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• 1. Gaussian Quadratures • Newton-Cotes Formulae – use evenly-spaced functional values – Did not use the flexibility we have to select the quadrature points • In fact a quadrature point has several degrees of freedom. Q(f)=∑i=1m ci f(xi) A formula with m function evaluations requires specification of 2m numbers ci and xi • Gaussian Quadratures – select both these weights and locations so that a higher order polynomial can be integrated (alternatively the error is proportional to a higher derivatives) • Price: functional values must now be evaluated at nonuniformly distributed points to achieve higher accuracy • Weights are no longer simple numbers • Usually derived for an interval such as [-1,1] • Other intervals [a,b] determined by mapping to [-1,1]
• 2. Gaussian Quadrature on [-1, 1] ∫ 1 −1 n f ( x )dx ≈ ∑ c i f ( x i ) = c 1 f ( x 1 ) + c 2 f ( x 2 ) + L + c n f ( x n ) i =1 • Two function evaluations: – Choose (c1, c2, x1, x2) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3 n=2: ∫ 1 −1 f(x)dx = c 1 f(x 1 ) + c 2 f(x 2 ) -1 x1 x2 1
• 3. Finding quadrature nodes and weights • One way is through the theory of orthogonal polynomials. • Here we will do it via brute force • Set up equations by requiring that the 2m points guarantee that a polynomial of degree 2m-1 is integrated exactly. • In general process is non-linear – (involves a polynomial function involving the unknown point and its product with unknown weight) – Can be solved by using a multidimensional nonlinear solver – Alternatively can sometimes be done step by step
• 4. Gaussian Quadrature on [-1, 1] n=2: ∫ 1 −1 f(x)dx = c 1 f(x 1 ) + c 2 f(x 2 ) Exact integral for f = x0, x1, x2, x3 – Four equations for four unknowns ⎧f ⎪ ⎪ ⎪f ⎨ ⎪f ⎪ ⎪f ⎩ 1 = 1 ⇒ ∫ 1dx = 2 = c 1 + c 2 −1 1 = x ⇒ ∫ xdx = 0 = c 1 x 1 + c 2 x 2 −1 2 2 2 = x ⇒ ∫ x dx = = c 1 x 1 + c 2 x 2 −1 3 2 1 2 1 3 3 = x ⇒ ∫ x 3 dx = 0 = c 1 x 1 + c 2 x 2 3 −1 1 I = ∫ f ( x )dx = f ( − −1 ⎧c 1 = 1 ⎪c = 1 ⎪ 2 −1 ⎪ ⇒ ⎨ x1 = 3 ⎪ 1 ⎪ ⎪ x2 = 3 ⎩ 1 3 )+ f ( 1 3 )
• 5. Error • If we approximate a function with a Gaussian quadrature formula we cause an error proportional to 2n th derivative
• 6. Gaussian Quadrature on [-1, 1] n=3: ∫ 1 −1 f ( x )dx = c 1 f ( x 1 ) + c 2 f ( x 2 ) + c 3 f ( x 3 ) x2 x3 -1 x1 1 • Choose (c1, c2, c3, x1, x2, x3) such that the method yields “exact integral” for f(x) = x0, x1, x2, x3,x4, x5
• 7. Gaussian Quadrature on [-1, 1] 1 f = 1 ⇒ ∫ xdx = 2 = c1 + c2 + c3 −1 1 f = x ⇒ ∫ xdx = 0 = c1 x1 + c2 x2 + c3 x3 −1 1 2 2 2 f = x ⇒ ∫ x dx = = c1 x12 + c2 x2 + c3 x3 3 −1 2 2 1 3 3 f = x 3 ⇒ ∫ x 3dx = 0 = c1 x13 + c2 x2 + c3 x3 −1 1 2 4 4 f = x ⇒ ∫ x dx = = c1 x14 + c2 x2 + c3 x3 5 −1 4 4 1 5 5 f = x 5 ⇒ ∫ x 5 dx = 0 = c1 x15 + c2 x2 + c3 x3 −1 ⎧c1 = 5 / 9 ⎪c = 8 / 9 ⎪ 2 ⎪c3 = 5 / 9 ⎪ ⇒⎨ ⎪ x1 = − 3 / 5 ⎪ x2 = 0 ⎪ ⎪ x3 = 3 / 5 ⎩
• 8. Gaussian Quadrature on [-1, 1] Exact integral for f = x0, x1, x2, x3, x4, x5 I=∫ 1 −1 3 5 3 8 5 ) f ( x )dx = f ( − )+ f (0 )+ f ( 9 5 9 5 9
• 9. Gaussian Quadrature on [a, b] Coordinate transformation from [a,b] to [-1,1] b−a b+a t= x+ 2 2 ⎧ x = −1 ⇒ t = a ⎨ ⎩x = 1 ⇒ t = b a ∫ b a t1 f ( t )dt = ∫ 1 −1 t2 b 1 b−a b+a b−a f( x+ )( )dx = ∫ g ( x )dx −1 2 2 2
• 10. Example: Gaussian Quadrature 4 I = ∫ te dt = 5216.926477 Evaluate 2t 0 Coordinate transformation b−a b+a t= x+ = 2 x + 2; dt = 2dx 2 2 4 1 I = ∫ te dt = ∫ (4 x + 4)e 2t 4 x+4 −1 0 1 dx = ∫ f ( x)dx −1 Two-point formula 1 I = ∫ f ( x )dx = f ( −1 −1 )+ f ( 1 )= (4 − 4 )e 3 3 3 = 9.167657324 + 3468.376279 = 3477.543936 4− 4 3 +(4 + 4 )e 3 ( ε = 33.34%) 4+ 4 3
• 11. Example: Gaussian Quadrature Three-point formula 1 5 8 5 I = ∫ f ( x )dx = f ( − 0.6 ) + f ( 0 ) + f ( 0.6 ) −1 9 9 9 5 8 5 4 − 0 .6 4 = ( 4 − 4 0.6 )e + ( 4 )e + ( 4 + 4 0.6 )e 4 + 0.6 9 9 9 5 8 5 = ( 2.221191545 ) + ( 218.3926001 ) + ( 8589.142689 ) 9 9 9 = 4967.106689 ( ε = 4.79%) Four-point formula I = ∫ f ( x )dx = 0.34785[ f ( −0.861136 ) + f ( 0.861136 )] 1 −1 + 0.652145 [ f ( −0.339981 ) + f ( 0.339981 )] = 5197.54375 ( ε = 0.37%)
• 12. Other rules • Gauss-Lobatto: – requiring end points be included in the formula • Gauss-Radau – Require one end point be in the formula