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



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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

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DESCRIPTION
       This package implements several simple mathematical algorithms:

       o      The integration of a function over an interval

       o      The numerical integration of a system of ordinary differential
              equations.

       o      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 integral2D 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 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.

              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:

       o      Other types of boundary conditions (zero gradient, zero flux)

       o      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://sourceforge.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)
