Given a system , the IterativeFormula command computes an equivalent fixed-point system of the form .

The IterativeFormula command can compute the iteration matrix  and vector  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 .

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

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 ; this vector is used to generate a sequence of approximate solution vectors  using the iterative formula .  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 .

method = gaussseidel, jacobi, SOR(numeric)

One of these three iterative methods for which to compute the matrix  and vector  in the iterative formula .  By default, the Gauss-Seidel method is used.

Gauss-Seidel method : ,

Jacobi method : ,

Successive Overrelaxation method : ,

In the above expressions,  is the negated upper-triangular part of ,  is the negated lower-triangular part of ,  is the diagonal part of , and  is extrapolation factor. We always have .

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  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  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.

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.