• NAME
• SYNPOSIS
• DESCRIPTION
  • Interpolation methods
    • Plotting a list of points (piecewise linear interpolation)
    • Horner polynomial/ Newton polynomial
    • Hermite polynomials
    • Hermite splines
    • Cubic spline approximation
  • Numerical Integration methods
    • Integration by trapezoid rule
    • Romberg method of integration
    • Inverse Romberg

NAME

    Numerical methods for the PG language

SYNPOSIS

DESCRIPTION

Interpolation methods

Plotting a list of points (piecewise linear interpolation)

    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);

Horner polynomial/ Newton polynomial

    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.

Hermite polynomials

    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.

Hermite splines

    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

Cubic spline approximation

    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);

Numerical Integration methods

Integration by trapezoid rule

    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.

Romberg method of integration

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

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

Inverse Romberg

    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.

File path = /ww/webwork/pg/macros/PGnumericalmacros.pl

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