I need help for 6 questions for numerical analysis

Exam #1: MA-4313/6313, Due Fri 6/18, 2021, 2:00PM 1

MA-4313/6313 Numerical Analysis I

Exam #1: Due Fri 6/18, 2021, 2:00PM

Name: Net ID:

• Each problem is worth 20 points. • Solve all problems using formulas and your calculator, if necessary; do not make

computer implementation. • Print out this problem set and show your solutions and answers on it; if you need

more space, you may add extra pages to the report.

1. Answer the assertions by “Yes" or “No" for each of {f, g}-pairs under the given conditions

{f, g} f = O(g) f = o(g){ f(x) =

18

x +

e−x

6 , g(x) =

1

x−2021

} , as x →∞

{f(x) = x− sin x, g(x) = x} , as x → 0

{ f(h) = h(1−eh), g(h) = h

} , as h → 0

{ f(n) = n4 −5n, g(n) = 3n2

} , as n →∞

Exam #1: MA-4313/6313, Due Fri 6/18, 2021, 2:00PM 2

2. Consider the sequence xn defined recursively as  x0 = 1xn+1 = 3

2 +

1

6 cos(2xn), n ≥ 0

(a) Use the fixed-point theorem to prove the sequence is convergent for any x0 ∈ [0, 2]. (b) How many iterations are required to get an approximate solution within a four-

digit accuracy?

Hint : You may use the inequality |p−xn| ≤ Kn|p−x0|, where p is the fixed point.

Exam #1: MA-4313/6313, Due Fri 6/18, 2021, 2:00PM 3

3. Consider the polynomial P(x) = 3x3 −3x2 −4x + 13

(a) Use Horner’s algorithm to find P(−1) and P ′(−1). (b) Perform two iterations of the Newton’s method to find a real-valued root, start-

ing with x0 = −1 and applying Horner’s algorithm (synthetic division) for the eval- uation of P(xk) and P ′(xk).

Exam #1: MA-4313/6313, Due Fri 6/18, 2021, 2:00PM 4

4. Using Newton’s method, the square root of Q can be found as follows: Let f(x) = x2 −Q. Then f ′(x) = 2x so that

xn+1 = xn − f(xn)

f ′(xn) = xn −

x2n −Q 2xn

= 1

2

( xn +

Q

xn

) .

For example, when Q = 2, the above formula becomes

xn+1 = xn 2

+ 1

xn . (1)

In order to prove the sequence generated by (1) converges, verify the following.

(a) When 0 < x0 < √ 2, we have x1 >

√ 2.

(b) When xn > √ 2, we have

√ 2 < xn+1 < xn.

So the sequence converges. (c) Apply lim

n→∞ to both side of (1) to find the limit.

Hint : (a) Consider x21−2 and show it positive. (b) For xn+1 < xn, consider xn−xn+1 = xn( 1 2 − 1

x2n ),

which you have to show positive. For the other part, try to verify x2n+1 −2 > 0.

Exam #1: MA-4313/6313, Due Fri 6/18, 2021, 2:00PM 5

5. Find the natural cubic spline S = {S1, S2} for the data:

x −1 0 2 y 1 3 19

Hint : Formulas: Let zi = S′′(xi), i = 0, 1, · · · , n. Then

Si(x) = zi−1(xi −x)3

6hi +

zi(x−xi−1)3

6hi

+ (yi hi −

1

6 zihi

) (x−xi−1) +

(yi−1 hi −

1

6 zi−1hi

) (xi −x).

where hi = xi −xi−1 and

hi zi−1 + 2(hi + hi+1) zi + hi+1 zi+1 = 6(yi+1 −yi)

hi+1 −

6(yi −yi−1) hi

.

Exam #1: MA-4313/6313, Due Fri 6/18, 2021, 2:00PM 6

6. Let f ∈ C1[a, b] and xi = a + i h, h =

b−a n

,

a partition of the interval [a, b]. Suppose that H is the piecewise cubic Hermite in- terpolation of f at {xi}. Then, what can you say about the error |f(x) − H(x)|? Show your analysis in detail.

Hint : Use the Hermite Interpolation Theorem, Theorem 3.38.