By Eduard L. Stiefel (Auth.)
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Additional info for An Introduction to Numerical Mathematics
It must be observed that in the corresponding matrix (ai, ai) (ai, a 2) (a 2, ai) («2, a2) both elements (ai, a 2) and (a2, a\)—located symmetrically with respect to the diagonal—are equal. We therefore speak of a symmetric matrix. As first rule for the formation of the normal equations, one should remember 11 equations are scalar products of the coefficient that the coefficients in the normal columns of the error equations. , if m = 2, then the solutions of the normal equations are identical with the solutions of the equations «21#1 + ci = 0 + «12^2 + «22#2 + C2 = 0.
Example, m error equations with three unknowns x\, x2i # 3 : Consider, for aaxi + ai2x2 + disxs + a = rt) i = 1, 2, ... , m. § + 2(αι, c)x± + 2(a2, c)x2 + 2(a3, c)x3+ (c, c). Here, for instance, (a2, a3) is the scalar product of the second and third column of coefficients in the error equations and (αχ, c) is the scalar product of the first column of coefficients and the column of constants. Again, the function (r, r) is composed of a quadratic form in the first three rows of (102) and a linear function in the fourth row.
Hence, in the transition from the approximation point to the new point (xi, x2), the maximum absolute value of the residuals has certainly decreased, and this is the case in all the following EX-steps. The procedure therefore consists in decreasing methodically and successively the largest residual. 7. CHEBYSHEV APPROXIMATION 49 and (89) 6 = y o - l. Now the simplex algorithm should actually begin. B y coincidence, how ever, the last column is already positive, and we are therefore finished. Setting the variables on the left border equal to zero, we find according to (84) that ri = —r, 7*2 = —r, 7*3 = + r .