# Errata for Functions of Matrices: Theory and Computation

• Page 3, (1.4): there is an unwanted right parenthesis in the $(1,m_k)$ position.
• Page 4, line 6: replace $E_k$ by $N_k$.
• Page 10, second paragraph of Section 1.2.6: $\psi(t)$ is the minimum polynomial for $n>5$ but not for $n=4$, because $F_4$ has a repeated eigenvalue 1.
• Page 27, line -8: replace “it” by “its”.
• Page 31, Problem 1.33: the displayed matrix should have an extra superdiagonal of 1s inserted immediately above the main diagonal, so that

$\notag A = \begin{bmatrix} 1 & 1 & \frac{1}{2!} & \frac{1}{3!} & \dots & \frac{1}{(n-2)!} \\ & 1 & 1 & \frac{1}{2!} & & \vdots \\ & & 1 & \ddots & \ddots & \vdots \\ & & & \ddots & \ddots &\frac{1}{2!}\\ & & & & \ddots & 1 \\ & & & & & 1 \end{bmatrix}.$

• Page 77, line -10: replace upper limit of sum by $k$.
• Page 80, line 1: should read “$[0/0]$, $[1/1]$, $[2/2]$, …”.
• Page 86: In lines 1 and 2 of Algorithm 4.14, n should be m, where T is block m-by-m.
• Page 97, last displayed equation should read

$\notag \|E_k\| \le c \|E_0\| + c \|E_{k-1}\| c \sum_{i=1}^{k-1} o( \|E_{k-1-i}\| ) \le c \|E_0\| + (1+(k-1)c) \cdot o( \|E_0\| ).$

• Page 101: in the proof of Theorem 4.23 the inequality $\rho(|A| + |B|) \le \rho(|A|) + \rho(|B|)$ is used but this inequality is incorrect. The proof can be fixed by making use of a norm for which $\|\, |A|\, \|$ is arbitrarily close to $\rho(|A|)$.
• Page 102: in (4.37), and three lines below it, $f(\rho(A))$ should be replaced by $\max_i |f(\lambda_i)|$.
• Page 114, line 6: remove the period inside parentheses and replace “It” by “it” on the following line.
• Page 120, line 5: should read “the $\lambda_i$ are”.
• Page 148: in the last equation of (6.30), the two occurrences of $E_k$ should be $\widetilde{E_k}$.
• Page 160, line -14 should read “if $A$ is a nonsingular $M$-matrix with $0< a_{ii} \le 1$ for all $i$ then all the iterates $X_k$ (or $Z_k$ in the case of iteration (6.19)) have the same properties.
• Page 165, line -3: $u_{ij} = {n-i+1 \choose n-j+1}$ should read $u_{ij} = {n-i \choose n-j}$.
• Page 161, Theorem 6.10 should say $A\in\mathbb{R}^{n\times n}$ and the sentence before the theorem should say “real $H$-matrices”.
• Page 196, insert “complete orthogonal decomposition” at the end of line -5.
• Page 217, line -3: the parentheses should read (“Berberian’s trick”).
• Page 224, line 9 of Algorithm 9.4 should read ω_{s+1+r} and on line 10 add, just before “quit”, “F = F_s”.
• Page 222, line 9: $M^n=0$ should read $N^n=0$.
• Page 228, in line 6 of Algorithm 9.6 i should be j; in line 8 of Algorithm 9.6 j should be i.
• Page 234, Table 10.1: formula in (3,2) position should read $Q e^T Q^*$.
• Page 241, second line of Sec. 10.3: “$\sigma\in\mathbb{C}$” should read “for integer $\sigma$”.
• Page 246, line 7: the first right parenthesis after $I$ should be deleted.
• Page 247, third line above Theorem 10.21: “scaling” should read “squaring”.
• Page 253, line 2: “larger” should read “smaller”.
• Page 261, in the first line of Section 10.7.3 a space is needed after “single shift”.
• Page 265, Problem 10.6 should read “cond_abs(exp,A)”.
• Page 266, Problem 10.16 should read “rounding errors in the squaring phase”.
• Page 272, second display from bottom: on left-hand side, $\kappa_{\log}(A)$ should read $\mathrm{cond}_{\mathrm{rel}}(\log A)$.
• Page 274, line 8 should read “$m$-point Gauss-Legendre quadrature rule”
• Page 275, line -10: extra right parenthesis needed before the final “|”.
• Page 280, line 1: the U should be in the mathcal font.
• Page 280: in (11.28), the conditions for the second case should be replaced by $|z| > \frac{1}{\delta} ~\mathrm{or}~~ |z+1| < \delta ~\mathrm{or}~ |z-1| < \delta$ for some suitable $\delta$ that depends on the unit roundoff.
• Page 283, line 11 of Algorithm 11.12: $L = 2G_0$ should read $L = 2G_1$.
• Page 283, 7 lines before Section 11.9: should read “Algorithm 11.2 can be combined”.
• Page 330, line 5: should read “on Hermitian matrices”.
• Page 332, 9 lines before Section B.16: should read “a complex addition requires 2 real additions”.
• Page 333, in equation (B.25) a right parenthesis should be added after $x_{j-1}$.
• Page 336:
• delete the “LU” line (appears in Table C.2).
• HM line should read “$2n^3$“.
• TY = B line should read “$n^3/3$“.
• HY = B line should read “$5n^3/3$“.
• Page 370, in solution to Problem 8.26, in (b) the operation count for (8.37) should be $3mn^2 + n^3/3$.