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math::calculus(n)                             Tcl Math Library                             math::calculus(n)



____________________________________________________________________________________________________________

NAME
       math::calculus - Integration and ordinary differential equations

SYNOPSIS
       package require Tcl  8.4

       package require math::calculus  0.7.1

       ::math::calculus::integral begin end nosteps func

       ::math::calculus::integralExpr begin end nosteps expression

       ::math::calculus::integral2D xinterval yinterval func

       ::math::calculus::integral2D_accurate xinterval yinterval func

       ::math::calculus::integral3D xinterval yinterval zinterval func

       ::math::calculus::integral3D_accurate 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

____________________________________________________________________________________________________________

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


       ::math::calculus::integral2D xinterval yinterval func

       ::math::calculus::integral2D_accurate xinterval yinterval func
              The commands integral2D and integral2D_accurate calculate 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 command integral2D evaluates the function at the centre of  each  rectangle,  whereas  the
              command  integral2D_accurate  uses  a  four-point quadrature formula. This results in an exact
              integration of polynomials of third degree or less.

              The function must take two arguments and return the function value.

       ::math::calculus::integral3D xinterval yinterval zinterval func

       ::math::calculus::integral3D_accurate xinterval yinterval zinterval func
              The commands integral3D and integral3D_accurate are the three-dimensional equivalent of  inte-gral2D integral2D
              gral2D  and  integral3D_accurate.   The  function func takes three arguments and 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 equa-tion equation
              tion 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  straightfor-ward straightforward
              ward  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  dif-ferent different
              ferent sign than the value at the end.

              In  contrast  to Newton-Raphson there is no need for the computation of the function's deriva-tive. derivative.
              tive.

              command f
                     Name of the command that evaluates the function for which the zero is to be returned

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

              float xe
                     End of the interval

              float eps
                     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).


EXAMPLES
       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]"

       For simple functions, the alternative could be:

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

       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 < 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]


BUGS, IDEAS, FEEDBACK
       This document, and the package it describes,  will  undoubtedly  contain  bugs  and  other  problems.
       Please  report  such  in  the  category  math  ::  calculus of the Tcllib SF Trackers [http://source -
       forge.net/tracker/? group_id=12883].  Please also report any ideas for enhancements you may  have  for
       either package and/or documentation.

SEE ALSO
       romberg

KEYWORDS
       calculus, differential equations, integration, math, roots

CATEGORY
       Mathematics

COPYRIGHT
       Copyright (c) 2002,2003,2004 Arjen Markus




math                                                0.7.1                                  math::calculus(n)