By Manish Goyal
Advances in fields comparable to bioengineering, commercial engineering, and robot layout now require engineers to have a valid heritage in statistical the right way to optimize functionality and reduce blunders in problem-solving purposes. through becoming a member of statistical research with computer-based numerical equipment, this publication bridges the space among thought and perform with software-based examples, movement charts, and purposes. Designed for engineering scholars in addition to practising engineers and scientists, the publication has a variety of examples with in-text ideas. by way of content material, it covers the series of mathematical themes wanted by way of the vast majority of collage classes, together with calculus, error-handling, and ODEs; additionally, the ebook covers statistical computation and trying out of speculation frequently passed over from numerical equipment texts. utilizing circulate charts and desktop courses, the authors exhibit how the mathematical techniques can be applied in functional purposes equivalent to circuits, sign processing, and extra. A CD-ROM with the resource code for the in-text computing device courses comprises calculation workouts and simulations.
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3) Truncation errors Truncation errors are caused by using approximate results or by replacing an infinite process with a finite one. 65. , If S = ∑ ax i i ∑ a x , then the is replaced by or truncated to S = i i i=1 1 error developed is a truncation error. A truncation error is a type of algorithm error. Also, if ex = 1 + x + 1+x+ x2 x3 x 4 + + + ...... ∞ = X (say) is truncated to 2! 3! 4! x2 x3 + = X′ (say), then truncation error = X – X′ 2! 3! 1 if 5 (i) The first three terms are retained in expansion.
The maximum relative error is Δx1 Δx2 Δxn ΔX . ≤ + + ...... + X X X X 44 COMPUTER-BASED NUMERICAL AND STATISTICAL TECHNIQUES It is clear that if two numbers are added then the magnitude of absolute error in the result is the sum of the magnitudes of the absolute errors in the two numbers. NOTE While adding up several numbers of different absolute accuracies, the following procedure is adopted: (i) Isolate the number with the greatest absolute error. (ii) Round-off all other numbers, retaining in them one digit more than in the isolated number.
009. 4 Example 2. Find the relative error in the function y = ax1 m1 x 2 m2 ...... xn mn . Sol. We have ∴ ∴ log y = log a + m1 log x1 + m2 log x2 + ...... + mn log xn FG IJ H K m m 1 ∂y 1 ∂y = 1 = = 2 , ...... etc. y ∂x1 x1 y ∂ x2 x2 er = ∂ y δ xn ∂y δx2 ∂y δx1 + ...... + + . . ∂xn ∂x2 ∂ x1 y y y = m1 δx1 m2 δx + δx2 + ...... + mn . S. This gives, (er)max. ≤ m1 δx1 δx2 δ xn + m2 + ...... + mn . x2 xn x1 48 COMPUTER-BASED NUMERICAL Corollary. , xn er ≈ δx1 δx2 δx + + ...... + n x1 x2 xn ∴ The relative error of a product of n numbers is approximately equal to the algebraic sum of their relative errors.