Y p consequently there exists a nonzero vector x m n

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Unformatted text preview: + ui,j +1 + O(h4 ) 4 Y P i + 1, j Figure 7.6.5 4 If the O(h ) terms are neglected, the resulting five-point difference equations, O C 4uij − (ui−1,j + ui+1,j + ui,j −1 + ui,j +1 ) = 0 for i, j = 1, 2, . . . , n, constitute an n2 × n2 linear system Lu = g in which the unknowns are the uij ’s, and the right-hand side contains boundary values. For example, a mesh with nine interior points produces the 9 × 9 system in Figure 7.6.6. 00 01 02 03 04 10 11 12 13 14 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 ⎛ 4 −1 0 4 −1 ⎜ −1 ⎜ 4 ⎜ 0 −1 ⎜ ⎜ −1 0 0 ⎜ ⎜ 0 ⎜ 0 −1 ⎜ 0 −1 ⎜0 ⎜ ⎜ 0 0 ⎜0 ⎝0 0 0 0 0 0 −1 0 0 0 −1 0 0 0 −1 0 0 0 0 0 0 4 −1 0 −1 4 −1 0 −1 4 −1 0 0 −1 0 0 −1 0 0 0 −1 0 0 0 −1 4 −1 −1 4 0 −1 ⎞⎛ ⎞ ⎞⎛ 0 g01 + g10 u11 0 ⎟ ⎜ u12 ⎟ ⎜ g02 ⎟ ⎟⎜ ⎟ ⎟⎜ 0 ⎟ ⎜ u13 ⎟ ⎜ g03 + g14 ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟⎜ ⎟ ⎟⎜ 0 ⎟ ⎜ u21 ⎟ ⎜ g20 ⎟ ⎟⎜ ⎟ ⎟⎜ 0 0 ⎟ ⎜ u22 ⎟ = ⎜ ⎟ ⎟⎜ ⎟ ⎟⎜ −1 ⎟ ⎜ u23 ⎟ ⎜ g24 ⎟ ⎟⎜ ⎟ ⎟⎜ ⎟⎜u ⎟ ⎜g + g ⎟ 0 ⎟ ⎜ 31 ⎟ ⎜ 30 41 ⎟ ⎠ ⎠ ⎝ u32 ⎠ ⎝ g −1 42 u33 g43 + g34 4 Figure 7.6.6 Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.6 Positive Definite Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 565 It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu The coefficient matrix of this system is the discrete Laplacian, and in general it has the symmetric block-tridiagonal form ⎛ ⎞ ⎛ ⎞ T −I 4 −1 ⎜ −I T −I ⎟ ⎜ −1 ⎟ 4 −1 ⎜ ⎟ ⎜ ⎟ .. .. .. .. .. .. ⎜ ⎟ with T = ⎜ ⎟. L=⎜ . . . . . . ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ −I T −I −1 4 −1 ⎠ −I T n2 ×n2 −1 4 n×n In addition, L is positive definite. In fact, the discrete Laplacian is a primary example of how positive definite matrices arise in practice. N...
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