By Eduard L. Stiefel (Auth.)

ISBN-10: 1483200388

ISBN-13: 9781483200385

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Extra resources for An Introduction to Numerical Mathematics

Example text

Our chain of edges can therefore not continue along Β Β', and we avoid the danger of running through the cycle Β Β' Β" Β'" of the four horizontal edges which leads back to corner B. Otherwise it would simply mean wasted effort. Since the edge Β A leads downward, neither y\ nor y% will be exchanged, and, for this reason, y2 will have to be exchanged. In view of the fact that the remaining planes y\ = 0, ys = 0 do not form an edge of the polyhedron, this will be possible only if y2 is exchanged with 1/4.

P m is also called a (92) vector. 1. THE METHOD OF L E A S T - S Q U A R E S 53 From (90) it now follows that (r,r) = (αϊ, a{)x\ + 2(αι, a2)xix2 + («2, a2)x\ (93) + 2(ai, φ ι + 2(a 2, Φ 2 + (c, c). Here, for example, (αϊ, a 2) is the scalar product of the first two columns of (90); hence, (αϊ, a2) = a n a i 2 + «21^22 + ... + am\am2 and (αϊ, ai) is the scalar product of the first column with itself: + ... + a ^ . (αϊ, αϊ) = αξχ + According to (93), the quantity (r, r) to be minimized is a quadratic function of the independent variables #i, # 2.

I. Example. 1 = η are to be smoothed by the method of least squares. 3278. 3004. 3444. 1656. 56 3. 3333. When formulating applied approximation problems, one must pay careful attention to a suitable choice of unknowns. First and foremost, one must introduce only as many unknowns as needed and no more. , no relation independent of the measurements, shall exist between the unknowns. In the more advanced development of approximation theory, such conditions are permitted (approximation with "side conditions"), but we will have to forbid them in our own elementary considerations.