ambiguity being resolved. (This is not a problem if you process your expressions through the parser itself, only if you are writing expressions in perl directly. Note that since student answers are processed by the parser, not perl directly, they can write -pi without problems.)

Another useful command is Compute(), which evaluates a formula and returns its value. This is one way to create point or vector-valued constants, but there is an easier way discussed below.

Specifying the Context

You may have noticed that "i" was used in two different ways in the examples above. In the first example, it was treated as a complex number and the second as a coordinate unit vector. To control which interpretation is used, you specify a parser "context".

The context controls what operations and functions are defined in the parser, what variables and constants to allow, how to interpret various paretheses, and so on. Changing the context can completely change the way a formula is interpreted.

There are several predefined contexts: Numeric, Complex, Vector, Interval and Full. (You can also define your own contexts, but that will be described elsewhere.) To select a context, use the Context() function, e.g.

Context("Numeric");

selects the numeric context, where i, j and k have no special meaning, points and vectors can't be used, and the only predefined variable is 'x'.

On the other hand, Context("Vector") makes i, j and k represent the unit coordinate vectors, and defines variables 'x', 'y' and 'z'.

Context("Interval") is like numeric context, but it also defines the parentheses so that they will form intervals (rather than points or lists).

Once you have selected a context, you can modify it to suit the particular needs of your problem. The command

$context = Context();

gets you a reference to the current context object (you can also use something like

$context = Context("Numeric");

to set the context and get its reference at the same time). Once you have this reference, you can call the Context methods to change values in the context. These are discussed in more detail in the documentation of the Context object [not yet written], but some of the more common actions are described here.

To add a variable, use, for example,

$context->variables->add(y=>'Real');

To delete any existing variables and replace them with new ones, use

$context->variables->are(t=>'Real');

To remove a variable, use

$context->variables->remove('t');

To get the names of the defind variables, use

@names = $context->variables->names;

Similarly, you can add a named constant via

$context->constants->add(M=>1/log(10));

and can change, remove or list the constants via methods like those used for variables above. The command

$M = $context->constants->get('M');

will return the value of the consant M. (See the pg/lib/Value/Context/Data.pm file for more information on the methods you can call for the various types of context data.)

To add new predefined words (like 'NONE' and 'DNE'), use something like

$context->strings->add(TRUE=>{},FALSE=>{});

Strings are case-insensitive, unless you say otherwise. To mark a string as being case-senstive, use

$context->strings->add(TRUE => {caseSensitive=>1});

You may want to privide several forms for the same word; to do so, make the additional words into aliases:

$context->strings->add(
    T => {alias=>'TRUE'},
    F => {alias=>'FALSE'},
);

so that either "TRUE" or "T" will be interpreted as TRUE, and similarly for "FALSE" and "F";

There are a number of values stored in the context that control things like the tolerance used when comparing numbers, and so on. You control these via commands like:

$context->flags->set(tolerance=>.00001);