Systems of Simultaneous Equations
Sometimes a scientific or engineering problem can be represented by a set of n linear equations in n unknowns, for example x + 2y = 15 3jc + 8y= 57
or, in the general case ax + aX2x 2 + al3x3 + ■ ■ ■ + alnx„ = cx a2\x i + a22x2 + a23x3 + ■ ■ ■ + a2,jc„ = c2
anXx 1 + a„2x2 + a„3x3 + ■ ■ ■ + annxn = cn where x\, X2, x^,..., xn are the experimental unknowns, c is the experimentally measured quantity, and the a,y are known coefficients. The equations must be linearly independent; in other words, no equation is simply a multiple of another equation, or the sum of other equations.
A familiar example is the spectrophotometric determination of the concentrations of a mixture of n components by absorbance measurements at n different wavelengths. The coefficients ay are the e, the molar absorptivities of the components at different wavelengths (for simplicity, the cell path length, usually 1.00 cm, has been omitted from these equations). For example, for a mixture of three species P, Q and R, where absorbance measurements are made at A,j, A-2 and ^ the equations are
This chapter describes direct methods (involving the use of matrices) and indirect (iterative) methods for the solution of such systems. The chapter begins by describing methods for the solution of systems of linear equations, and concludes by describing a method for handling nonlinear systems of equations.
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