Iterating we nd the solution of the perturbed problem

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Unformatted text preview: rithmetic was done on a simulated computer where oating point numbers have 21-bit fractions. The computed solutions and errors at = 1 are presented in Table 2.1.4 for ranging from 0 to 16. A ~ signi es results computed with 21-bit rounded arithmetic. Roundo errors are introduced at each step of the computation because of the imprecise addition, multiplication, and evaluation of ( ). They can, and typically do, accumulate to limit accuracy. In the present case, approximately four digits of accuracy can be achieved with a step size of approximately 2 14 . Decreasing the step size further will not increase accuracy. In order to study this, let yt N k ::: e t h =N T k fty ; ( ) be the solution of the IVP at = n , y tn t t 14 = 2k 0 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1024 11 2048 12 4096 13 8192 14 16384 15 32768 16 65536 Table 2.1.4: Solutions of = 21-bit rounded arithmetic. k N y yn 0 =1 1.00000000 0.50000000 0.25000000 0.12500000 0.06250000 0.03125000 0.01562500 0.00781250 0.00390625 0.00195312 0.00097656 0.00048828 0.00024414 0.00012207 0.00006104 0.00003052 0.00001526 , (0) = 1, h y =N y ~ ~N 2.00000000 -0.71828181 2.25000000 -0.46828184 2.44140625 -0.27687559 2.56578445 -0.15249738 2.63792896 -0.08035287 2.67698956 -0.04129227 2.69734669 -0.02093514 2.70773602 -0.01054581 2.71297836 -0.00530347 2.71561337 -0.00266846 2.71694279 -0.00133904 2.71764278 -0.00063904 2.71795559 -0.00032624 2.71811104 -0.00017079 2.71814919 -0.00013264 2.71804428 -0.00023755 2.71732903 -0.00095280 at = 1 obtained by Euler's method with yN e t be the in nite-precision solution of the di erence equation at = n, and t t ~ be the computed solution of the di erence equation at = n. yn t t The total error ~n may be written as e j~nj = j ( n) ; ~nj j ( n) ; nj + j n ; ~nj e yt y yt y y y : (2.1.17a) As usual, let en = ( n) ; yt (2.1.17b) yn and also let rn = yn ; ~n (2.1.17c) y: Thus, j~nj j nj + j nj e e 15 r : (2.1.17d) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 −3 x 10 Figure 2.1.3: Total error ~n (solid), discretization error n (dash-dot) as functions of step size . e r en (dashed), and roundo error h As previously noted, n is the global discretization error. We'll call n the round o error. According to Theorem 2.1.1, j nj . An analysis similar to the one used in Theorem 2.1.1 reveals that j nj ( 1], pp. 21-23). Thus, while decreasing decreases the discretization error it increases the bound on the roundo error. As shown in Figure 2.1.3, there is an optimal value of , OP T , that produces the minimum bound on the total error. Fortunately, roundo error accumulation is not typically a problem when solving ODEs on realistic computers with the practical numerical methods that we shall study in subsequent sections. e r e r kh K =h h h h Problems 1. Error bounds of the form (2.1.13) are called a priori estimates because they are 16 obtained without using the computed solution. Such a priori estimates are often very conservative. Indeed, we u...
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