math::calculus - Integration and ordinary differential equations

package require Tcl 8 package require math::calculus 0.5.1 ::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 t 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 ::math::calculus::regula_falsi f xb xe eps

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.

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 "x" as the only variable (the "integrate")

::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 "func" 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 "func" must match).

::math::calculus::rungeKuttaStep t 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 "func" 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 "conservative" 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)

::math::calculus::regula_falsi f xb xe eps

Return an estimate of the zero or one of the zeros of the function contained in the interval [xb,xe]. The error in this estimate is of the order of eps*abs(xe-xb), the actual error may be slightly larger. The method used is the so-called regula falsi or false position method. It is a straightforward implementation. The method is robust, but requires that the interval brackets a zero or at least an uneven number of zeros, so that the value of the function at the start has a different sign than the value at the end. In contrast to Newton-Raphson there is no need for the computation of the function's derivative.

f command Name of the command that evaluates the function for

which the zero is to be returned

xb float Start of the interval in which the zero is supposed

to lie

xe float End of the interval

eps float Relative allowed error (defaults to 1.0e-4)

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).

Let us take a few simple examples:

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

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

puts "Integral: [::math::calculus::integralExpr 0 100 20 {$x}]"

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 < 20 } { incr i } { set result [::math::calculus::eulerStep $t $tstep $xvec dampened_oscillator] puts "Result ($t): $result" set t [expr {$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]

romberg

calculus, differential equations, integration, math, roots

Copyright (c) 2002,2003,2004 Arjen Markus