Gaussian Quadrature. James Keesling. 1 Quadrature Using Points with Unequal Spacing

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1 Gussin Qudrture Jmes Keesling 1 Qudrture Using Points with Unequl Spcing In Newton-Cotes Integrtion we used points tht were eqully spced. However, there ws no need for the points to hve ny specil spcing. If we wish to estimte the integrl f(x) dx nd hve ny set of points {x 0, x 1,..., x n }, then we cn estimte the integrl by the formul f(x) dx A 0 f(x 0 ) + A 1 f(x 1 ) + + A n f(x n ). We cn solve for the constnts {A 0, A 1,..., A n } by mking the formul exct for the functions f(x) = 1, x, x, x 3,..., x n. This will give us n + 1 equtions tht we cn use to solve for the constnts {A 0, A 1, A,..., A n }. In Gussin Qudrture we use the intervl [, 1] s the stndrd nd the points {x 0, x 1,..., x n } will ll be contined in this intervl. There is mtrix eqution for the normlized constnts { 0, 1,..., n }. 1 x 0 x 0 x n 0 1 x 1 x 1 x n 1 M = Vndermonde([x 0, x 1,..., x n ]) = 1 x x x n x n x n x n n Let M T be the trnspose of M. Let A be the column vector with entries i. Let B be column vector with entries Then we get b i = nd solving for A we get the following. x i dx = 1 ()i+1. i + 1 M T A = B A = ( M T ) B 1

2 Choosing the Points We now hve flexibility to choose the points {x 0, x 1,..., x n } in wy tht will mke the estimte of the integrl even more ccurte. The theory behind the choice of points involves the Legendre polynomils. Let us denote these polynomils by {P n (x) n = 0, 1,,... }. These polynomils hve the following properties. (1) P n (x) hs degree n. () If i j, then P i (x) P j (x) dx = 0. (3) The vector spn of {P 0 (x), P 1 (x),..., P n (x)} is the sme s tht of {1, x, x,..., x n }. (4) For ech n Pn(x) dx = n + 1. Property (3) implies tht P n(x) P (x) dx = 0 for ny polynomil P (x) of degree less thn n. Here re the first few Legendre polynomils. P 0 (x) = 1 P 1 (x) = x P (x) = 1 (3x 1) P 3 (x) = 1 (5x3 3x) P 4 (x) = 1 8 (35x4 30x + 3) P 5 (x) = 1 8 (63x5 70x x) P 6 (x) = 1 16 (31x6 315x x 5) P 6 (x) = 1 16 (49x7 693x x 3 35x). There is simple formul tht gives these polynomils. P n (x) = 1 n n! d n ( (x dx n 1) n)

3 The roots of these polynomils re ll rel nd distinct. They re contined in the intervl [, 1]. The points used in Gussin Qudrture re the roots of P n+1, {x 0, x 1,..., x n }. Becuse of the properties of the Legendre polynomils, it turns out tht if P (x) is ny polynomil of degree k up to n + 1, then the Gussin Qudrture estimte of the integrl of P (x) is exct. We now prove this fct. Theorem.1. Suppose tht {x 0, x 1, x,..., x n } re the roots of the Legendre polynomil of degree n + 1. Suppose tht { 0, 1,..., n } re the normlized coefficients for Gussin Qudrture for these points. Then for ny intervl [, b], P (x) dx = b ( b i f x i + + b Proof. We will give the proof just for the intervl [, 1]. Adpting the proof to generl intervl just requires creful chnge of vribles. Let P n+1 (x) be the Legendre polynomil with its roots {x 0, x 1,..., x n }. Assume tht P (x) hs degree n + 1. Let P (x) = Q(x) P n+1 (x) + R(x) be the quotient nd reminder. Then the degree of Q(x) is n nd the degree of R(x) lso hs degree n. Consider the integrl P (x) dx = Q(x) P n+1 (x) + R(x) dx. Since the degree of Q(x) is less thn tht of P n+1 (x) we will hve So, we get the following. P (x) dx = Q(x) P n+1 (x) dx = 0. = P n+1 (x) Q(x) + R(x) dx R(x) dx On the other hnd we hve these equlities. R(x) dx = R(x i ) = P n+1 (x i ) Q(x i ) + R(x i ) The first equlity is becuse the degree of R(x) is less thn the number of points used in the qudrture formul. The second equlity is becuse the points x i were chosen to be the zeros of P n+1. So, we hve shown tht the qudrture formul is exct for polynomils of degree n ).

4 3 Implementtion of Gussin Qudrture For given Legendre polynomil P n+1 (x), finding the roots is not such n esy tsk. So, the wy this is usully done is to determine these vlues {x 0, x 1,..., x n } for certin vlue of n nd determine the coefficients { 0, 1,..., n }. Then store these vlues nd cll them from memory or embed the vlues in progrm implementing Gussin Qudrture. Here re the x i nd i vlues for n = 9. x 4 = x 5 = x 3 = x 6 = x = x 7 = x 1 = x 8 = x 0 = x 9 = The coefficients re the following. 4 = 5 = = 6 = = 7 = = 8 = = 9 = If we used the exct vlues of these points nd weights, then the Gussin Qudrture formul would be exct for polynomils of degree 19. In the next section we implement progrm with fewer points just for convenience. 4 TI-89 Progrm for Gussin Qudrture Here is progrm with eight points, n = 7. The points nd their weights re given below. Be creful to enter the vlues correctly. Obviously, the progrm will not give good estimtes if the numbers re not correct. 4

5 x 0 = = x 1 = = x = = x 3 = = x 4 = = x 5 = = x 6 = = x 7 = = :gussq(f,,b) :Prgm :[ , , , , , , , ] points :[ , , , , , , , ] weights :0 p :For i,1,8 :p + (b-)/*weights[1,i]*f (x=(b-)/*points[1,i] + (b+)/) p :EndFor :EndPrgm The vrible f is the function with ssumed vrible x. The inputs nd b re the left nd right endpoints of the intervl of integrtion, respectively. The vlue of the integrl is stored in the vrible p. This progrm will give the exct integrl for polynomils up to degree fifteen except for round-off error. So, to test the progrm compute the following integrl. 0 x 15 dx = 1 16 =.065 5