distanceplotoptions = [list]

The plot options for the  column graph of distances when output=plotdistance.

initialapprox = Vector

Initial approximation vector with which to begin the iteration; this vector must be numeric. This is a required keyword parameter.

maxiterations = posint

The maximum number of iterations to perform while approximating the solution to A.x=b.  If the maximum number of iterations is reached and the solution is not within the specified tolerance, a plot of distances can still be returned. This is a required keyword parameter.

method = gaussseidel, jacobi, SOR(numeric)

The method to be used when approximating the solution to A.x=b. See the Notes section below for some of the sufficient conditions for convergence.

gaussseidel = Gauss-Seidel method

jacobi = Jacobi method

SOR(numeric) = successive over-relaxation (SOR) method

Note that the SOR method is specified by the symbol SOR followed by the relaxation factor in parentheses.  The relaxation factor must be strictly between 0 and 2; otherwise, the generated sequence will diverge. For the purpose of demonstrating this divergence, however, values of the relaxation factor outside this range are still accepted by this procedure.

By default, method=gaussseidel.

output = solution, approximates, distances, plotdistance, plotsolution, or list

The return value of the function.  The default is solution. For more than one output, specify a list of the output in the order desired.

output=solution returns the final approximation of x.

output=approximates returns the approximation at each iteration in a list.

output=distances returns the error at each iteration in a list.

output=plotdistance returns a column graph of the errors at each iteration.

output=plotsolution returns a 3-D plot of the path of the approximations of x. This output is only available when A and b are 3-dimensional.

plotoptions=list

The plot options for the 3-D plot when output=plotsolution.

showsteps = true or false

Whether to print helpful messages in the interface as the IterativeApproximate command executes.

stoppingcriterion = function

The stopping criterion for the approximation of x in the form stoppingcriterion=distance(norm), where distance is either relative or absolute and norm is one of: posint, infinity ($\mathrm{\infty }$), or Euclidean. By default, stoppingcriterion=relative(infinity).

The stopping criterion for the approximation of x in the form stoppingcriterion=distance(norm), where distance is either relative or absolute and norm is one of: posint, infinity, or Euclidean. By default, stoppingcriterion=relative(infinity).

tolerance = positive

The tolerance of the approximation. The tolerance must be provided.

The IterativeApproximate command numerically approximates the solution to the linear system A.x=b, using one of these iterative methods: Gauss-Seidel, Jacobi, and successive over-relaxation.

It is possible to return both the approximation and the error at each iteration with this command; see the output and stoppingcriterion options under the Options section for more details.

It is also possible to view a column graph of the distances (errors) at each step, showing whether convergence is achieved.

When A and b are 3-dimensional, it is possible to obtain a plot tracing the path of the approximation sequence.

The entries of A and b must be expressions that can be evaluated to floating-point numbers.

The initialapprox, tolerance and maxiterations are all required keyword parameters; they must be given when the IterativeApproximate command is used.

If A is positive definite or strictly diagonally dominant, then A is invertible, and so the system A.x = b has a unique solution. Use IsMatrixShape to check if a matrix has one of these properties.

If the matrix A is strictly diagonally dominant, both the Jacobi and Gauss-Seidel methods produce a sequence of approximation vectors converging to the solution, for any initial approximation vector.

If A is positive definite, the Gauss-Seidel method produces a sequence converging to the solution, for any initial approximation vector; the same holds for the successive over-relaxation method, provided that the relaxation factor w is strictly between 0 and 2.

In general, if A gives rise to an iteration matrix T such that the spectral radius of T is strictly less than 1, the resulting sequence is guaranteed to converge to a solution, for any initial approximation vector.

This procedure operates numerically; that is, if the inputs are not already numeric, they are first evaluated to floating-point quantities before computations proceed. The outputs will be numeric as well. Note that exact rationals are considered numeric and are preserved whenever possible throughout the computation; therefore, one must specify floating-point inputs instead of exact rationals to obtain floating-point outputs.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[10\,-1\,2\,0\right]\,\left[-1\,11\,-1\,3\right]\,\left[2\,-1\,10\,-1\right]\,\left[0\,3\,-1\,8\right]\right]\right)\:$

$b≔\mathrm{Vector}\left(\left[6\,25\,-11\,15\right]\right)\:$

$\mathrm{IterativeApproximate}\left(A\,b\,\mathrm{initialapprox}=\mathrm{Vector}\left(\left[0.\,0.\,0.\,0.\right]\right)\,\mathrm{tolerance}={10}^{-3}\,\mathrm{maxiterations}=20\,\mathrm{stoppingcriterion}=\mathrm{relative}\left(\mathrm{\infty }\right)\,\mathrm{method}=\mathrm{jacobi}\right)$

$\left[\begin{array}{c}0.9996741452\\ 2.000447672\\ \mathrm{-1.000369158}\\ 1.000619190\end{array}\right]$

$\mathrm{IterativeApproximate}\left(A\,b\,\mathrm{initialapprox}=\mathrm{Vector}\left(\left[0.\,0.\,0.\,0.\right]\right)\,\mathrm{tolerance}={10}^{-3}\,\mathrm{maxiterations}=20\,\mathrm{stoppingcriterion}=\mathrm{relative}\left(\mathrm{\infty }\right)\,\mathrm{method}=\mathrm{jacobi}\,\mathrm{output}=\left[\mathrm{approximates}\,\mathrm{distances}\right]\right)$

$\left[\left[\begin{array}{c}0.\\ 0.\\ 0.\\ 0.\end{array}\right]\,\left[\begin{array}{c}0.6000000000\\ 2.272727273\\ \mathrm{-1.100000000}\\ 1.875000000\end{array}\right]\,\left[\begin{array}{c}1.047272727\\ 1.715909091\\ \mathrm{-0.8052272727}\\ 0.8852272726\end{array}\right]\,\left[\begin{array}{c}0.9326363636\\ 2.053305785\\ \mathrm{-1.049340909}\\ 1.130880682\end{array}\right]\,\left[\begin{array}{c}1.015198760\\ 1.953695765\\ \mathrm{-0.9681086260}\\ 0.9738427170\end{array}\right]\,\left[\begin{array}{c}0.9889913017\\ 2.011414725\\ \mathrm{-1.010285904}\\ 1.021350510\end{array}\right]\,\left[\begin{array}{c}1.003198653\\ 1.992241261\\ \mathrm{-0.9945217368}\\ 0.9944337401\end{array}\right]\,\left[\begin{array}{c}0.9981284735\\ 2.002306882\\ \mathrm{-1.001972230}\\ 1.003594310\end{array}\right]\,\left[\begin{array}{c}1.000625134\\ 1.998670301\\ \mathrm{-0.9990355755}\\ 0.9988883905\end{array}\right]\,\left[\begin{array}{c}0.9996741452\\ 2.000447672\\ \mathrm{-1.000369158}\\ 1.000619190\end{array}\right]\right],\left[1.000000000\,0.5768211921\,0.1643187763\,0.08037994851\,0.02869570322\,0.01351079833\,0.005027012138\,0.002354525155\,0.0008884866247\right]$

$\mathrm{IterativeApproximate}\left(A\,b\,\mathrm{initialapprox}=\mathrm{Vector}\left(\left[0.\,0.\,0.\,0.\right]\right)\,\mathrm{tolerance}={10}^{-3}\,\mathrm{maxiterations}=20\,\mathrm{stoppingcriterion}=\mathrm{relative}\left(\mathrm{\infty }\right)\,\mathrm{method}=\mathrm{jacobi}\,\mathrm{output}=\mathrm{plotdistance}\right)$

$A≔\mathrm{Matrix}\left(\left[\left[3.32\,1.43\,4.01\right]\,\left[2.03\,5.93\,2.03\right]\right]\right)$

$A≔\left[\begin{array}{ccc}3.32& 1.43& 4.01\\ 2.03& 5.93& 2.03\end{array}\right]$

$\mathrm{IterativeApproximate}\left(A\,\mathrm{initialapprox}=\mathrm{Vector}\left(\left[0.\,0.\right]\right)\,\mathrm{tolerance}={10}^{-5}\,\mathrm{maxiterations}=20\,\mathrm{method}=\mathrm{SOR}\left(1.25\right)\right)$

$\left[\begin{array}{c}1.243771952\\ \mathrm{-0.08345160693}\end{array}\right]$