numerically approximate the solution to a linear system
IterativeApproximate(A, b, opts)
Matrix; a square nxn matrix or an augmented (A|b) nxm matrix, where m=n+1
(optional) Vector; a vector of length n
equations of the form keyword=value, where keyword is one of distanceplotoptions, initialapprox, maxiterations, method, output, plotoptions, showsteps, stoppingcriterion, tolerance; the options for numerically approximating the solution to Ax=b
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.
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 (∞), 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.
A ≔ Matrix⁡10,−1,2,0,−1,11,−1,3,2,−1,10,−1,0,3,−1,8:
b ≔ Vector⁡6,25,−11,15:
View the approximate solution using the Jacobi method.
View the approximate solution with the error at each iteration.
View the approximate solution with the error at each iteration as a column graph.
The linear system may be input as an augmented matrix
A ≔ Matrix⁡3.32,1.43,4.01,2.03,5.93,2.03
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