# Results for: “Manish Goyal”

1 eBook

## ALLC3-3 |
Manish Goyal | Laxmi Publications | |||||

103 ALGEBRAIC AND TRANSCENDENTAL EQUATIONS = .56 − FG .56716 − .56 IJ (− .01121) H .00002619 + .01121K = .567143 Since, x2 and x3 agree upto four decimal places, the required root correct to three decimal places is 0.567. (ii) Let f(x) = xex – 2 Since, f(.852) = – .00263 and f(.853) = .001715 Hence, root lies between .852 and .853. Let x0 = .852 and x1 = .853 Using method of false position, RS x − x UV f (x ) T f (x ) − f (x ) W R .853 − .852 UV (− .00263) = .852 − S T.001715 − (− .00263) W 1 x2 = x 0 − 0 1 0 0 = .852605293 Now f(x2) = – .00000090833 Hence, root lies between .852605293 and .853 Using method of false position, RS x − x UV f (x ) | Replacing x by x T f (x ) − f (x ) W R . 853 − . 852605293 UV (− .00000090833) = (.852605293) – S T.001715 − (−.00000090833) W x3 = x 2 − 1 2 1 2 2 0 2 = 0.852605501 Since, x2 and x3 agree upto 6 decimal places, hence the required root correct to 4 decimal places is 0.8526. Example 8. (i) Solve x3 – 5x + 3 = 0 by using Regula-Falsi method. (ii) Use the method of false position to solve x3 – x – 4 = 0. See All Chapters |
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## ALLC10-3 |
Manish Goyal | Laxmi Publications | |||||

593 CURVE-FITTING, CUBIC SPLINE AND APPROXIMATION Sol. The normal equations to the curve are Σy = aΣx 2 + bΣx + 5c Σxy = aΣx 3 + bΣx 2 + cΣx Σx 2 y = aΣx 4 + bΣx 3 + cΣx 2 and U| V| W …(1) The values of Σx, Σx2,...... etc., are calculated by means of the following table : x y x2 x3 x4 xy x2 y 10 12 15 23 20 14 17 23 25 21 100 144 225 529 400 1000 1728 3375 12167 8000 10000 20736 50625 279841 160000 140 204 345 575 420 1400 2448 5175 13225 8400 Σx = 80 Σy = 100 Σx2 = 1398 Σx3 = 26270 Σx4 = 521202 Σxy = 1684 Σx2y = 30648 Substituting the obtained values from the table in normal equation (1), we have 100 = 1398a + 80b + 5c 1684 = 26270a + 1398b + 80c 30648 = 521202a + 26270b + 1398c On solving, a = – 0.07, b = 3.03, c = – 8.89 ∴ The required equation is y = – 0.07x2 + 3.03x – 8.89. Example 3. Fit a parabolic curve of regression of y on x to the following data : x: 1.0 1.5 2.0 2.5 3.0 3.5 4.0 y: 1.1 1.3 1.6 2.0 2.7 3.4 4.1 Sol. Here m = 7 (odd) x − 2.5 = 2x – 5 0.5 Results in tabular form are Let u= and v = y x y u v u2 uv u2 v u3 u4 1.0 1.1 –3 1.1 9 – 3.3 9.9 – 27 81 1.5 1.3 –2 1.3 4 – 2.6 5.2 –8 16 2.0 1.6 See All Chapters |
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## ALLC5-1 |
Manish Goyal | Laxmi Publications | |||||

5 Numerical Integration and Differentiation 5.1. INTRODUCTION Consider a function of a single variable y = f(x). If f(x) is defined as an expression, its derivative or integral may often be determined using the techniques of calculus. However, when f(x) is a complicated function or when it is given in a tabular form, we use numerical methods. In this chapter, we will discuss numerical methods for approximating the derivative(s) f (r)(x), r ≥ 1 of a given function f(x) and for the evaluation of the integral z b a f ( x) dx where a, b may be finite or infinite. The accuracy attainable by these methods would depend on the given function and the order of the polynomial used. If the polynomial fitted is exact then the error would be theoretically zero. In practice, however, rounding errors will introduce errors in the calculated values. The error introduced in obtaining derivatives is in general much worse than that introduced in determining integrals. It may be observed that any errors in approximating a function are amplified while taking the derivative whereas they are smoothed out in integration. See All Chapters |
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## ALLC10-2 |
Manish Goyal | Laxmi Publications | |||||

578 NUMERICAL METHODS AND STATISTICAL TECHNIQUES USING ‘C’ 10.17. FITTING OF THE CURVE 2x = ax2 + bx + c Normal equations are Σ 2xx2 = aΣx4+ bΣx3 + cΣx2 Σ 2x x = aΣx3 + bΣx2 + cΣx and Σ 2x = aΣx2 + bΣx + mc where m is number of points (xi, yi) 10.18. FITTING OF THE CURVE y = ae�3x + be�2x and Normal equations are Σye–3x = a Σe–6x + b Σe–5x Σye–2x = a Σe–5x + b Σe–4x EXAMPLES Example 1. Find the curve of best fit of the type y = aebx to the following data by the method of Least squares : x: 1 5 7 9 12 y: 10 15 12 15 21. Sol. The curve to be fitted is y = aebx or Y = A + Bx, where Y = log10 y, A = log10 a, and B = b log10 e ∴ The normal equations are ΣY = 5A + BΣx and ΣxY = AΣx + BΣx2 x y Y = log10 y x2 xy 1 5 7 9 12 10 15 12 15 21 1.0000 1.1761 1.0792 1.1761 1.3222 1 25 49 81 144 1 5.8805 7.5544 10.5849 15.8664 Σx = 34 and ΣY = 5.7536 Σx2 = 300 ΣxY = 40.8862 Substituting the above values in the normal equations, we get 5.7536 = 5A + 34B 40.8862 = 34A + 300B On solving A = 0.9766 ; B = 0.02561 ∴ a = antilog10 A = 9.4754 ; b = Hence the required curve is See All Chapters |
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## ALLC10-1 |
Manish Goyal | Laxmi Publications | |||||

10 Curve-Fitting, Cubic Spline and Approximation 10.1. CURVE FITTING Let there be two variables x and y which give us a set of n pairs of numerical values (x1, y1), (x2, y2).......(xn, yn). In order to have an approximate idea about the relationship of these two variables, we plot these n paired points on a graph thus, we get a diagram showing the simultaneous variation in values of both the variables called scatter or dot diagram. From scatter diagram, we get only an approximate non-mathematical relation between two variables. Curve fitting means an exact relationship between two variables by algebraic equations, infact this relationship is the equation of the curve. Therefore, curve fitting means to form an equation of the curve from the given data. Curve fitting is considered of immense importance both from the point of view of theoretical and practical statistics. Theoretically, it is useful in the study of correlation and regression. Practically, it enables us to represent the relationship between two variables by simple algebraic expressions e.g., polynomials, exponential or logarithmic functions. See All Chapters |