Approximations

Question 1

what is a flop?

Answer 1

Floating Point OPERATION

Question 2

What is a flop per second?

Answer 2

a unit of measurement used to approximate the time it takes for an operation to be executed

Question 3

how do computers display n on integers?

Answer 3

they have to display floating points using a string of binary digits.

• the first binary digit stores the sign of the number: 0 if positive, 1 if negative
• the proceeding string of binary digits stores the value of the exponent; 2^(N-M), where N represents the number represented by the string of digits, and M=2^(N-1)-1
• the last string of digits stores the value of the mantissa(multiplying factor)

Question 4

how do you find the actual floating point value?

Answer 4

multiply the exponent and the mantissa, using the sign from the first digit to determine the sign of the number

Question 5

What do O(n^k) and o(n^k) represent?

Answer 5

they represent the Big O and Little O of a certain function - the value of k depending on whether n approaches zero or infinity.

Question 6

What does k present for the Big O function when n goes to infinity?

Answer 6

for the Big O notation, limit as n goes to infinity of F(n)/n^k = constant, so k represents the power of n that dominates the function

Question 7

What does k present for the Little O function when n goes to infinity?

Answer 7

limit as n goes to infinity of f(n)/n^k = 0

so k presents the smaller integer greater than the power of n that domminates the function

Question 8

What does k present for the Big O function when n goes to zero?

Answer 8

k presents the power of n that least dominates the function

Question 9

What does k present for the Little O function when n goes to zero?

Answer 9

k represents the greatest integer smaller than the power of n that least dominates the function

Question 10

suppose that x, is the computed approximation of Xo, what is the absolute error of x?

Answer 10

would be given by AE(X1)=abs(X1-Xo)

Question 11

whats the formula for the relative error of x1?

Answer 11

RE(x1)=abs(x1-xo)/abs(xo)

Question 12

what gives an estimate for the relative error?

Answer 12

epsilon * abs(x) , where epsilon is the relative precision on a computer

Question 13

what is a rounding error?

Answer 13

refers to the error associated with rounding a value to n number of sig figs

Question 14

what is confidence?

Answer 14

the no. of sig figs we can accuately say that the value of something is

Question 15

what can happen if we do arithmetic involving a large number and a comparably smaller number, and how can we alleviate this?

Answer 15

will cause catastrophic cancellation, can rearrange the expression with the catatropshic cncellation into a mathematically equivalent but numerically preferable expression