compute an iterative formula to approximate the solution to a linear system numerically - Maple Help

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Student[NumericalAnalysis][IterativeFormula] - compute an iterative formula to approximate the solution to a linear system numerically

Calling Sequence

IterativeFormula(A, b, opts)

IterativeFormula(A, opts)

Parameters

A

-

Matrix; a square (n-by-n) matrix or an augmented n-by-n+1 matrix of the form (A|b)

b

-

(optional) Vector; a vector of length n

opts

-

(optional) equation(s) of the form keyword=value where keyword is one of digits, initialapprox, iterations, method, output, showsteps; the options for computing the iterative formula

Description

• 

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 xk+1=T.xk+c.

• 

Note that this iterative formula need not produce a converging sequence of vectors xk. 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.

Notes

• 

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.

Examples

withStudent[NumericalAnalysis]:

A:=Matrix1.0,0.1,2.,0.,0.1,1.1,0.1,3.,0.2,0.1,1.0,0.1,0.,0.3,0.1,0.8

A:=1.00.12.0.0.11.10.13.0.20.11.00.10.0.30.10.8

(1)

b:=Vector0.6,2.5,1.1,1.5

b:=0.62.51.11.5

(2)

IterativeFormulaA,b,method=jacobi,digits=3,output='T','c'

0.0.1002.000.0.09090.0.09092.730.2000.1000.0.1000.0.3750.1250.,0.6002.271.101.88

(3)

IterativeFormulaA,b,method=SOR1.25,iterations=5,initialapprox=Vector0.,0.,0.,0.,digits=4,output='iterates'

0.0.0.0.,0.75002.9261.1970.7850,3.9200.2571.9901.958,4.7133.4612.2443.127,4.7496.6692.4444.309,4.8399.9142.6735.498

(4)

See Also

Student[LinearAlgebra], Student[NumericalAnalysis], Student[NumericalAnalysis][ComputationOverview], Student[NumericalAnalysis][IsConvergent], Student[NumericalAnalysis][IterativeApproximate], Student[NumericalAnalysis][LinearSolve]


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