Chapter 11.3, Problem 9ES

Prove Theorem 11.3. [Hint: To use Theorem 6.31, first show that | $\frac{h}{2}$ (x_i)| < 1 implies that $\left|-1-\frac{h}{2}p\left(x_i\right)\right|+\left|-1+\frac{h}{2}p\left(x_i\right)\right|=2.]$

Theorem 11.3 Suppose that p, q, and r are continuous on [a, b]. If q(x) ≥ 0 on [a, b], then the tridiagonal linear system (11.19) has a unique solution provided that h < 2/L, where L = max_{a ≤ x ≤ b} |p(x)|.