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  remain-
	      ing 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  "con-
	      servative" 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 val-
		     ues 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)