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.