numerically approximate the solution to a linear system

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Student[NumericalAnalysis][LinearSolve] - numerically approximate the solution to a linear system

	Calling Sequence
	LinearSolve(A, b, opts) LinearSolve(A, opts)

Parameters

A	-	Matrix; a square matrix or an augmented (A\|b) matrix where
b	-	(optional) Vector or Matrix; a vector of length or a matrix of column length
opts	-	(optional) equation(s) of the form keyword = value, where keyword is one of initialapprox, maxiterations, method, stoppingcriterion, tolerance; options for numerically approximating the solution to a linear system

Description

•	The LinearSolve command numerically approximates the solution to the linear system , using the specified method.

•	The IterativeApproximate command and the MatrixDecomposition command are both used by the LinearSolve command.

•	If b is a matrix, then the systems will be solved for each column of , and hence there will be multiple solutions returned.

•	Different options are required to be specified in opts, depending on the method. These dependencies are outlined below.

•	The Notes section in the Student[NumericalAnalysis][IterativeApproximate] help page lists conditions under which the Jacobi, Gauss-Seidel, and successive over-relaxation iterative methods produce a solution.

Options

•	initialapprox = Vector or Matrix

The initial approximation. To obtain a float solution instead of an exact solution, the initialapprox should contain floats instead of integers. By default, a zero vector is used.

•	maxiterations = posint

The maximum number of iterations to perform while approximating the solution to . 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. By default, maxiterations = 20.

•	method = jacobi, gaussseidel, SOR(numeric), LU, LU[tridiagonal], PLU, or PLU[scaled]

The method to use when solving the linear system . Please see the Notes section for sufficient conditions for convergence. This option is required. Each method is described below:

–	jacobi : The Jacobi method. Optionally, the stoppingcriterion, maxiterations, initialapprox and tolerance options may be specified as well.

–	gaussseidel : The Gauss-Seidel method. Optionally, the stoppingcriterion, maxiterations, initialapprox and tolerance options may be specified as well.

–	SOR(w) : The Successive Overrelaxation method with w as its extrapolation factor. Optionally, the stoppingcriterion, maxiterations, initialapprox and tolerance options may be specified as well.

–	LU and LU[tridiagonal] : LU Decomposition. This method performs LU factorization on A and then solves the subsequent systems. None of the remaining options are used with this method. An error will be raised if the LU[tridiagonal] method is specified and A is not tridiagonal.

–	PLU and PLU[scaled] : PLU Decomposition. This method performs PLU factorization on A and then solves the subsequent systems. None of the remaining options are used with this method.

•	stoppingcriterion = distance(norm)

The stopping criterion for an iterative technique; it is of the form stoppingcriterion=distance(norm), where distance is either relative or absolute and norm is a nonnegative integer, infinity, Euclidean, or Frobenius. By default, stoppingcriterion=relative(infinity).

•	tolerance = positive

The tolerance of the approximation. By default, a tolerance of is used.

Examples

A := Matrix([[10., -1., 2., 0.], [-1., 11., -1., 3.], [2., -1., 10., -1.], [0., 3., -1., 8.]])

(1)

b := Vector[column]([[6.], [25.], [-11.], [15.]])

(2)

Vector[column]([[.9999776440], [2.000001578], [-.9999942334], [.9999867498]])

(3)

Vector[column]([[1.000000000], [2.000000000], [-1.000000000], [.9999999999]])

(4)

Try solving multiple systems (but with the same coefficient Matrix)

B := Matrix([[6., 25., -11., 15.], [7., 8., 16., 4.], [4., 2., 9., 5.], [17., 6., 3., 22.]])

(5)

[Vector[column]([[.5095334686], [.1482082486], [.5264367816], [2.135226505]]), Vector[column]([[2.623529412], [.8352941177], [-.2000000000], [.4117647058]]), Vector[column]([[-1.210953347], [1.465179175], [1.287356321], [-0.1352265042e-1]]), Vector[column]([[1.376064909], [-.2600405681], [.4896551724], [2.908722110]])]

(6)

	See Also
	Student[LinearAlgebra], Student[NumericalAnalysis], Student[NumericalAnalysis][BackSubstitution], Student[NumericalAnalysis][ComputationOverview], Student[NumericalAnalysis][ForwardSubstitution], Student[NumericalAnalysis][IterativeApproximate], Student[NumericalAnalysis][MatrixDecomposition]