Unformatted text preview: resulting upper triangular linear system for c(M ) . Thus, let RM be the M M
matrix obtained by deleting the last row of RM and gM be the M -vector obtained by
deleting the last element of gM , then c(M ) = R;1gM :
M (9.4.18) Example 9.4.3. Saad 4] solves the partial di erential equation ;uxx ; uyy ; uzz + 10(exy u)x + 10(e;xy u)y = f (x y z)
on a unit square with trivial Dirichlet boundary conditions. He chooses a 17 17 17
mesh and uses centered di erencing. The function f (x y z) is chosen so that the right
side b of the discrete problem (9.1.1) satis es b = A 1 1 : : : 1]T :
With Dirichlet boundary conditions, the dimension of the discrete system is (16)3 = 4096.
Saad 4] solves this system by GMRES procedures with and without preconditioning. The
results of some of his calculations are reported in Table 9.4. The column labeled Iter
reports the number of matrix-by-vector multiplications to reduce the initial residual by
a factor ot 107 in the L2 norm. The parameter Kflops reports the number of oating point operations performed divided by 1000. The parameters Residual and Error
record the residual and error in the L2 norm. The rst row of Table 9.4 reports data for
restarted GMRES with a Krylov space of dimension M = 10. The second row displays
data for GMRES calculations performed with a truncated orthogonalization using k = 10
columns. The third and fourth rows contain results of preconditioned GMRES calculations using, respectively, SSOR and ILU(0) preconditioners. The acceleration parameter 9.4. Krylov Subspace Methods 67 ! was set to unity for the SSOR technique. The preconditioned versions increased the
dimension of the Krylov space until convergence was attained. All procedures were initiated with a random initial guess.
ILU(0)-GMRES Iter K ops
17 4004 Residual
0.30(-3) Table 9.4.1: Number of iterations (Iter) to convergence, number of oat...
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