By A.K. Jaiswal, Anju Khandelwal
Salient positive aspects of this paintings contain: mathematical derivation of every technique is given to construct the scholars figuring out of numerical research; numerous solved examples are given; laptop courses for the majority numerical equipment mentioned were offered in 'C' language; errors research for the majority tools are provided; each one bankruptcy starts off with an advent of involved subject; and, workout questions provide a chance to the scholars to check their knowing of the thoughts.
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Extra info for A Textbook of Computer Based Numerical and Statistical Techniques
6875 approximately. 1 1. Find the smallest root lying in the interval (1, 2) up to four decimal places for the equation x6 – x4 – x3 – 1 = 0 by Bisection Method [Ans. 4036] 3 2. Find the smallest root of x – 9x + 1 = 0, using Bisection Method correct to three decimal places. [Ans. 111] [Ans. 5121375] 3. Find the real root of ex = 3x by Bisection Method. 4. Find the positive real root of x – cos x = 0 by Bisection Method, correct to four decimal places between 0 and 1. [Ans. 7393] 5. Find a root of x3 – x – 11 = 0 using Bisection Method correct to three decimal places which lies between 2 and 3.
Ex = 1 + x + 21 ERRORS AND FLOATING POINT x Sol. Given that e = 1 + x + x2 x3 x n −1 xn ξ + + ....... + + e ,0 < ξ < x 2! 3! (n − 1)! n ! Then the remainder term is, xn ξ e n! So that ξ = x gives maximum absolute error Rn(x) = and Ea(max) = xn x e n! Er(max) = xn n! 1 1 −8 < 10 ⇒ n = 12 n! 2 Hence we have 12 terms of the expansion in order that its sum is correct to 8 decimal places. For an 8 decimal accuracy at x = 1, Example 38. Find the number of terms of the exponential series such that their sum gives the value of ex correct to six decimal places at x = 1.
Find the smaller root of the equation x2 – 400x + 1 = 0 using four-digit arithmetic. Sol. 0000. 0025. 8000 × 10 3 This is the exact root of the given equation. Remark: When two nearly equal numbers are subtracted then there is a loss of significant figures. 211 is correct to three figures only. Similarly numbers 12450 and 12360 are correct to four figures and their difference 90 is correct to one figure only. The error due to loss of significant figures sometimes renders the result of computation worthless.