Systems of Linear Algebraic Equations - Numerical Methods in Engineering with MATLAB - page 31

Numerical solutionsofill-conditioned equations are not to betrusted. The reason
isthat the inevitable roundoff errors during the solutionprocess areequivalenttoin-
troducing small changes into the coefficientmatrix. This inturnintroduces largeerrors
into the solution, the magnitudeofwhich dependson the severity of ill-conditioning.
In suspect cases the determinant of the coefficient matrix shouldbecomputed so that
the degree of ill-conditioning can be estimated. Thiscan be done during orafter the
solutionwith onlyasmall computational effort.

Linear Systems

Linear, algebraicequations occur in almost all branches of numerical analysis.But
their most visible applicationinengineering is in the analysisoflinear systems(any
systemwhose response is proportionaltotheinput is deemed to belinear). Linear
systems includestructures, elastic solids, heat flow, seepageoffluids, electromagnetic
fields and electriccircuits;i.e., most topics taught in an engineering curriculum.
If the systemis discrete, such as a truss oranelectriccircuit, thenits analysis
leads directly to linear algebraicequations. In the case of a staticallydeterminate
truss, for example, the equations arise when the equilibrium conditions of the joints
are writtendown. The unknowns x 1 ,

x n represent the forces in the members
and the support reactions, and the constants b 1 ,

x 2 ,...,

b 2 ,..., b n are the prescribed external

loads.
The behavior of continuoussystems is describedbydifferentialequations, rather
than algebraicequations. However, because numerical analysiscan dealonlywith
discrete variables, it is first necessary to approximate adifferentialequationwith a
system of algebraicequations. The well-known finite difference, finite element and
boundary element methodsofanalysis work in this manner. Theyuse different ap-
proximationstoachieve the “di scretization,”but in each case the finaltask is the same:
solve a system (oftenavery largesystem)oflinear, algebraicequations.
In summary, the modeling of linear systems invariably gives rise to equationsof
the form Ax b , where b is the input and x represents the response of the system.

The coefficient matrix A , which reflects the characteristics of the system, is inde-
pendent of the input. In otherwords, if the input ischanged, the equationshaveto
be solved again with adifferent b , but the same A . Therefore, it is desirable to have
an equation-solving algorithm thatcan handle any number of constant vectors with
minimalcomputational effort.

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Methods of Solution

There aretwoclasses of methodsfor solving systemsoflinear, algebraicequations:
direct and iterativemethods. The common characteristicof directmethods isthat they