Linux & Unix Commands - Search Man Pages

math::calculus(n) Math math::calculus(n)NAMEmath::calculus - Integration and ordinary differential equationsSYNOPSISpackage 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 toleranceDESCRIPTIONThis 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.PROCEDURESThis 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 dA(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).--EXAMPLESLet 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' =- 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]-ryKEYWORDSmath, calculus, integration, differential equations, rootsmath1.0 math::calculus(n)

All times are GMT -4. The time now is 05:06 AM.