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Differential Equations Solutions 173

Differential Equations Solutions 173 - U i j = C i j − L...

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Chapter 30 Solutions: Case Study: Fast Solvers and Sylvester Equations: Both Sides Now CHALLENGE 30.1. Equating the ( j, k ) element on each side of the equation ( B y U + UB x ) = F , we obtain f ( x j , y k ) = 1 h 2 ( u ( x j 1 , y k )+2 u ( x j , y k ) u ( x j +1 , y k ) u ( x j , y k 1 )+2 u ( x j , y k ) u ( x j , y k +1 )) , which is the same as equation ( k 1) n + j of ( A x + A y ) u = f . CHALLENGE 30.2. (a) Using MATLAB notation for subvectors, the algorithm is: for i = 1 : n , for j = 1 :
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Unformatted text preview: , U ( i, j ) = ( C ( i, j ) − L ( i, 1 : i − 1) ∗ U (1 : i − 1 , j ) − U ( i, 1 : j − 1) ∗ R (1 : j − 1 , j )) / ( L ( i, i ) + R ( j, j )) end end The number of multiplications is n X i =1 n X j =1 ( i − 1 + j − 1) = n 2 ( n − 1) , and the other operations are also easy to count. 183...
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