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Solution of Linear Algebraic Equations part 3

Notice the essential difference between equation (2.1.8) and equation (2.1.6). In the latter case, the C’s must be applied to b in the reverse order from that in which they become known. That is, they must all be stored along the way. | 2.2 Gaussian Elimination with Backsubstitution 41 which peeling of the C 1 s one at a time implies a solution x Ci C C3 b 2.1.8 Notice the essential difference between equation 2.1.8 and equation 2.1.6 . In the latter case the C s must be applied to b in the reverse order from that in which they become known. That is they must all be stored along the way. This requirement greatly reduces the usefulness of column operations generally restricting them to simple permutations for example in support of full pivoting. CITED REFERENCES AND FURTHER READING Wilkinson J.H. 1965 TheAlgebraic Eigenvalue Problem New York Oxford University Press . 1 Carnahan B. Luther H.A. and Wilkes J.O. 1969 Applied Numerical Methods New York Wiley Example 5.2 p. 282. Bevington P.R. 1969 Data Reduction and ErrorAnalysis for the Physical Sciences New York McGraw-Hill Program B-2 p. 298. Westlake J.R. 1968 A Handbook ofNumerical Matrix Inversion and Solution ofLinear Equations New York Wiley . Ralston A. and Rabinowitz P. 1978 A First Course in NumericalAnalysis 2nd ed. New York McGraw-Hill 9.3-1. 2.2 Gaussian Elimination with Backsubstitution The usefulness of Gaussian elimination with backsubstitution is primarily pedagogical. It stands between full elimination schemes such as Gauss-Jordan and triangular decomposition schemes such as will be discussed in the next section. Gaussian elimination reduces a matrix not all the way to the identity matrix but only halfway to a matrix whose components on the diagonal and above say remain nontrivial. Let us now see what advantages accrue. Suppose that in doing Gauss-Jordan elimination as described in 2.1 we at each stage subtract away rows only below the then-current pivot element. When a22 is the pivot element for example we divide the second row by its value as before but now use the pivot row to zero only a32 and a42 not a12 see equation 2.1.1 . Suppose also that we do only partial pivoting never interchanging columns so that the order of the .