This is Intermediate Numerical Analysis I.
I am Stephen Preston from the Math department.
Stephen.Preston@colorado.edu
The course web site is http://math.colorado.edu/~prestos/math4650
It will be linked from both the math and applied math pages.
Office hours (tentative): Math bldg. 302, MWF 12-1.
I am also available other times, or by appointment.
Course syllabus (also somewhat tentative; will be finalized this week)
Course schedule
Today's plan: Overview of numerical analysis.
Basic principle: Know what the answer is before you run the program.
Example:
The harmonic series is
This is infinity!
A simple calculus proof (the integral test):
The area of the yellow rectangles
is the harmonic series.
The area in green is the area under
the curve y = 1/x,
which we know is infinite.
Hence the harmonic series must
be at least as large,
and thus it's also infinite.
What happens if we program a computer to compute this sum?
| > | # A bad Maple program to compute the sum of the harmonic series. |
| > | # We simulate k-digit arithmetic and keep going until the digits stop changing. |
| > | k := 6: # Set number of significant digits.
total := 0: # Initialize the sum. n := 1: # Initialize the index. err := 1: # Initialize the error to something nonzero. while err > 0 do newtotal := evalf[k](total + 1/n): # Keep adding 1/n to the sum and round off to six digits. err := evalf[k](newtotal-total): # Check whether the digits have changed. total := newtotal: # Reset the sum. n := n+1: # Increment the index. end do: print(n,total); # Print results. |
- What general problems do you see with this program?
- Why did the program stop at n = 20001?
- What is the significance of the output 10.7624?
- Could we program this so that the computer actually finds out the answer doesn't exist?