Let f(x) = 1 + x − (sin(x)/(x · (e^x) )). Propose a numerically sound algorithm...
Let f(x) = 1 + x − (sin(x)/(x · (e^x) )). Propose a numerically sound algorithm for evaluating f(x) for x ~ 0. Please provide all steps/details, including your thought process so I can follow along and learn, thanks!
Homework Answers
You can use what you already did and perform the long division. This would lead to
f(x)=1+x−sin(x)xex=2x−x23+O(x3)f(x)=1+x−sin(x)xex=2x−x23+O(x3)
Using more terms for numerator would lead to
f(x)=2x−x23+x430−x590+O(x6)f(x)=2x−x23+x430−x590+O(x6)
To avoid the long division, as suggested by Antonio Vargas, the simplest is effectively to write
f(x)=1+x−sin(x)xe−x=1+x−(1−x26+x4120−x65040+O(x8))×f(x)=1+x−sin(x)xe−x=1+x−(1−x26+x4120−x65040+O(x8))×
(1−x+x22−x36+x424−x5120+x6720+O(x7))(1−x+x22−x36+x424−x5120+x6720+O(x7))
and multiply.
For illustration purposes, I generated a table (values of xx and f(x)f(x) computed and approximated) (Fortran code using double precision). You could notice that some chaos appears with f(x)f(x) for x<10−10x<10−10 and that the results are definitely wrong when xx approaches 10−2010−20 which is the order of magnitude of machine accuracy.
1.00D-01 1.9666989048D-01 1.9666988889D-01
1.00D-02 1.9966666999D-02 1.9966666999D-02
1.00D-03 1.9996666667D-03 1.9996666667D-03
1.00D-04 1.9999666667D-04 1.9999666667D-04
1.00D-05 1.9999966667D-05 1.9999966667D-05
1.00D-06 1.9999996667D-06 1.9999996667D-06
1.00D-07 1.9999999667D-07 1.9999999667D-07
1.00D-08 1.9999999967D-08 1.9999999967D-08
1.00D-09 1.9999999997D-09 1.9999999997D-09
1.00D-10 2.0000000001D-10 2.0000000000D-10
1.00D-11 2.0000000029D-11 2.0000000000D-11
1.00D-12 1.9999999920D-12 2.0000000000D-12
1.00D-13 1.9999995583D-13 2.0000000000D-13
1.00D-14 2.0000006425D-14 2.0000000000D-14
1.00D-15 1.9999735375D-15 2.0000000000D-15
1.00D-16 1.9998109071D-16 2.0000000000D-16
1.00D-17 2.0003530082D-17 2.0000000000D-17
1.00D-18 1.8973538018D-18 2.0000000000D-18
1.00D-19 2.1684043450D-19 2.0000000000D-19
1.00D-20 0.0000000000D+00 2.0000000000D-20
1.00D-21 0.0000000000D+00 2.0000000000D-21
1.00D-22 0.0000000000D+00 2.0000000000D-22
1.00D-23 0.0000000000D+00 2.0000000000D-23
1.00D-24 0.0000000000D+00 2.0000000000D-24
1.00D-25 0.0000000000D+00 2.0000000000D-25
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