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math::calculus(3tcl)	       Tcl Math Library		  math::calculus(3tcl)

______________________________________________________________________________

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	 left‐
       bnd 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 integra‐
	      tion 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	 left‐
       bnd 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-coordi‐
		     nate.

	      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 bound‐
		     ary 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 approx‐
		     imated 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 fol‐
	      lowing 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	 func‐
	      tion  contained in the interval [xb,xe]. The error in this esti‐
	      mate 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	 func‐
	      tion  at	the  start  has a different sign than the value at the
	      end.

	      In contrast to Newton-Raphson there is no need for the  computa‐
	      tion 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 argu‐
       ments. To avoid problems with the visibility of these  procedures,  the
       fully-qualified	name of these procedures is determined inside the cal‐
       culus 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 dif‐
	      ferences 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(3tcl)
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