HW1 - ( u- 2 , N u ) L 2 . 5. Problem 1 in Page 78. 6....

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MATH 692: HW 1 Due date: Feb. 2 1. Write a subroutine to compute the zeroes of orthogonal polynomials defined by a three-term recurrence relations. (i) Use the subroutine to compute the Chebyshev-Gauss points with N = 16 and compare it with the explicit formula in Page 17. (ii) Use the subroutine to compute the Legendre-Gauss-Lobatto points with N = 16. 2. Derive the explicit formula for Chebyshev-Gauss-Radau points and weights with “ - 1” being a point. 3. Prove 1.3.23. 4. (optional for non-math students) Let I = ( - 1 , 1), X N = { u P N : u ( ± 1) = u ± ( ± 1) = 0 } and define π 2 , 0 N : H 2 0 ( I ) = { u H 2 ( I ) : u ( ± 1) = u ± ( ± 1) = 0 } → X N by Z I ( u - π 2 , 0 N u ) ±± v ±± N dx = 0 , v N X N . Derive an estimate for ²
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Unformatted text preview: ( u- 2 , N u ) L 2 . 5. Problem 1 in Page 78. 6. (optional) Let { p k } be a sequence of orthogonal polynomials, with respect to the weight function ( x ) and the interval ( a, b ), generated by the three-term recurrence relation (setting p-1 ( x ) 0) p k +1 ( x ) = ( a k x-b k ) p k ( x )-c k p k-1 ( x ) , k . Find the three-term recurrence relation corresponding to q k ( x ) = p k +1 ( x )- k p k ( x ) x-a with k = p k +1 ( a ) p k ( a ) which is orthogonal with respect to the weight function ( x )( x-a ). 1...
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