math::calculus - Math

# math::calculus(n) 1.0 math "Math"

## NAME

math::calculus - Integration and ordinary differential equations

## SYNOPSIS

package require Tcl 8
package require math::calculus 0.5

::math::calculus::integral begin end nosteps func
::math::calculus::integralExpr begin end nosteps expression
::math::calculus::integral2D xinterval yinterval func
::math::calculus::integral3D xinterval yinterval zinterval func
::math::calculus::eulerStep t tstep xvec func
::math::calculus::heunStep t tstep xvec func
::math::calculus::rungeKuttaStep tstep xvec func
::math::calculus::boundaryValueSecondOrder coeff_func force_func leftbnd rightbnd nostep
::math::calculus::solveTriDiagonal acoeff bcoeff ccoeff dvalue
::math::calculus::newtonRaphson func deriv initval
::math::calculus::newtonRaphsonParameters maxiter tolerance

## DESCRIPTION

This package implements several simple mathematical algorithms:

- The integration of a function over an interval
- The numerical integration of a system of ordinary differential equations.
- Estimating the root(s) of an equation of one variable.

The package is fully implemented in Tcl. No particular attention has been paid to the accuracy of the calculations. Instead, well-known algorithms have been used in a straightforward manner.

This document describes the procedures and explains their usage.

## PROCEDURES

This package defines the following public procedures:

::math::calculus::integral begin end nosteps func
  Determine the integral of the given function using the Simpson rule. The interval for the integration is [begin, end]. The remaining arguments are:
  nosteps
    Number of steps in which the interval is divided.
  func
    Function to be integrated. It should take one single argument.
::math::calculus::integralExpr begin end nosteps expression
  Similar to the previous proc, this one determines the integral of the given expression using the Simpson rule. The interval for the integration is [begin, end]. The remaining arguments are:
  nosteps
    Number of steps in which the interval is divided.
  expression
    Expression to be integrated. It should use the variable &quot;x&quot; as the only variable (the &quot;integrate&quot;)
::math::calculus::integral2D xinterval yinterval func
  The command integral2D calculates the integral of a function of two variables over the rectangle given by the first two arguments, each a list of three items, the start and stop interval for the variable and the number of steps.
  The currently implemented integration is simple: the function is evaluated at the centre of each rectangle and the content of this block is added to the integral. In future this will be replaced by a bilinear interpolation.
  The function must take two arguments and return the function value.
::math::calculus::integral3D xinterval yinterval zinterval func
  The command Integral3D is the three-dimensional equivalent of integral2D. The function taking three arguments is integrated over the block in 3D space given by three intervals.
::math::calculus::eulerStep t tstep xvec func
  Set a single step in the numerical integration of a system of differential equations. The method used is Euler's.
  t
    Value of the independent variable (typically time) at the beginning of the step.
  tstep
    Step size for the independent variable.
  xvec
    List (vector) of dependent values
  func
    Function of t and the dependent values, returning a list of the derivatives of the dependent values. (The lengths of xvec and the return value of &quot;func&quot; must match).
::math::calculus::heunStep t tstep xvec func
  Set a single step in the numerical integration of a system of differential equations. The method used is Heun's.
  t
    Value of the independent variable (typically time) at the beginning of the step.
  tstep
    Step size for the independent variable.
  xvec
    List (vector) of dependent values
  func
    Function of t and the dependent values, returning a list of the derivatives of the dependent values. (The lengths of xvec and the return value of &quot;func&quot; must match).
::math::calculus::rungeKuttaStep tstep xvec func
  Set a single step in the numerical integration of a system of differential equations. The method used is Runge-Kutta 4th order.
  t
    Value of the independent variable (typically time) at the beginning of the step.
  tstep
    Step size for the independent variable.
  xvec
    List (vector) of dependent values
  func
    Function of t and the dependent values, returning a list of the derivatives of the dependent values. (The lengths of xvec and the return value of &quot;func&quot; must match).
::math::calculus::boundaryValueSecondOrder coeff_func force_func leftbnd rightbnd nostep
  Solve a second order linear differential equation with boundary values at two sides. The equation has to be of the form (the &quot;conservative&quot; form):

               d      dy     d
               -- A(x)--  +  -- B(x)y + C(x)y  =  D(x)
               dx     dx     dx

  Ordinarily, such an equation would be written as:

                   d2y        dy
               a(x)---  + b(x)-- + c(x) y  =  D(x)
                   dx2        dx

  The first form is easier to discretise (by integrating over a finite volume) than the second form. The relation between the two forms is fairly straightforward:

               A(x)  =  a(x)
               B(x)  =  b(x) - a'(x)
               C(x)  =  c(x) - B'(x)  =  c(x) - b'(x) + a''(x)

  Because of the differentiation, however, it is much easier to ask the user to provide the functions A, B and C directly.
  coeff_func
    Procedure returning the three coefficients (A, B, C) of the equation, taking as its one argument the x-coordinate.
  force_func
    Procedure returning the right-hand side (D) as a function of the x-coordinate.
  leftbnd
    A list of two values: the x-coordinate of the left boundary and the value at that boundary.
  rightbnd
    A list of two values: the x-coordinate of the right boundary and the value at that boundary.
  nostep
    Number of steps by which to discretise the interval. The procedure returns a list of x-coordinates and the approximated values of the solution.
::math::calculus::solveTriDiagonal acoeff bcoeff ccoeff dvalue
  Solve a system of linear equations Ax = b with A a tridiagonal matrix. Returns the solution as a list.
  acoeff
    List of values on the lower diagonal
  bcoeff
    List of values on the main diagonal
  ccoeff
    List of values on the upper diagonal
  dvalue
    List of values on the righthand-side
::math::calculus::newtonRaphson func deriv initval
  Determine the root of an equation given by

          func(x) = 0

  using the method of Newton-Raphson. The procedure takes the following arguments:
  func
    Procedure that returns the value the function at x
  deriv
    Procedure that returns the derivative of the function at x
  initval
    Initial value for x
::math::calculus::newtonRaphsonParameters maxiter tolerance
  Set the numerical parameters for the Newton-Raphson method:
  maxiter
    Maximum number of iteration steps (defaults to 20)
  tolerance
    Relative precision (defaults to 0.001)

Notes:

Several of the above procedures take the names of procedures as arguments. To avoid problems with the visibility of these procedures, the fully-qualified name of these procedures is determined inside the calculus routines. For the user this has only one consequence: the named procedure must be visible in the calling procedure. For instance:

        namespace eval ::mySpace {
           namespace export calcfunc
           proc calcfunc { x } { return $x }
        }
        #
        # Use a fully-qualified name
        #
        namespace eval ::myCalc {
           proc detIntegral { begin end } {
              return [integral $begin $end 100 ::mySpace::calcfunc]
           }
        }
        #
        # Import the name
        #
        namespace eval ::myCalc {
           namespace import ::mySpace::calcfunc
           proc detIntegral { begin end } {
              return [integral $begin $end 100 calcfunc]
           }
        }

Enhancements for the second-order boundary value problem:

- Other types of boundary conditions (zero gradient, zero flux)
- Other schematisation of the first-order term (now central differences are used, but upstream differences might be useful too).

## EXAMPLES

Let us take a few simple examples:

Integrate x over the interval [0,100] (20 steps):

    proc linear_func { x } { return $x }
    puts &quot;Integral: [::math::calculus::integral 0 100 20 linear_func]&quot;

For simple functions, the alternative could be:

    puts &quot;Integral: [::math::calculus::integralExpr 0 100 20 {$x}]&quot;

Do not forget the braces!

The differential equation for a dampened oscillator:

    x'' + rx' + wx = 0

can be split into a system of first-order equations:

    x' = y
    y' = -ry - wx

Then this system can be solved with code like this:

    proc dampened_oscillator { t xvec } {
       set x  [lindex $xvec 0]
       set x1 [lindex $xvec 1]
       return [list $x1 [expr {-$x1-$x}]]
    }

    set xvec   { 1.0 0.0 }
    set t      0.0
    set tstep  0.1
    for { set i 0 } { $i &lt; 20 } { incr i } {
       set result [::math::calculus::eulerStep $t $tstep $xvec dampened_oscillator]
       puts &quot;Result ($t): $result&quot;
       set t      [expr {$t+$tstep}]
       set xvec   $result
    }

Suppose we have the boundary value problem:

        Dy'' + ky = 0
        x = 0: y = 1
        x = L: y = 0

This boundary value problem could originate from the diffusion of a decaying substance.

It can be solved with the following fragment:

       proc coeffs { x } { return [list $::Diff 0.0 $::decay] }
       proc force  { x } { return 0.0 }

       set Diff   1.0e-2
       set decay  0.0001
       set length 100.0

       set y [::math::calculus::boundaryValueSecondOrder \
          coeffs force {0.0 1.0} [list $length 0.0] 100]

## KEYWORDS

math, calculus, integration, differential equations, roots