Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

8 CHAPTER 1. INTRODUCTION
Exercises
(1.1) Suppose f ∈ O ( h k) . Show that f ∈ O (h m ) for any m with 0 < m < k. (Hint: Take
h ∗ < 1.) Note this may appear counterintuitive, unless you remember that O ( h k) is a better
approximation than O (h m ) when m < k.
(1.2) Suppose f ∈ O ( h k) , and g ∈ O (h m ) . Show that fg ∈ O ( h k+m) .
(1.3) Suppose f ∈ O ( h k) , and g ∈ O (h m ) , with m < k. Show that f + g ∈ O (h m ) .
(1.4) Prove that f(h) = −3h 5 is in O ( h 5) .
(1.5) Prove that f(h) = h 2 + 5h 17 is in O ( h 2) .
(1.6) Prove that f(h) = h 3 is not in O ( h 4) (Hint: Proof by contradiction.)
(1.7) Prove that sin(h) is in O (h).
(1.8) Find a O ( h 3) approximation to sin h.
(1.9) Find a O ( h 4) approximation to ln(1+h). Compare the approximate value to the actual when
h = 0.1. How does this approximation compare to the O ( h 3) approximate from Example 1.7
for h = 0.1?
(1.10) Suppose that f ∈ O ( h k) . Can you show that f ′ ∈ O ( h k−1) ?
(1.11) Rewrite √ x + 1 − √ 1 to get rid of subtractive cancellation when x ≈ 0.
(1.12) Rewrite √ x + 1 − √ x to get rid of subtractive cancellation when x is very large.
(1.13) Use a Taylor’s expansion to rid the expression 1 − cos x of subtractive cancellation for x
small. Use a O ( x 5) approximate.
(1.14) Use a Taylor’s expansion to rid the expression 1 − cos 2 x of subtractive cancellation for x
small. Use a O ( x 6) approximate.
(1.15) Calculate cos(π/2 + 0.001) to within 8 decimal places by using the Taylor’s expansion.
(1.16) Prove that if x is an eigenvector of A then αx is also an eigenvector of A, for the same
eigenvalue. Here α is a nonzero real number.
(1.17) Prove, by induction, that if λ is an eigenvalue of A then λ k is an eigenvalue of A k for integer
k > 1. The base case was done in Example Problem 1.13.
(1.18) Let B = ∑ k
i=0 α iA i , where A 0 = I. Prove that if λ is an eigenvalue of A, then ∑ k
i=0 α iλ i is
an eigenvalue of B. Thus for polynomial p(x), p(λ) is an eigenvalue of p(A).
(1.19) Suppose A is an invertible matrix with eigenvalue λ. Prove that λ −1 is an eigenvalue for
A −1 .
(1.20) Suppose that the eigenvalues of A are 1, 10, 100. Give the eigenvalues of B = 3A 3 − 4A 2 + I.
Show that B is singular.
(1.21) Show that if ‖x‖ 2
= r, then x is on a sphere centered at the origin of radius r, in R n .
(1.22) If ‖x‖ 2
= 0, what does this say about vector x?
(1.23) Letting x = [3 4 12] ⊤ , what is ‖x‖ 2
?
(1.24) What is the norm of
⎡
⎤
1 0 0 · · · 0
0 1/2 0 · · · 0
A =
0 0 1/3 · · · 0
?
⎢
⎣
.
. . . ..
⎥
. ⎦
0 0 0 · · · 1/n
(1.25) Show that ‖A‖ 2
= 0 implies that A is the matrix of all zeros.
(1.26) Show that ∥ ∥ A
−1 ∥ ∥
2
equals (1/|λ min |) , where λ min is the smallest, in absolute value, eigenvalue
of A.
(1.27) Suppose there is some µ > 0 such that, for a given A,
‖Av‖ 2
≥ µ‖v‖ 2
,