How many digits are recognized in binary code and what are these digits?
2 digits - 0 and 1.
What is 8 in binary?
1000
What is one of the main differences between double and single precision.
Double precision has 64 bits and single only has 32 bits, hence, making double precision more accurate.
In single precision, how many bits are set aside for the sign, exponents+bias, and mantissa, respectively?
1, 8, 23
In double precision, how many bits are set aside for the sign, exponents+bias, and mantissa, respectively?
1, 11, 52
In base 2, what is infinity stored as?
$$ (1.00...0)_2 \times 2^{(11...1)_2} $$
In base 2, what is the largest positive number that can be stored?
$$ (1.11...1)_2 \times 2^{(11...0)_2} $$
In base 2, what is the smallest possible number that can be stored?
$$ (1.00...0)_2 \times 2^{(00...1)_2} $$
What is machine precision, or eps?
The smallest recognizable difference between two floating point numbers of order.
What is larger, the machine epsilon of a double precision system or the machine epsilon of a single precision system?
The single precision system will have a larger machine epsilon.
For a given computer with $ d $ number of mantissa, what is the machine precision of that computer?
$$ eps = 2^{-d} $$
What are round-off errors?
Errors that occur because computers have a finite precision for storing numbers and making calculations.
What are truncation errors?
These are errors that occur when a computer is forced to approximate a mathematical procedure that would require infinite operations.
What is the difference between chopping and rounding?
In chopping, the computer stores up to a digit and leaves out the rest whereas rounding stores up to a digit in position P and if the following digit in position P+1 is greater than or equal to 5, the digit in P is rounded up.
What is total absolute error?
It is the combination of the rounding and truncation errors.$$ E^T = E^R + E^{TR} $$
What is the true error?
True error is the true solution minus the numerical solution.
What is the true relative error?
It is the true solution minus the numerical solution all divided by the true solution.
What is the percent error?
It is the true relative error times $ 100\% $.