1.3. Propagation of errors the function rule.
The function rule.
After one makes an approximation, there are often further computations that use these approximate values.
We would like to estimate the error in a computed value from the errors in the
1.8 Binary floating point numbers
Computers often use base 2 for their representation of floating point numbers. A number x is expressed in
binary (base 2) floating point form if it is written as a signed number with magnitude between 1 and 2
1.5 Floating point numbers and round-off errors.
Floating point numbers.
Round-off errors are due to the fact that people, calculators, and computers usually do not keep track of or
store numbers exactly in the course of a series of calculations. Sc
Name: _ This is a closed book exam. You may use a calculator and the formulas handed out
along with the exam. Show your work so I can see how you arrived at your answers. (This is particularly important
if I am to be able
(25 points) The New Youth Plastic Surgery Clinic performs face-lifts. Currently the clinic
performs 50 face-lifts / week for which they charge $10,000 per face-lift. Currently for each
face lift the clinic pays a plastic surge
Approximations and Errors
x = true value of something
xa = approximate value of something (e.g. measured or computed)
x = x - xa = (signed) error in xa
= (x) = (x,xa) = | x - xa | = absolute error
= (x) = (x,xa) = | | = relativ
1.7 Other ways to specify errors.
Number of significant digits and intervals.
In section 1.3 we discussed how one could specify the error in an approximate value by means of the
absolute error and the relative error. In this section we look at two o
1.4 The algebra of errors.
The function rule estimates the error in a computed value from the errors in the values it is computed from
in a single step. However, for simple functions it may be simpler to use the following rules.
Proposition 1. Suppose xa
Approximations and Errors
1.1 Taylor series.
Taylor series and their error.
Mathematical approximations are necessary because on most computers the only built-in operations are
addition, subtraction, multiplication, and division. In order to do ot
1.6 Round-off errors in floating point computations.
When people or computers do computations with floating point numbers, they usually round the result of
each arithmetic operation to a certain fixed number of digits of precision.
1.2 Absolute and relative errors.
In mathematics, science, and engineering we calculate various numbers, such as the current in an electric
circuit, or the viscosity of the transmission fluid in a car, or the price of Ford Motor Company stock a year