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Unformatted text preview: tional error oF M . In N operations, roundo errors may accumulate ran-domly (error N M ), systematically (error N M ), or catastrophically (1 / ( b-a ) may InF). (2) Truncation or discretization error: An infnite series has to be approxi-mated by a fnite number oF terms on the computer. or example, a contin-uous real interval [ a, b ] is approximated by a fnite set oF gridpoints x = a , x 1 = x + x , x 2 = x + 2 x , . . . , x n = x + n x = b ; or, as another example, e x N n =0 x n n ! , | x | < 1 (3) Numerical instability: roundo error introduced early in the computation can be exponentially amplifed and swamp the true solution. or example, the Forward Euler method For the heat equation u t = u xx is stable only iF t x 2 / 2. orward Euler computed solutions to the heat equation with 20 x . 2...
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- Fall '11