MATH 4650-001 & SC 4656-001: Numerical Analysis I. Fall 2002

University of Colorado Denver INSTRUCTOR Dr. Andrew Knyazev

NA I Trial Final on Chapters 4-6. December 4, 2002.

1. Find the polynomial of the least degree that interpolates this table:

x	0	1
y	0	1

[A]   x  [B]  x² [C]  x³ [D] x+x² [E] None

Solution: See p. 137.

2. Let f(x)=x³. Compute f[0,1]:

[A]   x [B]     0 [C]    1 [D] 2 [E] None

Solution: See p. 145.

3. Let f(x)=x⁵ . Compute f[0,1,2,3,4,5,6,7,8,9]:

[A]   x [B]     0 [C]   1 [D] 2 [E] None

Solution: See p. 168.

4. Find the best error estimate of  interpolation error of function sin(x) on the interval [0,1] by using a polynomial of degree 2 and equally spaced nodes in [0,1] including the endpoints:

[A] cos(x) [B] 0 [C] 1/2 [D]  3/8 [E] None

Solution: See p. 166.

5. Which of the following formulas can serve as an approximation of f´(x):

[A] f(x)  [B]  f(x+h)-f(x) [C] (f(x+h)-f(x))/h   

[D] (f(x+h)-2f(x)+f(x-h))/h² [E] None

Solution: See p. 172.

6. Which of the following expressions describes the precision of the trapezoid rule with uniform spacing h applied to a function with continuous second derivative on a finite interval:

[A] 0  [B] o(h) [C] O(h) [D] O(h²) [E] None

Solution: See p. 198.

7.  Compute R(1,1) in the Romberg algorithm applied to the function f(x) = x on the interval [0,2]

[A] 1 [B]     2 [C] 5 [D] 10 [E]   None

Solution: See pp. 210-211.

8.  Apply the basic Simpson's rule to approximate an integral of f(x) = sin(x) on [0, 2 pi]:  

[A] sin(x) [B] 0 [C] 1 [D] pi [E] none

Solution: See p. 222.

9.  The Gaussian quadrature with 5 nodes will be exact for all polynomial of degree at most

[A] 4   [B]    5 [C]   8 [D]     9 [E] None

Solution: See p. 230. Here, n+1=5 nodes, so n=4.

10.  The number of multiplications or divisions in the forward elimination phase of the Gaussian elimination with scaled partial pivoting applied to a general k-by-k matrix is approximately

[A]  k [B]  k² [C] k³ [D] k³/3  [E] None

Solution: See p. 264.

11.  The number of multiplications or divisions in the forward elimination phase of the naive Gaussian elimination applied to a tridiagonal k-by-k matrix is proportional to 

[A]  k [B]  k² [C] k³ [D] k³/3  [E] None

Solution: See Problem 6.3.2.