ll Turkcell 02:14 ? %30 learn.bilg..edu.tr 14. (30 points) For the following dat
ID: 3899412 • Letter: L
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ll Turkcell 02:14 ? %30 learn.bilg..edu.tr 14. (30 points) For the following data set generated by the function f(y)y y ou the domain 0,3 x (0,4 with interval length in r-axis 1 and interval length in y-axis k- 2, estimate by hand f, and fy with forward, backward and central derivatives on every point of x possible. Also caleulate the exact derivatives of the function. Present your findings on a single table and compare results, compared to exact results which one was better? 0 2 3 r-10 1 2 3| y-2 2 2 2| ??0 4 12 24 0 8 24 48 0 23 15. (20 points) Explain and compare truncation error and round-off error with examples. 16. (20 points) Write a Matlab script program that calculates sum stops changing. How do you know that it would stop? sini by adding terms until the Page 3Explanation / Answer
15)Answer:
Truncation error:
Truncation error is the difference between a truncated value and the actual value. A truncated quantity is represented by a numeral with a fixed number of allowed digits, with any excess digits "chopped off" (hence the expression "truncated").
As an example of truncation error, consider the speed of light in a vacuum. The official value is 299,792,458 meters per second. In scientific (power-of-10) notation, that quantity is expressed as 2.99792458 x 108. Truncating it to two decimal places yields 2.99 x 108. The truncation error is the difference between the actual value and the truncated value, or 0.00792458 x 108. Expressed properly in scientific notation, it is 7.92458 x 105
->In computing applications, truncation error is the discrepancy that arises from executing a finite number of steps to approximate an infinite process. For example, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ... adds up to exactly 1. However, if we truncate the series to only the first four terms, we get 1/2 + 1/4 + 1/8 + 1/16 = 15/16, producing a truncation error of 1 - 15/16, or 1/16
Round off error:
Roundoff errors arise because digital computers cannot represent some quantities exactly. There are two major facets of roundoff errors involved in numerical
calculations: –
Digital computers have size and precision limits on their ability to represent numbers
– Certain numerical manipulations are highly sensitive to roundoff errors
Roundoff error is the difference between an approximation of a number used in computation and its exact (correct) value. In certain types of computation, roundoff error can be magnified as any initial errors are carried through one or more intermediate steps
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