Simpson's Rule

12 Key excerpts on "Simpson's Rule"

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  Numerical Methods in Chemical Engineering Using Python® and Simulink®

  • Nayef Ghasem(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  In Simpson’s rule, we estimate the areas of regions under curves by using rectangles. Simpson’s rule is a method for approximating a definite integral using a multidefined quadratic function; the method uses parabolas rather than a straight line. In numerical analysis, Simpson’s 1/3 rule is a method for the numerical approximation of definite integrals. Specifically, it is the following approximation: ∫ a b f (x) d x = h 3 { f (x o) + 4 f (x 1) + 2 f (x 2) + 4 f (x 3) + … + f (x n) } (6.4) To obtain an approximate value of the definite integral, split the main interval into subintervals with an even number of n subintervals (Figure 6.1). Each step size (width), h, can be defined as follows: FIGURE 6.1 Graphical representation of Simpson’s rule. h = b − a n (6.5) Simpson’s rule works best when the function has a similar shape to the approximation device within a narrow interval. Example 6.1 Application of Simpson’s Rule Use Simpson’s rule to integrate the following definite integral, dividing it into two subintervals. ∫ 0 1 x 3 d x Evaluate the definite integral manually and confirm the manual calculations with the following approaches: Simulink/MATLAB programming of Simpson’s rule with n subinterval. Write a Python program to evaluate a definite integral using Simpson’s rule with n subdivisions. Solution Divide the main interval [0,1] into two subintervals with a width, h, h = 1 − 0 2 = 1 2 Substitute in Simpson’s rule formula as. follows: S n = h 3 (f (0) + 4 f (1 2) + f (1)) Substitute the values of the function at the subdivisions S n = (1 2) 3 (0 + 4 1 8 + 1) The approximate numerical value of the definite integral is S n = 1 4 = 0.25 The exact value of the definite integral is ∫ 0 1 x 3 d x = x 4 4 = 1 4 − 0 = 1 4 = 0.25 The approximate numerical value agreed with the exact value of the. definite integral. Simulink Solution Using the Simpson rule, the Simulink block diagram for solving Example 6.1 is described in Figure 6.2

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  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  [Image: c  John Wiley & Sons, Inc; drawn by Patrick Francisco] 7.7 Numerical Integration; Simpson’s Rule 461 GEOMETRIC INTERPRETATION OF SIMPSON’S RULE The midpoint (or tangent line) approximation and the trapezoidal approximation for a def- inite integral are based on approximating a segment of the curve y = f (x) by line segments. Intuition suggests that we might improve on these approximations using parabolic arcs rather than line segments, thereby accounting for concavity of the curve y = f (x) more closely. Figure 7.7.5 At the heart of this idea is a formula, sometimes called the one-third rule. The one-third rule expresses a definite integral of a quadratic function g(x) = Ax 2 + Bx + C in terms of the values Y 0 , Y 1 , and Y 2 of g at the left endpoint, midpoint, and right endpoint, respectively, of the interval of integration [m − Δx, m + Δx] (see Figure 7.7.5):  m+Δx m−Δx (Ax 2 + Bx + C) dx = Δx 3 [Y 0 + 4Y 1 + Y 2 ] (11) Verification of the one-third rule is left for the reader (Exercise 53). By applying the one-third rule to subintervals [x 2k−2 , x 2k ], k = 1, . . . , n / 2, one arrives at Formula (8) for Simpson’s rule (Exercise 54). Thus, Simpson’s rule corresponds to the integral of a piecewise-quadratic approximation to f (x). ERROR BOUNDS With all the methods studied in this section, there are two sources of error: the intrinsic or truncation error due to the approximation formula, and the roundoff error introduced in the calculations. In general, increasing n reduces the truncation error but increases the roundoff error, since more computations are required for larger n. In practical applications, it is important to know how large n must be taken to ensure that a specified degree of accuracy is obtained. The analysis of roundoff error is complicated and will not be consid- ered here.

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  An Introduction to Numerical Methods and Analysis

  • James F. Epperson(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  b)/2, the midpoint. Define the basic Simpson’s Rule, then, as

  where the Lagrange functions are here defined as Then the quadrature rule is given by where

  To compute these we define h = b − c = c − a = (b − a)/2, so that we have

  Then

  where we have used the change of variable u = x − a − h in the last step. The computation is now fairly straighforward:

  Similar computations (the student should verify these) yield B = A and C = 4h/3. Thus, Simpson’s rule becomes

  where h = (b − a)/2; thus, h is the distance between points in the discretization of the interval [a, b].

  EXAMPLE 5.3

  To apply this crude version of Simpson’s rule to the function f(x) = + sin πx, over the interval , we have

  The exact value is I(f) = 0.9501581580, showing that our approximation is decently accurate even though we used only a single approximating parabola. Figure 5.2 shows the application of Simpson’s rule to this example.

  Figure 5.2

  Illustration of Basic Simpson’s Rule.

  The composite rule is easily constructed by adding up individual instances of Simpson’s rule applied to pairs of subintervals. (Note that this means that Simpson’s rule requires an even number of subintervals.) We get

  (5.4)

  where h

  i

  = (x

  2i

  − x

  2i−2

  )/2. If we assume a uniform spacing of the mesh points, i.e., h

  i

  = h for all i, then all this simplifies a great deal and we have

  (5.5)

  EXAMPLE 5.4

  If we consider the same example that we used previously, but now use two approximating parabolas, then we get

  which is much more accurate than the S2 value. Figure 5.3 shows Simpson’s rule applied in this case, and the decrease in the error is evident.

  Figure 5.3

  Illustration of Simpson’s Rule.

  Simpson’s Rule can be constructed for a nonuniform grid, but the formula corresponding to either of (5.4) or (5.5)

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  Monte Carlo Simulations Of Disordered Systems

  • Sudhir Jain(Author)
  • 1992(Publication Date)
  • World Scientific

    (Publisher)

  This gives us the trapezoidal rule : IT = [/(so) + /(*i)]ty2 + [/(*i) + /(*a)]ty2 + ';- + [f{*N-i) + f{xs)]h/2 (3.7) = fc/2 x [/(*o) + 2/(*i) + 2/(x 2 ) + • • • + 2/(*tf-i) + f{x N )} (3.8) Numerical Integration 21 fc=0 Once again we can write I = I T + A(/j), where, A(Jr), the error in the trape-zoidal rule, is given by A(/ T ) oc =j£(B -A ) / ( 0 , where A < £ < 5 . (3.10) Since the error is now proportional to the second derivative of /(#), the rule given by Eqn.(3.9) is exact for all polynomials of degree < 1. §3.4 Simpson's Rule Simpson's Rule is perhaps one of the most widely used techniques for numerical integration. This time we shall divide the interval [A,B] into N equal intervals, where N is even. A possible division of the interval into 4 sub-divisions is shown in Fig. 3.4. The idea this time is to approximate f(x) by parabolic arcs through 3 succes-sive ordinates. So, for the situation depicted in Fig. 3.4 the arcs would go through 2/0»2/1»2/2 5 2/1»2/2 > 2/3 5 a nd 2/2»2/3 > 2/4 • Thus, we would require 3 separate parabolic fits. It is easy to see that I s = h/3 x {f(x 0 ) + f(x N ) + 4[/(*i) + /(x 3 ) + • • • + / ( X J V -I ) ] +2[/(* 2 ) + /(* 4 ) + • • • + f(*N-*)]} (3.H) , „ _ , , N (1 k = 0 y N; = L T r r i £ 5 fc /(x fc ), where 5 fc = { 4 A: odd; (3.12) 3 i V fc =o [ 2 fc(^iV)even. (Remember that N is even in Eqn.(3.12).) The error, A(Js), involved in this rule can be shown to be given by A(/s) oc d£{B -Atf^iQ, where A < £ < 5 . (3.13) Note that the error involved in Simpson's Rule is proportional to the 4th derivative of /(#), which, of course, means that this particular rule is exact for all polynomials 22 Monte Carlo Simulations... B Figure 3.4 Simpson's Rule with 4 sub-divisions. of degree < 3. It is this feature which makes Simpson's Rule a popular method for numerical integration.

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  Computational Methods in Physics and Engineering

  • Samuel S M Wong(Author)
  • 1997(Publication Date)
  • WSPC

    (Publisher)

  There are several variants of Simpson's Rule. For example, it is possible to change the relative weight of the contributions from different grid points and achieve an even higher-order accuracy than that of Eq. (2-15). Efficiency can also be improved by using extrapolation techniques, as we shall see later in §3-5 in connection with the Romberg method of integration. 38 Chap. 2 Integration and Differentiation 2-4 Gaussian quadrature The approximations used by trapezoidal and Simpson's Rules may be regarded as methods to replace the integrand in each subintervals by, respectively, second-and third-order polynomials. The idea of a polynomial approximation may also be applied to the interval [a, 6] as a whole. This leads us to the idea behind Gaussian quadrature, the method of Gauss for numerical integration. There are several different forms of this method. A typical one is the Gauss-Legendre integration. In general, a function f(x) in a given interval [a, 6] may be expanded in terms of a complete set of polynomials Pi(x), /(*) = f > P < ( * ) . (2-17) The definition of the expansion coefficients at depends somewhat on the polynomial Pe(x) used and we shall see examples later. For each polynomial, the subscript I indicates the order, the highest power of the argument x. For f(x) in Eq. (2-17), it is given by n, the upper limit of the summation. In general, it is more convenient to adopt a set of orthogonal polynomials to expand a function. In principle, any orthogonal polynomial can be used. For our purpose here, we shall take Legendre polynomials, discussed later in §4-2. The or-thogonality relation between these polynomials is £ ' P k (x)P e (x) dx = 2^TI«M-(2-18) Since Legendre polynomials are defined only in the interval [—1,-1-1], it is necessary for us to change the interval of integration for an arbitrary integral from [a, 6], for finite a and b, to [—1, +1] by making the substitution b — a b + a , x —♦ — x + — .

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  Fundamentals of Engineering Numerical Analysis

  • Parviz Moin(Author)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  The geometrical foundation of this formula is that the function f in the interval is approximated by a straight line passing through the end points, and the area under the curve in the interval is approximated by 30 3.2 ERROR ANALYSIS 31 x j x j +1 f(x) Figure 3.1 Trapezoidal rule; approximating f by a straight line between x j and x j + 1 . the area of the resulting trapezoid (see Figure 3.1 ). For the entire interval [ a , b ] the trapezoidal rule is obtained by adding the integrals over all sub-intervals: I ≈ h ⎛ ⎝ 1 2 f 0 + 1 2 f n + n − 1 j = 1 f j ⎞ ⎠ , (3.2) where uniform spacing x = h is assumed. If we approximate f in each interval by a parabola rather than a straight line, then the resulting quadrature formula is known as Simpson’s rule. To uniquely define a parabola as a fitting function, it must pass through three points (or two intervals). Thus, Simpson’s formula for the integral from x j to x j + 2 is given by x j + 2 x j f ( x ) dx ≈ x 3 f ( x j ) + 4 f ( x j + 1 ) + f ( x j + 2 ) . (3.3) Similarly, Simpson’s rule for the entire domain with uniform mesh spacing, x = h is given by I ≈ h 3 ⎛ ⎜ ⎜ ⎝ f 0 + f n + 4 n − 1 j = 1 j = odd f j + 2 n − 2 j = 2 j = even f j ⎞ ⎟ ⎟ ⎠ . (3.4) Note that in order to use Simpson’s rule for the entire interval of integration, the total number of points ( n + 1) must be odd (even number of panels). Before we discuss the accuracy of these formulae, notice that they both can be written in the compact form: I = b a f ( x ) dx ≈ n i = 0 w i f ( x i ) (3.5) where w i are the weights. For example, for the trapezoidal rule w 0 = w n = h 2 and w i = h for i = 1 , 2 , . . . , n − 1. 3.2 Error Analysis We will now establish the accuracy of these formulas using Taylor series ex-pansions. It turns out that it is easier to build our analysis around the so-called 32 NUMERICAL INTEGRATION x j x j +1 f(x) y j Figure 3.2 Rectangle rule; approximating f in the interval between x j and x j + 1 by its value at the midpoint.

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  An Introduction to Numerical Methods and Analysis

  • James F. Epperson(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Ja Thus Simpson's Rule appears to be more accurate than the underlying theory (from interpolation) would suggest. What's going on here that causes this? Table 5.3 Simpson's Rule applied to f(x) = e x , [a, b] = [0,1]. n 2 4 8 16 32 64 128 256 512 1024 2048 Sn(f) 1.718861151877 1.718318841922 1.718284154700 1.718281974052 1.718281837562 1.718281829028 1.718281828495 1.718281828461 1.718281828459 1.718281828459 1.718281828459 /(/) - s n (f) -0.579323E-03 -0.370135E-04 -0.232624E-05 -0.145593E-06 -0.910273E-08 -0.568969E-09 -0.355611E-10 -0.222178E-11 -0.137890E-12 -0.910383E-14 0.444089E-15 Error Ratio N/A 15.6517 15.9113 15.9777 15.9944 15.9986 15.9998 16.0057 16.1127 15.1463 -20.5000 It is possible to directly prove an error estimate for Simpson's Rule, in a manner similar to that for the trapezoid rule. But a more illuminating explanation of why Simpson's Rule is "more accurate than it ought to be" can be had by looking at the extent to which it integrates polynomials exactly. This leads us to the notion of degree of precision for a quadrature rule. Simpson's Rule AND DEGREE OF PRECISION 275 Definition 5.1 (Degree of Precision) Let I n be a quadrature rule, and assume that it is exact for all polynomials of degree < p. Then we say that I n has degree ofprecision p. We ought to expect that we get better accuracy from integration methods that have a higher degree of precision. As an example, we note that Problem 21 of §2.5 shows that the trapezoid rule has degree of precision p = 1, because it integrates all linear polynomials exactly. Simpson's Rule, by construction, clearly ought to have degree of precision p = 2. We will show that it actually has degree of precision p = 3; i.e., Simpson's Rule integrates cubic polynomials exactly, which is something of a bonus beyond what we ought to expect from the construction. We will then show that having degree of precision p = 3 leads to an error estimate that is 0(h 4 ).

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  Ship Hydrostatics and Stability

  • Adrian Biran(Author)
  • 2003(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Example 3.2 We refer again to Example 3.1 using this time Simpson’s rule. We can experiment with decreasing subintervals and obtain the results shown inTable 3.1, where they are compared with the results yielded by the trapezoidal rule. The convergence is considerably faster than that obtained in the case of the trapezoidal rule. The per cent errors are shown in Table 3.2. As predicted by Eq. (3.13), each time we divide the subinterval h by 2, the error decreases approximately in the ratio 1 / 16 . Note also that only two subintervals yield better results with Simpson’s rule than eight with the trapezoidal rule. 3.4 Calculating points on the integral curve The trapezoidal and Simpson’s rules produce one number for an interval of ordinates, i.e. I ( a, b ) = b a f ( x ) d x Sometimes we are interested not only in one number, but also in a sequence of numbers that describe the integral as a function within the given interval, I ( x ) = x a f ( x ) d x, a ≤ x ≤ b (3.14) Thus, in certain hydrostatic calculations we may need to know the areas of transverse sections (stations) as functions of draught (see Chapter 4). Another example is that of calculations of dynamic stability which require the knowledge of the area under the curve of the righting arm as function of the heel angle. The latter subject is discussed in Chapters 6 and 8. An appropriate name for a procedure that yields such an integral is integral with variable upper limit . Numerical integration in naval architecture 81 Let us consider a sequence of points, x 1 , x 2 , . . . , x n , and a sequence of values f ( x 1 ) , f ( x 2 ) , . . . , f ( x n ) . In the first example above, the values of the independent variable, x i , represent draught, the functions f ( x i ) , half-breadth and the integral, the area of the station up to that draught. In the second example, x i is a heel angle, f ( x i ) , the righting arm, GZ and the integral, the area under the righting-arm curve up to the respective angle.

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  An Introduction to Numerical Methods and Analysis, Solutions Manual

  • James F. Epperson(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  n ≥ 10206208. Clearly Simpson’s rule is more efficient.

  10. Use Simpson’s rule to produce a graph of E (x ), defined to be the length of the exponential curve from 0 to x, for 0 < x < 3. See Problem 14 of § 2.5.

  Solution: If we use a fixed number of points, say n = 128, for any value of x , then we get the plot shown in Figure 5.1 .

  Figure 5.1 Solution for Exercise 5.3.10.

  11. Let f (x ) = |x |; use the trapezoid rule (with n = 1), corrected trapezoid rule (with n = 1), and Simpson’s rule (with n = 2), to compute

  and compare your results to the exact value. Explain what happens in the light of our error estimates for the trapezoid and Simpson’s rules.

  Solution: Because the function is piecewise linear, and one grid point matches the “kink” in the graph, the trapezoid rule will produce the exact value of the integral, but neither Simpson’s rule nor the corrected trapezoid rule will be exact.

  12. Write out the expression for Simpson’s rule when c is not the midpoint of the interval [a , b ]. To simplify matters, take c – a = h , b – c = θh .

  Solution: We get

  where

  13. What is the degree of precision of the corrected trapezoid rule, What about the n subinterval version,

  Solution: Let and consider the computation

  When simplified, and compared to the exact values of

  we get that

  Ik

  = I for k = 0,1,2,3, but not for k = 4, thus showing that the degree of precision is p = 3. The same result holds for the n subinterval version.

  14. Prove that if we want to show that the quadrature rule

  In (f

  ) has degree of precision p , it suffices to show that it will exactly integrate

  xk

  , 0 ≤ k ≤ p over the integral (0, 1).

  Solution: Let q (x ) be an arbitrary polynomial of degree p . Then

  so

  Therefore, we need only concern ourselves with exactly integrating simple powers instead of general polynomials. A change of variable can now be employed to transform the interval of interest to [0,1] from the more general [a , b ].

  15. Construct the analogue of Simpson’s rule based on exactly integrating a cubic interpolate at equally spaced points on the interval [a , b ].

  Solution:

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  Mathematics for Engineers IV

  • Gerd Baumann(Author)
  • 2010(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  When n is doubled the error in CS n f decreases by a factor of about 68. 5.2.5 Tests and Exercises The following two subsections serve to test your understanding of the last section. Work first on the test examples and then try to solve the exercises. 5.2.5.1 Test Problems T1. What is the basic idea of the trapezoidal rule? T2. How does Simpson's Rule work? T3. How can the accuracy of a simple integration rule be improved? T4. What is the idea of introducing correction terms? 5.2.5.2 Exercises E1. Use the simple integration formulas like Trapezoidal and Simpson to approximate the following integrals : a. 0 2 sin x x b. 0 1 x 1 x x c. 0 1 ln 1 x sin x x d. 0 1 x 5 x 2 x E2. Evaluate the following integrals using composite Simpson' s rule with n 3 and n 5 subintervals: a. 0 1 x x x b. 0 0.984 x x x E3. Derive the Simpson's 1 3 rule and 3 8 rule, using integrating the appropriate interpolating polynomial. E4. Evaluate the following integrals accurately, using simple Trapezoidal and Simpson's formulas: a. 0 1 5 x cos 10 x x x b. 0 1 x m ln 1 x n x for m 0, 1 2, 1.1 and n 1, 2, 3 195 IV - Chapter 5: Numerical Inte g ration 0 1 ln 1 x sin x x d. 0 1 1 b sin 20 x x for b 2, 1.1, 1.01, 1.001, … E5. Consider the integral (1) I 0 1 1 1 x x and solve it either with the error correction formulas for Trapezoidal or Simpson's Rule. E6. Suppose a body of mass m is traveling vertically upward starting at the surface of the earth. If all resistance except gravity is neglected, the escape velocity v is given by (2) v 2 2 g R 1 2 where x R , R 6 378 137 m is the radius of the earth, and g 9.81 m s 2 is the force of gravity at the earth's surface. Approximate the escape velocity v . E7. The Laguerre polynomials L 0 x , L 1 x , … form an orthogonal set on 0, and satisfy 0 x L i x L j x x 0 for i j . The polynomial L n x has n distinct zeros x 1 , x 2 ,…, x n in 0, .

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  Applied Mathematics for Science and Engineering

  • Larry A. Glasgow(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  After all, there is nothing to prevent us from using a polynomial to approximate f (x), as opposed to a straight line. Before we attempt that, however, let us examine one other case where the value of the definite integral is known so we can explore the impact of increasing the number of intervals using both single- and double-precision arithmetic. Again we will use the trapezoid rule to consider (4.2) n Single Precision, I (x) Double Precision, I (x) 125 1.570797 1.5707963268 250 1.570800 1.5707874013 500 1.570789 1.5707952107 1000 1.570781 1.5707961870 2000 1.570817 1.5707963270 4000 1.570853 1.5707963250 8000 1.570871 1.5707963264 Notice that once we exceed n ≈ 2000 with the single-precision calculations, the result deviates increasingly from the correct value given in eq. (4.2). We will consider this unwelcome development—cumulative roundoff error—more fully later in this chapter. Simpson's Rule We saw that we could employ a linear approximation with the trapezoid rule and get satisfactory results for integrals that were not particularly “difficult.” An obvious extension that might yield better results could be obtained by merely increasing the degree of the polynomial. For example, we might let f (x) be approximated by (4.3) If we place the three necessary function evaluations (for y 1, y 2, and y 3) at x = −Δ x, x = 0, and x = +Δ x, then we find (4.4) (4.5) (4.6) Now we will actually integrate the polynomial (eq. 4.3) from −Δ x to +Δ x : (4.7) We use the above-mentioned three function evaluations to obtain a, b, and c. The resulting algorithm for Simpson's Rule is (4.8) For an easy comparison, let us apply this method to the integral (eq. 4.2) examined previously,

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  Elements of Numerical Analysis

  • Radhey S. Gupta(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  6.4 Trapezoidal Rule In the trapezoidal rule, the function f ( x ) is approximated over the interval ( x 0 , x 1 ) by a straight line (polynomial of degree one) passing through the points ( x 0 , y 0 ) and ( x 1 , y 1 ) , See Fig. 6.2. Figure 6.2. Approximation by trapezium . Numerical Integration • 229 Thus the area under the curve y = f ( x ) is approximated by the area of the trapezium, i . e., x 1 Z x 0 f ( x ) dx = h y 0 + y 1 2 . (6.14) The value of the integral I over ( a , b ) may be obtained repeating the above process for all the n intervals. Now let us discuss it analytically. ( i ) Based on Lagrange’s Method By Lagrange’s method, the equation of the straight line passing through the points ( x 0 , y 0 ) and ( x 1 , y 1 ) is given by, y = P ( x ) = x -x 1 x 0 -x 1 y 0 + x -x 0 x 1 -x 0 y 1 = 1 h [( x -x 0 ) y 1 -( x -x 1 ) y 0 ] , (6.15) with associated error R ( x ) = ( x -x 0 )( x -x 1 ) f 00 ( ξ ) 2! , x 0 ≤ ξ ≤ x 1 . (6.16) The area over the first interval is thus approximated as, x 1 Z x 0 f ( x ) dx = 1 h x 1 Z x 0 [( x -x 0 ) y 1 -( x -x 1 ) y 0 ] dx = 1 2 h ( x -x 0 ) 2 y 1 -( x -x 1 ) 2 y 0 x 1 x 0 = h 2 ( y 0 + y 1 ) . (6.17) This is ‘trapezoidal rule’. The error in trapezoidal rule will be given by integrating (6.16) from x 0 to x 1 , i . e., E = x 1 Z x 0 ( x -x 0 )( x -x 1 ) f 00 ( ξ ) 2 dx = f 00 ( ξ ) 2 · h 3 1 Z 0 p ( p -1 ) dp ∵ dx = hdp 230 • Elements of Numerical Analysis = -h 3 12 f 00 ( ξ ) , x 0 ≤ ξ ≤ x 1 . (6.18) The composite formula for integral I may be obtained by applying the trapezoidal rule over all the n intervals and adding, i . e., b = x n Z a = x 0 f ( x ) dx = h 2 [ y 0 + y n + 2 ( y 1 + y 2 + .... + y n -1 )] . (6.19) The upper bound for the error in the composite formula (6.19) may be taken as n times the largest error in any interval, i . e., TE = -nh 3 12 f 00 ( ξ ) , x 0 ≤ ξ ≤ x n (6.20) = -( b -a ) h 2 12 f 00 ( ξ ) .

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