1.2 Round-off Errors and Computer Arithmetic Flashcards
List the 3 machine number properties (base, precision, and range)
(a) Each machine number has an integer base, beta > 1
(b) Each machine number has an integer precision, k >= 1 (basically the # of sigfigs)
(c) Each machine has an integer exponent, n, which falls within a given range (n_min, n_(min+1), … , n_(max-1), n_max)
What is the floating point number system?
The set of numbers
x = (+- 0.d1d2d3d4…dk)beta^n
where
d1 = {1,2,3, ... , beta-1}
d2, d3, ..., dk = {0,1,2,3 ... , beta-1}
n_min <= n <= n_max
the number d1d2d3...dk is called the mantissa
Having an unevenly spaced number line bounds the ________ error
relative
Having an evenly spaced number line bounds the ________ error
absolute
What is the IEEE Standard for double precision floating point numbers?
(a) base is 2
(b) Exponent is 11 bits in offset binary
(c) Mantissa is 52 bits (k = 53)
What is overflow?
When the program terminates or sets the answer to infinity once the calculation surpasses the realmax
What is underflow
When the program sets the answer to zero because the calculations drop below the realmin
What is chopping and rounding?
Chopping is when you simply drop off numbers you are not interested in and rounding is … well… rounding
|p - p*| represents the _____ _____
absolute error
|p - p*|/|p| represents the _____ _____
relative error (basically the percentage error)
What is the maximum relative error for IEEE double precision floating point?
Chopping: |x-fl(x)| <= beta^(1-k)
Rounding: |x-fl(x)| <= (1/2)beta^(1-k)
What is machine epsilon and what does it represent?
2.2*10^(-16). If two numbers are closer together than this number, then the computer cannot tell the difference
What is a cancellation error?
An error caused by subtracting nearly equal numbers
What is an amplification of round-off error?
An error caused by multiplying your round-off error by a factor of >1
What is an accumulation of round-off error?
An error caused by round-off errors being repeatedly added together
