By Eduard L. Stiefel (Auth.)
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Extra resources for An Introduction to Numerical Mathematics
However, in more complex cases it is possible that the simplex algorithm leaves us in one corner. , which forms a cycle through which we keep running. Methods have been developed which avoid this case , . The upper edge of the polyhedron in Fig. 2 is horizontal. Each point of this edge therefore has the same value of the objective function as the end point E. Hence, each of these points is also a solution of the linear program. In other words, programs exist which have several solutions.
Hence, if in table (48) the variables xjc and yt are to be exchanged (pivot a w ) , the elements of the scheme will be transformed as follows: (49) From the first relation and the first requirement mentioned above, we imme diately conclude that a ^ > 0; hence, the pivot element has to be strictly positive. But the third relation and the second requirement then imply that ajc < 0. In other words, the pivot row must be chosen in such a way that its last element is negative. Finally, we obtain from the first requirement and the second relation (49) that for fixed i and all j Φ i.
LEAST-SQUARES APPROXIMATION therefore has to be made small. 7183 = η and the residual τχ assumes the meaning of the approximation error. 0527. (106) The residuals are (in units of the fourth decimal place) -54, 108, -6, -102, 49. The mean residual is 52. For the maximal approximation error r mx aafter a Chebyshev approximation, we obtain the bounds 52 < r mx a< 108. 8432*2. (108) It represents the exponential function in the interval (0; 1) with about the accuracy of a slide rule.
An Introduction to Numerical Mathematics by Eduard L. Stiefel (Auth.)