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# 3 2 915b 1 2 b i i i i 916a where l0

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Unformatted text preview: 9.1.6), we nd 2A3 3 6 r( ) 7 7 6 r( 11 12) 7 7 76 6 7 6 r( 2 1 ) 7 7 76 7 = 6 r( 2 2 ) 7 : 76.7 7 6 .. 7 7 76 7 6 r( ) 7 5 6 17 6 r( ) 7 4 27 5 2c 2L 3 6 d0 6 L01 R1 7 6 c10 6 .. 76 6 7 6 d1 .. .. 6 76 . 6 7 6 .. 6 4 L R 56 4 R +1 6 c N N N (9.1.6b) N (9.1.7) N d N N B Thus, the 2(N + 1) coe cients c , d , i = 0 1 : : : N , are determined as the solution of a block bidiagonal matrix of dimension 2N with 2 2 blocks. This system may be solved by the methods of Section 8.5. We can now ask if there is an optimal placement of the two collocation points on each subinterval that, e.g., minimizes the discretization error y(x) ; Y (x) in some norm. This question was answered in a landmark paper by de Boor and Swartz 3] and we will follow their analysis. Consider the inner product i i (v u) = Z b v(x)u(x)dx: (9.1.8) a Let us assume that u(x) and v(x) are smooth except, perhaps for jump discontinuities at z , j = 1 2 : : : M ; 1. Also let z0 = a and z = b. Consider j (v Lu) = Z M b v u + pu + qu]dx 00 (9.1.9) 0 a where the integral is interpreted as a...
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