A Family of Matrix Problems

where C(x) = 1 − 3x + x 2 /2 + x 3 /30. Note that C (x) is positive for x > 3 and C(9/2) = 53/80. Hence C(x) is positive for x > 9/2 and the bounds are established. We can now write F (z) > z 2 0 e −zt dt + z ∞ 2 2t 2 e −(z+1)t dt, and thus F (z) > 1 + e −2z−2 (w(z) − e 2), where w(z) = 8 − 4 (z + 1) 2 − 4 (z + 1) 3. Note that w(z) increases from 200/27 at z = 2 to 8 at z = ∞. Since 200/27 > e 2 , we are done. The following family of matrix problems arises in the design of some high-order local non-reflecting boundary conditions [1,2,3]:          1 0 1 2