Unformatted text preview: rithmetic was done on a simulated computer where oating
point numbers have 21bit 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
21bit 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 =
21bit 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 niteprecision 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 (dashdot) 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. 2123). 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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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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