# Errata for Numerical Solution of Ordinary Differential Equations, by Atkin, Han and Stewart¶

The textbook contains a number of small errors. As I find these, I post the corrections on this page. Other errors may also exist; these are only those that I have written down.

1. page 33, problem 2, should include a time interval to use when solving the problem numerically. I recommend a time interval $$t\in [0,5]$$.

2. page 35, problem 11, should also include a time interval to use when solving the problem numerically. I recommend a time interval $$t\in [0,4]$$.

Additionally, it should be more precise about how the solution errors should be calculated. I recommend an absolute error (as opposed to relative), and that the convergence orders should be computed using the maximum error over the nodes for each value of $$h$$.

3. page 53, middle of the page. The authors incorrectly state that

“Traditional root-finding methods (e.g., Newton’s method, the secant method, the bisection method) can be applied to (4.11) to find its root $$y_{n+1}$$; but often that is a very time-consuming process. Instead, (4.11) is usually solved by a simple iteration technique.”

The authors then go on to describe a simple fixed-point iteration.

This is not in fact true. The authors’ motivation for using backwards Euler is for stiff differential equations, and simple fixed-point iterations are ill-suited for stiff implicit systems, primarily because the iteration (4.12) is not a contraction unless the step size $$h$$ is incredibly small. Even the authors contradict themselves later on in the book, when in section 8.5 they state:

“We illustrate the difficulty in solving the finite difference equations by considering the backward Euler method. ... For convergence we would need to have

$\begin{split}\left| h\frac{\partial f(t_{n+1},y_{n+1})}{\partial y}\right| < 1.\end{split}$

But with stiff equations, this would again force $$h$$ to be very small, which we are trying to avoid. Thus another rootfinding method must be used to solve for $$y_{n+1}$$ in (8.42).”

In my experience, I have rarely seen the combination of an implicit time integration method coupled with a simple fixed-point iteration; for non-stiff problems explicit methods are vastly preferred, and for stiff problems more robust nonlinear solvers are required.

4. page 76, equation (5.40) should read:

$|e_{n+1}| \le (1+hL) |e_n| + h \tau_h(Y),\quad t_0\le t_N\le b.$
5. page 108, the final answer for problem 11 in chapter 6 should read:

$Y(t_{n+1}) - \left[Y(t_{n-1}) + 2h\,f(t_n, Y(t_n))\right] = \frac13 h^3 Y'''(t_n) + \mathcal O(h^4)$
6. page 151, equation (9.6) should read:

$D(r): \quad \sum_{i=1}^s b_i c_i^{k-1} a_{ij} = \frac{b_j}{k}(1-c_j^k), \qquad k=1,2,\ldots r, \quad j=1,2,\ldots,s.$
7. page 152, Table 9.5 has two incorrect coefficients for the Gauss method of order 6. The correct table should be

$\begin{split}\begin{array}{c|ccc} (5-\sqrt{15})/10 & 5/36 & 2/9-\sqrt{15}/15 & 5/36-\sqrt{15}/30\\ 1/2 & 5/36+\sqrt{15}/24 & 2/9 & 5/36-\sqrt{15}/24\\ (5+\sqrt{15})/10 & 5/36+\sqrt{15}/30 & 2/9+\sqrt{15}/15 & 5/36\\ \hline & 5/18 & 4/9 & 5/18\\ \end{array}\end{split}$
8. page 156, the final paragraph before section 9.3: The DIRK method in Table 9.8 (part b) is in fact B-stable, since it has eigenvalues $$\{1.55303,0,0\}$$, which are all non-negative.

9. page 161, problem 2, the first equation should read:

$z_{n,i} = y_n + h\sum_{j=1}^s a_{ij} f(t_n+c_j h, z_{n,j}), \quad i=1,2,\ldots s.$