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Notice that the subtraction in the calculation of r2 was carried out exactly. The cancellation in the subtraction only gives an indication of the unhappy consequence of a loss of information in previous steps, due to the rounding of one of the operands, and is not the cause of the inaccuracy. In numerical calculations, if possible one should try to avoid formulas that give rise to cancellation, as in the above example. For the quadratic equation this can be done by rewriting of the formulas. Comparing coefficients on both sides of x 2 + (b/a)x + c/a = (x − r1 )(x − r2 ) = x 2 − (r1 + r2 )x + r1 r2 , we get the relation between coefficients and roots r1 + r2 = −b/a, r1 r2 = c/a.

In y we got only two correct decimal digits. This is due to cancellation, which is an important cause of loss of accuracy; see Sec. 4. 5 at the end of this section. A very important equation of mathematical physics is Poisson’s equation:6 ∂ 2u ∂ 2u + 2 = f (x, y), ∂x 2 ∂y (x, y) ∈ . 21) Here the function f (x, y) is given together with some boundary condition on u(x, y). 21) with f (x, y) = 0. Finite difference approximations are useful for partial derivatives. Suppose that is a rectangular region and introduce a rectangular grid that covers the rectangle.

Many computer languages (for example, MATLAB) allow the definition of such recursive algorithms. The divide and conquer is a top down description of the algorithm in contrast to the bottom up description we gave first. 3. Sorting the items of a one-dimensional array in ascending or descending order is one of the most important problems in computer science. In numerical work, sorting is frequently needed when data need to be rearranged. One of the best known and most efficient sorting algorithms, quicksort by Hoare [202], is based on the divide and conquer paradigm.

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