digits = posint

Set the environment variable Digits to this value. By default, digits is set to the current value of Digits.

initialapprox = Vector

An initial approximate vector $x_0$; this vector is used to generate a sequence of approximate solution vectors $x_k$ using the iterative formula $x_{k+1}=T·x_k+c$.  If the iterations option is specified, then the initialapprox option must be specified as well, or an exception will be raised. In order to obtain a sequence of floating-point vectors (instead of exact rationals), the initialapprox vector should contain entries of type float rather than exact integers or rationals.

iterations = posint

The maximum number of iterations to calculate. By default, this is set to $10$.

method = gaussseidel, jacobi, SOR(numeric)

One of these three iterative methods for which to compute the matrix $T$ and vector $c$ in the iterative formula $x_{k+1}=T·x_k+c$.  By default, the Gauss-Seidel method is used.

Gauss-Seidel method : $T={\left(\mathrm{D}-L\right)}^{\mathrm{-1}}·U$, $c={\left(\mathrm{D}-L\right)}^{\mathrm{-1}}·b$

Jacobi method : $T={\mathrm{D}}^{\mathrm{-1}}·\left(L+U\right)$, $c={\mathrm{D}}^{\mathrm{-1}}·b$

Successive Overrelaxation method : $T={\left(-wL+\mathrm{D}\right)}^{\mathrm{-1}}·\left(\left(1-w\right)\mathrm{D}+wU\right)$, $c=\left(w{\left(-wL+\mathrm{D}\right)}^{\mathrm{-1}}\right)·b$

In the above expressions, $U$ is the negated upper-triangular part of $A$, $L$ is the negated lower-triangular part of $A$, $\mathrm{D}$ is the diagonal part of $A$, and $w$ is extrapolation factor. We always have $A=\mathrm{D}-L-U$.

output = one of L, U, D, T, c, spectralradius, iterates, or a list containing one of more of those

The return value of the command. For more than one output, specify a list of outputs in the order desired. By default, output = [L, U,D, T, c].

The outputs L, U, D, T, and c are as described in the above sections.

If the iterates option is specified, the list of approximate solution vectors $x_k$ is output. The total number of vectors returned is determined by the iterations option.

If the spectralradius option is specified, the spectral radius of the iteration matrix $T$ is output.

showsteps = truefalse

Whether to print helpful messages in the interface as the IterativeFormula command executes. By default, this option is set to false.

Given a system $A·x=b$, the IterativeFormula command computes an equivalent fixed-point system of the form $x=T·x+c$.

The IterativeFormula command can compute the iteration matrix $T$ and vector $c$ for the following methods: the Gauss-Seidel iterative, Jacobi iterative, and successive over-relaxation methods.

An initial vector is specified using the initialapprox option and then a sequence of approximate solution vectors is generated using the iterative formula $x_{k+1}=T·x_k+c$.

Note that this iterative formula need not produce a converging sequence of vectors $x_k$. It can be shown that such an iterative scheme converges if and only if the spectral radius of the matrix $T$ is strictly less than 1. This spectral radius can be returned as an output via the output option. See below for more details.

This procedure operates symbolically; that is, the inputs are not automatically evaluated to floating-point quantities, and computations proceed symbolically and exactly whenever possible. To obtain floating-point results, it is necessary to supply floating-point inputs.

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

$A≔\mathrm{Matrix}\left(\left[\left[1.0\,-0.1\,2.\,0.\right]\,\left[-0.1\,1.1\,-0.1\,3.\right]\,\left[0.2\,-0.1\,1.0\,-0.1\right]\,\left[0.\,0.3\,-0.1\,0.8\right]\right]\right)$

$A≔\left[\begin{array}{cccc}1.0& \mathrm{-0.1}& 2.& 0.\\ \mathrm{-0.1}& 1.1& \mathrm{-0.1}& 3.\\ 0.2& \mathrm{-0.1}& 1.0& \mathrm{-0.1}\\ 0.& 0.3& \mathrm{-0.1}& 0.8\end{array}\right]$

$b≔\mathrm{Vector}\left(\left[0.6\,2.5\,-1.1\,1.5\right]\right)$

$b≔\left[\begin{array}{c}0.6\\ 2.5\\ \mathrm{-1.1}\\ 1.5\end{array}\right]$

$\mathrm{IterativeFormula}\left(A\,b\,\mathrm{method}=\mathrm{jacobi}\,\mathrm{digits}=3\,\mathrm{output}=\left['T'\,'c'\right]\right)$

$\left[\begin{array}{cccc}0.& 0.100& \mathrm{-2.00}& 0.\\ 0.0909& 0.& 0.0909& \mathrm{-2.73}\\ \mathrm{-0.200}& 0.100& 0.& 0.100\\ 0.& \mathrm{-0.375}& 0.125& 0.\end{array}\right],\left[\begin{array}{c}0.600\\ 2.27\\ \mathrm{-1.10}\\ 1.88\end{array}\right]$

$\mathrm{IterativeFormula}\left(A\,b\,\mathrm{method}=\mathrm{SOR}\left(1.25\right)\,\mathrm{iterations}=5\,\mathrm{initialapprox}=\mathrm{Vector}\left(\left[0.\,0.\,0.\,0.\right]\right)\,\mathrm{digits}=4\,\mathrm{output}=\left['\mathrm{iterates}'\right]\right)$

$\left[\left[\begin{array}{c}0.\\ 0.\\ 0.\\ 0.\end{array}\right]\,\left[\begin{array}{c}0.7500\\ 2.926\\ \mathrm{-1.197}\\ 0.7850\end{array}\right]\,\left[\begin{array}{c}3.920\\ \mathrm{-0.257}\\ \mathrm{-1.990}\\ 1.958\end{array}\right]\,\left[\begin{array}{c}4.713\\ \mathrm{-3.461}\\ \mathrm{-2.244}\\ 3.127\end{array}\right]\,\left[\begin{array}{c}4.749\\ \mathrm{-6.669}\\ \mathrm{-2.444}\\ 4.309\end{array}\right]\,\left[\begin{array}{c}4.839\\ \mathrm{-9.914}\\ \mathrm{-2.673}\\ 5.498\end{array}\right]\right]$