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Matrix Formualtion of Interpolating Using Variational Methods, Lecture Notes - Mathematics

Calculus I

Post: October 9th, 2011
Description
matrix formulation of interpolation, unequally spaced mesh points, equally spaced mesh points, three points symmetric about center, restrictions of second derivative, infinite splines, difference equation, particular solution
matrix formulation of interpolation, unequally spaced mesh points, equally spaced mesh points, three points symmetric about center, restrictions of second derivative, infinite splines, difference equation, particular solution
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Matrix Formulation of Variational Method Interpolation Adrian Down March 03, 2006 1 1.1 Matrix formulation Unequally-spaced mesh points Recall the matrix formulation of the differential equation considered last lecture, 2   1 Y (x0 ) h0 h0 1 1 2 + 2   Y (x1 )  h1  h0 h0 h1      . .. .. .. . . . .    .    1 2 2 1   Y (xn−1 ) + hn−1 hn−1 hn−2 hn−2 1 2 Y (xn ) hn−1 hn−1   1 1  − h2 h2 Y (x0 ) 0 0 1  − 12 12 − 12   Y (x1 )   h0 h0 h1 h2  1    . ... ... ...   . = 3  .   1 1 1 1   − h2 − h2  Y (xn−1 )  h2 h2 n−2 n−2 n−1 n−1 1 Y (xn ) − 21 2 hn−1 hn−1 where hj = xj+1 − xj , for j ∈ { 0, . . . , n − 1 }. This is general formulation for non-uniform node spacing. 1.2 Equally-spaced mesh points Suppose that the mesh points are uniformly spaced, so that hi = hj = h, ∀i, j..

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