Final exam Numerical Analysis

How many significant digits does the floating point number 0.05670 × 10³ have?

• 5

• 4

• 6

• 3

Bit 1 is used for the sign of the number, bit 2 for the sign of the exponent, bits 3-4 for the magnitude of the exponent, and the remaining twelve bits for the magnitude of the mantissa. What is machine epsilon for this comp

• 2^-16

• 2^-12

• 2^-8

• 2^-4

Let ˆx be some approximation of x. Which of the following error definitions is correct?

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• 
• 
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In floating-point arithmetic, which of the following operations on two positive floating-point numbers can produce an overflow?

• addition

• subtraction

• multiplication

• division

How many zeroes does a polynomial of degree n have?

• n + 2

• n + 1

• n

• n − 1

Let f(x) = x 2 − 2x + 1. The bisection method can be used to approximate the root of the function f(x). see the picture.

• True

• False

Which of the statements below regarding the convergence of the bisection method is TRUE?

• The iteration is always guaranteed to converge.

• The order of convergence is linear

• The asymptotic rate of convergence is 1 2 .

• All true

• All false

You are given an interval [a, b] where f(a) x f(b) < 0. Which of the statements below regarding the bisection method is TRUE?

• There must be a root x ^∗ somewhere in the interval

  (a, b).

• 
• 

The irrational number √ 2 can be approximated by applying the secant method to the nonlinear equation x ² − 2 = 0. What is the iteration formula?

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• 
• 
• 

The convergence of the secant method to simple roots is quadratic.

• True

• False