### Numerical Analysis: Mathematics of Scientific Computing, 3rd Edition David Kincaid & Ward Cheney Errata List

PREFACE

• Dedicated, in memoriam, to our parents Sarah and Robert B. Kincaid, and Carleton Elliot Cheney.
• (Page x, Line +9)
• (Page xii, Line +6)
Replace Examples of general-purpose mathematical libraries ...''
with References to general-purpose mathematical libraries ...''
• (Page xii, Line -14)
• (Page xii, Line -6 to -5) omit sentence:
"Such sections are marked with an asterisk."

CHAPTER 1

• (Page 16, Lines 14-27) An example of a sequence that converges to $\sqrt{2}$ is $x_{n+1} = x_n - (x_n^2 - 2)[\frac{x_n - x_{n-1}}{x_n^2 - x_{n-1}^2}$ Selecting two initial values, we have \eqalign{x_1 &= 2\cr x_2 & 1.5\cr x_3 &= 1.42857\,1\cr x_4 &= 1.41463\,4\cr x_5 &= 1.41421\,6\cr x_6 &= 1.41421\,4 The convergence to $\sqrt{2}=1.41421\,3562\ldots$ is quite rapid. Using double-precision computations, we find numerical evidence that $\frac{|x_{n+1} - \sqrt{2}|}{|x_n - \sqrt{2}|^1.62}\le 0.77$ which corresponds to {\bf superLinear convergence}.
• Page 35, Problem 19: Should read: $x(\lambda)=[\lambda, \lambda^2, \lambda^3, \ldots]$

CHAPTER 3

• Page 127, Caption Figure 3.8
Instead of "... $p(x) = z^5 + 1$" it should read: "...$p(z) = z^5 -1"$
• Page. 128, Line 5
Instead of $p(z) = z^5+1$ should read: $p(z) = z^5-1$

CHAPTER 4

• (Page 207, indent next to last line in top psuedo-code) ... \qquad {\bf output} $k+1, x^{(k+1)}$\\ {\bf end}
• (Page 190, Line 5)
Replace "from Definition (6)" with "from Equation (6)"
• (Page 203, Line 5)
Replace exponent $m$ with $m+1$ to read: $||I - BA||^{m+1} ||x^{(0)}- x||$
• (Page 215, Theorem 5, Line 3,5)
Replace "For the iteration formula" with "In order that the iteration formula"
Replace "to produce a sequence" with "produce a sequence .."
• (Page 230, Problem 18) Should read: "Prove that if $A$ is symmetric positive definite, then ..."

CHAPTER 5

• (Page 258, in paragraph "To simplify, ...")
Note that $\varepsilon^{(k)}$ is a vector.
• (Page 260, Line -10)
add equation number (5) to last displayed equation proof of THEOREM 1.
• (Page 265, 1st line of Proof should read:
Let $A$ and $B$ be similar matrices; that is, for some nonsingular matrix $P$,
• (Page 268, (2,3) entry of matrix $T$ should be 125 not 1250 so that the second row of matrix $T$ reads:
0 465 125
• (Page 271, Problem 5.2.9)
Remove Hint: Use Gershgorin's Theorem.
• (Page 281) At the beginning of the last paragraph, the sentence should read: The process described above terminates after $\min (m-1, n)$ iterations for an $m \times n$ matrix $A$.
• (Page 282, Line -4)
Remove $\alpha$ to read: $v = (1.3166, -0.50637, -0.10127)^T$
• (Page 283, near top of page)
Note: For the $m \times n = 4 \times 3$ matrix, the number of iterations is $min (m-1, n) = min(3, 3) = 3$.
• (Page 288, Line +1)
Replace (See Problem 5.3.39, p. 287)'' with (See Problem 5.4.39, p. 298)''
• (Page 301, pseudocode) Run for-loop top down.
• (Page 305) The computation of the Hessenberg matrix by row and column operations is in error. All the matrices are correct except for the last one. There should be two steps in going between the last two matrices: the row operations between rows 3 and 4 and the inverse column operations between columns 4 and 3. Note: The entries $(1,4)=7$ and $(2,4)=-2$ should remain the same. For the matrix $$A = [\matrix{-3 3 7 2\\ 1 2 3 -5\\ 2 -1 0 3\\ 4 2 -2 4}]$$ results are $$H=[\matrix{-3 6.25 1.1935 7\\ 4 3.5 2.51613 -2\\ 0 -3.875 0.59677 3.5\\ 0 0 -2.10406 1.90323}]$$ or $$H=[\matrix{-3 25/4 37/31 7\\ 7/2 78/31 -2\\ 0 -31/8 37/62 7/2\\ 0 0 -2022/961 59/31}]$$ The original matrix $A$ and the Hessenberg matrix $H$ now have the same eigenvalues. Consequently, the Hessenberg matrix in the textbook is in error!!

CHAPTER 6

• (Page 326, Proof, second set displayed equations, bottom summation should be from $j=0$ to $n$ NOT $m$ and reads: $$sum_{j=0}^n b_j t_j^i = 0 \qquad (0\le i \le m)$$
• (Page 331, Solution Example 1, Line -2): Change to read: 5 2 | 2
• Page 354, Some errors in the numbers listed in the table. See p. 354 Now you will get the same results by using either an implementation of the algorithm in the book or pp=csaps(x,y,1) from the MATLAB spline toolbox.
• (Page 359, Line -4): Change upper limit in summation from m to n to read: $\sum_{j=0}^n b_j t_j^i = 0$
• (Page 361, Problem 6.4.1: Refer to the tridiagonal algorithm for determining the values of $z_i$. Prove that $u_i z_i + h_i z_{i+1} - v_i = 0$ for all $i=n-1,n-2,...,1$.
• (Page 450, Problem 4) Should be $f(x_j)=\langle g, E_{-j}\ranlge_n$

CHAPTER 7

• (Page 495, Line +3): Change to : $x^2 - \frac13$
• (Page 498, Problem 7.3.3, Last two displayed lines): Should be x_0 = -x_2 = ... and x_1 = -x_3 = ...
• (Page 510, Equation (10) right-hand side: change 30 to 60
• (Page 511, pseudo-code)
Line -12, right-hand side: change 30 to 60
Lines -5 to -2: move indentation left four spaces
Line -3: Third vector component should be $v_4^{(k-1)}$ not $v_5^{(k-1)}$
Lines -1: change "end do" to "end while"

CHAPTER 8

• (Page 525, Line 4 after (3)) Replace "we can expect that at some finite value of t there will be no solution; that is, $x(t)=+\infty$." with "we should be prepared for a solution that has a vertical asymptote."
• (Page 54, Problem 8.3.6, Line 1)
Change "order 4" to "order 5"
• (Page 547, Computer Problem 8.3.6, Line 3)
Change $-(2+\sqrt{2})F_2$ to $+(2-\sqrt{2})F_2$
• (Page 548, Computer Problem 8.3.9)
Line -2: Change 891/8329 to 891/8320
Line -1: Change -539/394 to -539/384
• (Page 571, Computer Problem 8.6.3) Change differential equations to read: x'_1 = -13x_1 + 6x_2; x'_2 = -13x_2 - 6x_1
• (Page 602, Equation (15))
Remove extra zero from right hand side vector.

CHAPTER 9

• (Page 617, Line 9)
Add: ... can be used are these from (3) p. 466, (8) p. 468, and (9) p. 469:
• (Page 617, Line 13)
Add: ... affording various degrees of precision. (See problems in Section 7.1.)
• (Page 618, Figure 9.3)
Horizontal axis should be labeled as $g(x)$ not $g(t)$.
• (Page 619, Line 7)
Replace sentence: "An analysis following the algorithm will show why this is true."
with "An analysis in the following paragraphs will show why this is true."
• (Page 620, Line 1)
Replace sentence: "Recall that $\rho(A)$ is the special radius of matrix $A$."
with "Recall that $\rho(A)$ is the spectral radius of matrix $A$."
• (Page 633, Line -1 above Problems 9.3)
Insert phrase: when programmed in double precision.
• (Page 642, restate problem 4)
Prove that if the Dirichlet problem defined by Equations (7) and (8) has a solution, then it has a solution that satisfies the symmetry conditions of Problem 3.

CHAPTER 10

• (Page 703, Line +6)
Remove transpose in displayed equation $A=AD^T$ to read: $A=AD$
• (Page 703, Lines -10,-4,-3,-1)
Change $d_q$ to $e_q$ to read:
where $y=x-\lambda D_q - \lambda e_q$ (Line -10)
$=c^Tx+\lambda(c_q-e_q)$ (Line -4)
.. we shall select $q$ so that $c_q>e_q.$... (Line -3)
to select $q$ so that $c_q-e_q$ is as large as possible. ... (Line -1)
• (Page 705, Line +2, +3)
Remove transpose on $D^T$ (3 times) to reads:
... Then $Ax=b=Au=A(Du).$ It follows that $x=Du$ because $Ax$ (twice in Line 2 )
and $ADu$ are Linear combinations ... (Line 3)
• (Page 705, Line -2)
Change $D_q$ to $D_3$ to read: $x - \lambda D_3 =$ ...
• (Page 706, Line +2)
Change $D_q$ to $D_3$ and $d_3$ to $e_3$ to read:
$y = x - \lambda D_3+\lambda e_3 = [0\quad 1\quad 1\quad 0\quad 0]^T$

CHAPTER 11

• (Page 711, Section 11.5 title, Line 8)
• (Page 711, Line -10)
Replace "Nocedal-Wright [1999] has written a recent textbook..." with "Nocedal-Wright [1999] is a recent textbook ..."
• (Page 714, Line -12)
For this case, these replacements should be carried out in strict order:
• (Page 714, Line -6)
Insert "in strict order" to read: v
The replacement needed are then in strict order:
• (Page 715, Problem 11.1.1, Line 2)
Insert "approximately" to read: ... by a factor of approximately 0.62.
• (Page 721, ALGORITHM 1, Step 1)
Insert "Compute" and replace equal sign ($=$) with left arrow ($\leftarrow$)
• (Page 722, Line 4)
Remove comma before where to read: "...on the ray $x^{(k)}+tv^{(k)}$ where $f$ has ..."
• (Page 722, Section 11.5: title, Line 1)
• (Page 723, Line 16)
Replace "the relative \textbf{flatness} is small, which is the quantity" with "the relative \textbf{flatness} is small. This is defined to be the quantity"
• (Page 723, end of Section 11.5, Line -5, -2)
• (Page 727, Line -6) Change Aubin [2000] to Aubin [1998]
• (Page 728, Line 10)
At beginning of displayed equation change $\lambda_i u$ to $\lambda u_i$ and add $=f_i(\omega)$ to the end to read:
$\lambda u_i + \theta v_i > \lambda f_i(x) + \theta f_i(y) \ge f_i(\lambda x + \theta y) = f_i(\omega)$

BIBLIOGRAPHY

• (Page 747, add missing reference) Aubin, J. P. 1998. Optima and Equilibria: An Introduction to NonLinear Analysis, 2nd ed. New York: Springer-Verlag
• (Page 761, Line -12)
Nelder, J.A., and R. Mead, 1965. ...

INDEX

• (Page 778, add to of top second column with overhang of two characters)
Lemma on (continued)
• (Page 780, column 1, Line 5)
• (Page 785, 786, add to of top second column with overhang of two characters)
Theorem on (continued)
• (Page 787, add to top of first column with overhang of two characters)
Theorem on (continued)
• (Page 787, add to second column, after line 9)
Traveling salesman problem, 724-725

### Instructor's Solution Manual for Numerical Analysis: Mathematics of Scientific Computing, 3rd Edition David Kincaid & Ward Cheney Errata List

Thus, there are $2\cdot 2^{23}(2^8 - 2) = 2^{25}(2^7 -1) =2^{25}(127) = 4,261,412,864 different numbers. • (Problem 2.2.3) Insert at beginning to solution: No. For example, consider the case$x = 5$. Add to end of Hint: See Abramowitz-Stegun Handbook online at {\tt www.math.ucla.edu/~cbm/aands/} or doing a Google search for it. • (Problems 2.2.4 \& 2.2.5) Interchange solutions to Pbs 2.2.4 and 2.2.5. • (Problem 3.1.2b) Should read$| r - c_n | \le 2^{-(n+1)} (b_0 - a_0) = 2^{n}\$