Problems for Obligatory 1

# Problems for Obligatory 1 - Spring 2010 MAT260 – 2...

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Unformatted text preview: Spring 2010 MAT260 – BEREGNINGSALGORITMER 2 Obligatory Exercise 1 Deadline: 16. March 15.00 Print your name and Candidate number on the front page, and deliver your solutions to the problems below in a box marked MAT260 in the ’Ekspedisjon’ office in the 5th floor Problem 1 Given a weighted inner-product on the space of real functions, ( f, g ) = integraldisplay ∞-∞ e- x 2 f ( x ) g ( x ) dx. Find the three first orthogonal polynomials p , p 1 and p 2 (of degree 0, 1 and 2) with respect to this inner-product. You may use without proof that integraldisplay ∞-∞ e- x 2 x p dx = 0 for p odd integraldisplay ∞-∞ e- x 2 dx = √ π integraldisplay ∞-∞ e- x 2 x 2 dx = 1 2 √ π integraldisplay ∞-∞ e- x 2 x 4 dx = 3 4 √ π Problem 2 Show that the Hilbert matrix with elements a ij = (1 + i + j )- 1 , ≤ i, j ≤ n is a Gram matrix for the functions 1 , x, x 2 , ..., x n for an appropriate choice of inner-product. In the text book the Hilbert matrix is called ’notorious’.of inner-product....
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Problems for Obligatory 1 - Spring 2010 MAT260 – 2...

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