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Solution of Linear Algebraic Equations part 1

A set of linear algebraic equations looks like this: a11 x1 + a12 x2 + a13 x3 + · · · + a1N xN = b1 a21 x1 + a22 x2 + a23 x3 + · · · + a2N xN = b2 a31 x1 + a32 x2 + a33 x3 + · · · + a3N xN = b3 ··· ··· (2.0.1) | Chapter 2. Solution of Linear Algebraic Equations 2.0 Introduction A set of linear algebraic equations looks like this aiixi 012x2 013x3 --- O1N xN bi a2ixi 022X2 023X3 --- O2N XN b2 031X1 032x2 033x3 --- O3N xn b3 2.0.1 om 1x1 0m 2x2 0m3x3 0mn xn bM Here the N unknowns xj j 1 2 . N are related by M equations. The coefficients Oj with i 1 2 . M and j 1 2 . N are known numbers as are the right-hand side quantities bj i 1 2 . M. Nonsingular versus Singular Sets of Equations If N M then there are as many equations as unknowns and there is a good chance of solving for a unique solution set of xj s. Analytically there can fail to be a unique solution if one or more of the M equations is a linear combination of the others a condition called row degeneracy or if all equations contain certain variables only in exactly the same linear combination called column degeneracy. For square matrices a row degeneracy implies a column degeneracy and vice versa. A set of equations that is degenerate is called singular. We will consider singular matrices in some detail in 2.6. Numerically at least two additional things can go wrong While not exact linear combinations of each other some of the equations may be so close to linearly dependent that roundoff errors in the machine render them linearly dependent at some stage in the solution process. In this case your numerical procedure will fail and it can tell you that it has failed. Sample page from NUMERICAL RECIPES IN C THE ART OF SCIENTIFIC COMPUTING ISBN 0-521-43108-5 32 2.0 Introduction 33 Accumulated roundoff errors in the solution process can swamp the true solution. This problem particularly emerges if N is too large. The numerical procedure does not fail algorithmically. However it returns a set of x s that are wrong as can be discovered by direct substitution back into the original equations. The closer a set of equations is to being singular the more likely this is to happen since increasingly close cancellations will occur .