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` Numerical methods for the PG language`

```
Usage: plot_list([x0,y0,x1,y1,...]);
plot_list([(x0,y0),(x1,y1),...]);
plot_list(\x_y_array);
plot_list([x0,x1,x2...], [y0,y1,y2,...]);
plot_list(\@xarray,\@yarray);
```

```
Usege: $fn = horner([x0,x1,x2],[q0,q1,q2]);
Produces the newton polynomial
&$fn(x) = q0 + q1*(x-x0) +q2*(x-x1)*(x-x0);
```

Generates a subroutine which evaluates a polynomial passing through the points `(x0,q0), (x1,q1), ... `

using Horner's method.

```
Usage: $poly = hermit([x0,x1...],[y0,y1...],[yp0,yp1,...]);
Produces a reference to polynomial function
with the specified values and first derivatives
at (x0,x1,...).
&$poly(34) gives a number
```

Generates a subroutine which evaluates a polynomial passing through the specified points with the specified derivatives: (x0,y0,yp0) ... The polynomial will be of high degree and may wobble unexpectedly. Use the Hermite splines described below and in Hermite.pm for most graphing purposes.

```
Usage: $spline = hermit_spline([x0,x1...],[y0,y1...],[yp0,yp1,...]);
Produces a reference to a piecewise cubic hermit spline
with the specified values and first derivatives
at (x0,x1,...).
&$spline(45) evaluates to a number.
```

Generates a subroutine which evaluates a piecewise cubic polynomial passing through the specified points with the specified derivatives: (x0,y0,yp0) ...

An object oriented version of this is defined in Hermite.pm

```
Usage:
$fun_ref = cubic_spline(~~@x_values, ~~@y_values);
```

Where the x and y value arrays come from the function to be approximated. The function reference will take a single value x and produce value y.

` $y = &$fun_ref($x);`

You can also generate javaScript which defines a cubic spline:

```
$function_string = javaScript_cubic_spline(~~@_x_values, ~~@y_values,
name => 'myfunction1',
llimit => -3,
rlimit => 3,
);
```

The string contains

```
<SCRIPT LANGUAGE="JavaScript">
<!-- Begin
function myfunction1(x) {
...etc...
}
</SCRIPT>
```

and can be placed in the header of the HTML output using

` HEADER_TEXT($function_string);`

` Usage: lefthandsum(function_reference, start, end, steps=>30 );`

Implements the Left Hand sum using 30 intervals between 'start' and 'end'. The first three arguments are required. The final argument (number of steps) is optional and defaults to 30.

` Usage: righthandsum(function_reference, start, end, steps=>30 );`

Implements the right hand sum using 30 intervals between 'start' and 'end'. The first three arguments are required. The final argument (number of steps) is optional and defaults to 30.

` Usage: midpoint(function_reference, start, end, steps=>30 );`

Implements the Midpoint rule using 30 intervals between 'start' and 'end'. The first three arguments are required. The final argument (number of steps) is optional and defaults to 30.

` Usage: simpson(function_reference, start, end, steps=>30 );`

Implements Simpson's rule using 30 intervals between 'start' and 'end'. The first three arguments are required. The final argument (number of steps) is optional and defaults to 30, but must be even.

` Usage: trapezoid(function_reference, start, end, steps=>30 );`

Implements the trapezoid rule using 30 intervals between 'start' and 'end'. The first three arguments are required. The final argument (number of steps) is optional and defaults to 30.

` Usage: romberg(function_reference, x0, x1, level);`

Implements the Romberg integration routine through 'level' recursive steps. Level defaults to 6.

` Usage: inv_romberg(function_reference, a, value);`

Finds b such that the integral of the function from a to b is equal to value. Assumes that the function is continuous and doesn't take on the zero value. Uses Newton's method of approximating roots of equations, and Romberg to evaluate definite integrals.

```
rungeKutta4
Finds integral curve of a vector field using the 4th order Runge Kutta method.
Useage: rungeKutta4( &vectorField(t,x),%options);
Returns: \@array of points [t,y})
Default %options:
'initial_t' => 1,
'initial_y' => 1,
'dt' => .01,
'num_of_points' => 10, #number of reported points
'interior_points' => 5, # number of 'interior' steps between reported points
'debug'
```