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7.6 Positive Deﬁnite Matrices
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Please report violations to [email protected] Discretize the problem by overlaying the square with a regular mesh containing n2 interior points at equally spaced intervals of length h as explained in Example 7.6.2 (p. 563). Let fij = f (xi , yj ), and deﬁne f to be
the vector f = (f11 , f12 , . . . , f1n |f21 , f22 . . . , f2n | · · · |fn1 , fn2 , . . . , fnn )T .
Show that the discretization of Poisson’s equation produces a system
of linear equations of the form Lu = g − h2 f , where L is the discrete
Laplacian and where u and g are as described in Example 7.6.2.
7.6.10. As deﬁned in Exercise 5.8.15 (p. 380) and discussed in Exercise 7.8.11
(p. 597) the Kronecker product (sometimes called tensor product , or
direct product ) of matrices Am×n and Bp×q is the mp × nq matrix
a2n B ⎟
. a11 B
⎜ a21 B
. a12 B
. am1 B am2 B · · · amn B D
H Verify that if In is the n × n identity matrix, and if
An = ⎜
−1 .. .
−1 ⎞ ⎟
2 n×n P is the nth -order ﬁnite diﬀerence matrix of Example 1.4.1 (p. 19), then
the discrete Laplacian is given by O
C Ln2 ×n2 = (In ⊗ An ) + (An ⊗ In ). Thus we have an elegant matrix connection between the ﬁnite diﬀerence
approximations of the one-dimensional and two-dimensional Laplacians.
This formula leads to a simple alternate derivation of (7.6.8)—see Exercise 7.8.12 (p. 598). As you might guess, the discrete three-dimensional
Ln3 ×n3 = (In ⊗ In ⊗ An ) + (In ⊗ An ⊗ In ) + (An ⊗ In ⊗ In ). Copyright c 2000 SIAM Buy online from SIAM
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