Please answer each question by entering the most nearly correct answer (a, b, c, d) in the blank on the left.

____1. The approximation cos(x) 1 - x2 /2 + x4 /24 is to be used over [-1,1]. The absolute value of the truncation error is approximately:

a. x6 /6! b. x4 /24 c. 1/6 x3 d. x5

____2. The polynomial that interpolates the data shown in the table can be written as:

a. 1 - x2 b. 1 + x 7/2 x2 + 1/6 x3

c. 1 - x2 + 1/6 x3 d. 1 - x2 - 1/6 x3

.

Xx x Y y

0 21 1

1 2/3

2 3 1/3

3 3 3 1

____3. Use Newtons method to solve the equation x - sin(x-1) -1 = 0, with the initial guess

of x0 = 0.5. After 2 iterations, the approximation, x2 , is about

a. 0.69712

b. 0.66072

c. 0.77912

d. None of the above

____4. The Newton form of the interpolating polynomial is:

a. usually faster to compute than the Lagrange form b. less useful for theorem proving

c. harder to use when the f values are subject to change d. all of the above

____5. The error in the approximation f(x) (f(x+h) f(x-h))/ (2h) is:

a. O(h) b. O(h2)

c. O(h3) d. O(h4)

6. Consider the following system of linear algebraic equations:

x + y z = 1

x y + z = 1

x + y = 2

Is it possible to use nave Gauss elimination to solve this system? _____

If it is not possible, state why. If it is possible, use it to solve the system.

____7. The value 1/3 is approximately represented as a floating point number in single precision (32 bits) as:

a.

0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

b.

1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

c.

0 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1

d.

1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1

____8. The 2-point Gauss quadrature formula is of degree:

a. one b. two

c. three d. four

____9. An adaptive quadrature method is:

a. hard to use when derivative of f(x) is not known

b. does not require an estimate of the error in the approximation at any time

c. a poor method to use when there is a wide variation in the behavior of f(x) over the range

d. none of the above

____10. Consider the following Java program:

public class Flot {

public static void main(String args[]) {

double a = 0.1;

double x = 0.0;

double error;

int i;

for (i=1; i<=10; i++) x = x + a;

error = x - 1.0;

System.out.println("x = " + x + ", error = " + error);

}

}

The output from the above program is:

x = 0.9999999999999999, error = -1.1102230246251565E-16

Why is the value of x not equal to one?

____11. If possible, when generating an interpolating polynomial, the points should be:

a. evenly spaced b. unevenly spaced with the highest density near the middle of the range

c. spaced with the highest density closest to the right limit of the range

d. spaced with the highest density of points near the limits of the range in proportion to the spacing of zeroes of the Chebyshev polynomials

____12. When using the bisection method to find the zero of the function shown below when the initial interval (a,b) is (-2,2), the final answer is the root near:

a. 0.2500 b. 1.0354 c. 1.8208 d. -1.3208 e. -0.5354

____13. The secant method generally converges slower (in terms of CPU time) than Newtons method when:

a. f (x) is ex - x

b. f(x) is e2x - x

c. f(x) requires less time than f(x) to evaluate

d. all of the above

____14. The Gauss-Seidel method is guaranteed to converge when the coefficient matrix of the linear system of equations is:

a. Tridiagonal b. Diagonally dominant c. Pentadiagonal d. Non-singular

____15. When trying to solve numerically the equation, x - 0.9 sin(x) - 0.2 = 0, using

x -1 = 0.5 and x0 = 0.75 as initial guesses, the Secant Method gives for the approximation x2:

a. 1.04618725

b. 1.70168725

c. 0.90160425

d. 0.80160425