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The Comparison of the Trapezoid Rule and the Gaussian Quadrature

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The Comparison of the Trapezoid Rule and the Gaussian Quadrature The Comparison of the Trapezoid Rule and the Gaussian Quadrature Margo Rothstein Georgia College and State University May 3, 2018 1 Abstract This paper provides a detailed exploration of the comparison of the Gaussian Quadrature and the Trapezoid Rule. They are both numerical methods that are used to approximate integrals. The Gaussian Quadrature is a much better approximation method than the Trapezoid Rule. We will prove and demonstrate the approximating power of both rules, and provide some practical applications of the rules as well. 2 Introduction Integration is the calculation of the the theorem states the following: area under the curve a function f(x). The integral of f produces a family of an- Z b tiderivatives F (x)+c that is used to calculate f(x)dx = F (b) − F (a): the area, denoted by, a Z f(x)dx: There are many methods developed to find F (x) that will produce the exact area. These A definite integral calculates the area under methods include integration by parts, par- the curve of a function f(x) on a specific in- tial fraction decomposition, change of vari- terval [a; b], denoted by, ables, and trig substitution. However, there are some functions that can not be inte- Z b grated exactly. They must be approximated. f(x)dx: We will discuss two approximation methods a in numerical analysis called the Trapezoid The Fundamental Theorem of Calculus is Rule and the Gaussian Quadrature. The how a definite integral should be calculated; definition of a Quadrature rule for nodes 1 x1 < x2 < x − 3 < ··· < xn is n Z b X f(x)dx ≈ cif(xi): a i=1 A closed quadrature rule means that f(x) is evaluated at the endpoints. An open quadrature rule is when f(x) is not evalu- ated at the endpoints. 3 Trapezoid Rule The Trapezoid Rule is a closed rule will format the area of the trapezoid so Quadrature rule where f(x) is only evalu- that it matches the definition. Note the fol- ated at the endpoints a and b on the inter- lowing: val [a; b]. Visually on a graph, we see the Z b f(a) + f(b) line that passes through points (a; f(a)) and f(x)dx = (b − a) 2 (b; f(b)), which forms a trapezoid with the a (b − a) x-axis. Thus, the area under the curve can = (f(a) + f(b)) be approximated by the area a Trapezoid, as 2 (b − a) (b − a) shown in Figure 1. = f(a) + f(b) 2 2 Figure 1 = c1f(a) + c2f(b) 2 y X = cif(xi): i=1 f(b) The area of a Trapezoid is now in the form of a Quadrature Rule with n = 2. Note that (b−a) c1 = c2 = 2 . That is, the coefficients f(a) are always that same. Since the Trapezoid Rule takes the area under a linear function, the value of the summation is exact only for polynomials of degree ≤ 1. For every poly- a b x nomial of degree > 1, the Trapezoid Rule will over- or under- approximate. Consider the Lagrange Polynomial interpolation for 2 points (x ; f(x )) and (x ; f(x )). The area of a Trapezoid is, 0 0 1 1 (x − x1) (x − x0) 00 Z b f(a) + f(b) P1(x) = f(x0) + f(x1) + Of (cx) f(x)dx = (b − a): x0 − x1 x1 − x0 a 2 00 where Of (cx) is the error term from the Since the Trapezoid Rule is a Quadrature approximation and cx depends continuously 2 on x0 and x1. For a linear function f(x) = tions, we will see how the coefficients and kx + C for some constants k and C, the nodes of the Gaussian Quadrature are spe- second derivative f 00(x) = 0. Therefore, cially chosen to produce high accuracy for whenever f(x) is a linear function, the er- higher degree polynomials. First, we will ex- ror will be zero and thus the approximation plore some underlying information about the will be exact. In the next couple of sec- Legendre Polynomials. 4 The Gaussian Quadrature 4.1 Legendre Polynomials Before we learn about the Gaussian why the property of orthogonal functions is Quadrature, we need to understand the Leg- valuable for the Gaussian Quadrature. endre Polynomials. The Legendre Poly- nomials are a set of nonzero polynomials Theorem (1): If fp1; p2; :::; png is a set of nonzero orthogonal polynomials on the S = fP1(x);P2(x);P3(x) ::: g that are or- thogonal on the interval [−1; 1]. By the def- closed interval [a; b] where the degree of inition of orthogonal, every pi = i for i 2 f1; 2; :::; ng, then every p has i distinct roots on the interval (a; b). ( i Z 1 = 0 if i 6= j Pi(x)Pj(x)dx = By Theorem (1), every P (x) 2 S has i dis- 6= 0 if i = j: i −1 tinct roots on [−1; 1]. We will refer back to The next Theorem will help us understand these roots in the following section. 4.2 Gaussian Quadrature Recall that the Trapezoid Rule is exactly nomials. Z 1 accurate for polynomials of degree ≤ 1. Sup- 1dx = c1 + c2; pose we wanted to develop a Quadrature rule −1 that is accurate for polynomials of a much Z 1 higher degree. Hence, we want a linear com- xdx = c1(x1) + c2(x2); ∗ −1 bination of values f(xi ) that is either ex- Z 1 actly or very close to the actual value of the 2 2 2 integral. Say that we want to integrate a x dx = c1(x1) + c2(x2) ; −1 polynomial function P (x) on [−1; 1] where i Z 1 n = 2. 3 3 3 x dx = c1(x1) + c2(x2) : −1 Z 1 2 X Since we have four unknowns c1, c2, x1, and P (x)dx = ciP (xi) = c1P (x1)+c2P (x2): x2, and 4 equations, we have a solvable sys- −1 i=1 tem. When this system is solved, we obtain q 1 q 1 Consider the integrals of the first four poly- c1 = c2 = 1, x1 = 3 and x2 = − 3 . Note 3 q q 1 1 nd a given n, the Gaussian Quadrature is exact that 3 and − 3 are the roots of the 2 degree Legendre polynomial. When we ex- up to polynomials of degree 2n − 1. Clearly, tend this example to the general nth case, this is a much more accurate method than th the Trapezoid Rule. Table 1 gives the roots we obtain that each xi is a root of the i Legendre polynomial. and coefficients up to n = 4. R 1 −1 dx = c1 + c2 + ::: +cn = 2 R 1 −1 xdx = c1x1 + c2x2 + ::: +cnxn = 0 Table 1 ::: n xi ci R 1 2n−2 2n−2 2n−2 1 x dx = c1x1 + ::: +cnxn = 2 p 1 −1 3 (−1)2n−1 1 − p−1 1 2n−1 2n−1 3 q 3 5 R 1 2n−1 2n−1 2n−1 3 − x dx = c1x + ::: +cnx = 5 9 −1 1 n 8 2n 0 1 − (−1) 9 2n 2n q 3 5 5 9 q p p We have that the Gaussian Quadrature 15+2 30 90−5 30 4 − 35 180 is a linear combination of the function f(x) q p p th 15−2 30 90+5 30 evaluated at the roots of the n Legendre − 35 180 th q p p polynomial. In the n case shown, we can 15−2 30 90+5 30 35 180 see that the degree of the last polynomial in q p p 15+2 30 90−5 30 the series of equations is 2n − 1. Hence, for 35 180 5 Comparison of Applications We will look at a simple example that Now, we show the result from the Gaussian will demonstrate the accuracy of the Gaus- Quadrature with two points. sian Quadrature and the Trapezoid Rule. r r Consider the integral of f(x) = x2 from Z 1 1 1 1 1 2 x2dx = ( )2 + (− )2 = + = : [−1; 1]. The exact value of this integral can −1 3 3 3 3 3 be found by the standard integration rule for polynomials. We see that the Gaussian Quadrature pro- duced the exact answer, while the Trapezoid Z 1 3 Rule was off by 4 . 2 x 1 1 −1 2 3 x dx = j−1 = − ( ) = : −1 3 3 3 3 Consider the integral of a function f(x) The next integral is the result from using the on [a; b] 6= [−1; 1]. The integral is not Trapezoid Rule. given on [−1; 1], and therefore the Gaussian Quadrature cannot be applied directly to it. Z 1 2 (1 − (−1)) 2 2 We must use a substitution for x in order x dx = ((−1) +(1) ) = (1)(2) =to 2 normalize: the function onto [−1; 1]. Let −1 2 4 a = k1t1 + k2 and b = k1t2 + k2, where t1 interval is half of [−1; 1]. and t are the upper and lower bounds of the 2 Z 1 Z 1 1 1 1 interval we want to normalized [a; b] to. In tan−1(x)dx = tan−1( t + ) dt 0 −1 2 2 2 this particular case, [t1; t2] = [−1; 1]; that is, 1 1 r1 1 1 1 r1 1 t1 = −1 and t2 = 1. Note that adding and = (tan−1( ( )) + ) + (tan−1( (− )) + ) 2 2 3 2 2 2 3 2 subtracting a and b yield the following: p p 1 3 + 3 1 − 3 + 3 = (tan−1( )) + (tan−1( )) 2 6 2 6 a + b = −k + k + k + k = 2k , 1 2 1 2 2 ≈ 0:4380: a − b = −k1 + k2 − k1 − k2 = −2k1.
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