dterm Examination CSCI 3231 Spring, 2005 ___________

Your Name ..

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

___/_1. The approximation e^x 1 + x + x² is to be used over [-1,1]. The truncation error is approximately:

a. 1/6 x⁴ b. 1/24 x⁴ c. 1/6 x³ d. 1/24 x³

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

a. -1 + 4/3 x x² + 1/6 x³ b. -1 + 7/3 x 7/2 x² + 1/6 x³

c. -1 + 13/3 x 3/2 x² + 1/6 x³ d. 3 + 4 x . x² + 1/6 x³

.

Xx x	Y y
1	21 2
2	3
3	3 3
3 4	3 3

____3. When using Newtons method for solving the equation x²- 2 sin(x) + sin(x)² = 0, with the initial guess of x₀ = 0.47, one obtains the following results for x₀, x₁, x₂, x_3, etc_. :

0.4700, -13.0147, -7.0581, -3.9329, -1.9824, 0.7333, 1.0952, 0.9938, and eventually converges to a solution near 0.986106 (Note: there is also a solution at 0.0). Why is Newtons method having such a hard time converging?

a. The initial guess, x₀, is not sufficiently good

b. The derivative of x²- 2 sin(x) + sin(x)² is zero near x₀

c. Both a and b

d. None of the above

____4. The Lagrange form of the interpolating polynomial is:

a. always easier to compute than the Newton form b. less useful for theorem proving

c. more useful when f(x) is available d. easier to use when the f values might change

____5. The Richardson extrapolation method is:

a. a method for integrating f(x) b. a method for determining a rational approximation

c. a method for determining an approximate value for f(x) outside of the range

d. a good method for approximating the derivative of f(x) at a point

__c__6. The value 1/3 is approximately represented as a floating point number in single precision (32 bits) as:

a.

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____7. The nave Gauss elimination method is:

a. good for when the determinant of the system is zero b. difficult to use for linear equations

c. quadratically convergent d. faster than Gauss elimination with scaled partial pivoting

____8. The 3-point Gauss quadrature formula is:

a. difficult to use when f(x) is a polynomial b. of degree three

c. of degree four d. the highest degree quadrature formula with less than 4 nodes

____9. The adaptive Simpson quadrature method is:

a. useful for differentiating trigonometric functions

b. useful for integrating functions with singularities

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

d. never as efficient as an adaptive scheme based on the compound Trapezoid rule

____10. The approximation formula f(x) = (f(x+h)-f(x))/h is:

a. a good choice for approximating the derivative of f(x)

b. a good method for approximating the integral of f(x)

c. a good choice for approximating the function when f is a periodic function

d. subject to the problem of large rounding errors as h 0.

____11. Which of the following numbers is computer representable?

a. 10/3 b. 3.1 c. 3.01 d. 3.125

____12. The Taylor series is:

a. useful for estimating rounding errors b. not of much practical use

c. useful for deriving many formulae d. useful for establishing computer representability

____13. 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

____14. The formula, I (2/3)(f(-1)+f(0)+f(1)) for numerical quadrature is of degree:

a. 1 b. 2 c. 3 d. none of the preceding

____15. Apply the above formula to the problem of integrating x⁴ over the interval [-1,1]. The error in the result is: a. 4/15 b. 4/5 c. 1/6 d. 14/15

____16. Applying Richardsons extrapolation method to the problem of finding f(x) when x=0 , f(x) = sin(x), and h=0.5 gives (as its first, most primitive approximation):

a. 0.00000 b. 0.923712 c. 1.00000 d. 0.958851

____17. When using the bisection method to find the zero of the function shown below, the final answer is the root near: a. -0.961355 b. -0.322108 c. 0.525058 d. 0.985091

____18. The secant method generally converges faster (in terms of CPU time) than Newtons method when:

a. f(x) requires less time than that required for a floating point multiplication operation

b. f(x) requires more time than f(x) to evaluate

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

d. f(x) requires less than half the time required to evaluate f(x)

____19. 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

____20. 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

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4. d, as its formula has difference

5. d, Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to thetrapezoid rule, and the BulirschStoer algorithm for solving ordinary differential equations.

7. a, nave Gauss elimination method is one of the most popular method for solving simultaneous linear equations.

8. c, The 3-point Gauss quadrature formula is a degree of four

9. b, Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of numerical integration

10. a, it is a differentiation formula

11.All numbers are computer representable

12. a