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RedHat 9 (Linux i386) - man page for calculus (redhat section n)

math::calculus(n)			       Math				math::calculus(n)

NAME
       math::calculus - Integration and ordinary differential equations

SYNOPSIS
       package require Tcl 8

       package require math::calculus 0.5

       ::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 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

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 straight-
       forward 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 expres-
	      sion  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 func-
	      tion 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 equa-
	      tions. 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 equa-
	      tions. 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 tstep xvec func
	      Set  a  single  step in the numerical integration of a system of differential equa-
	      tions. 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 pro-
	      vide 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-coordi-
		     nate.

	      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)

       Notes:

       Several of the above procedures take the names of procedures as arguments. To avoid  prob-
       lems 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]

KEYWORDS
       math, calculus, integration, differential equations, roots

math					       1.0				math::calculus(n)


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