The Cooper Union for the Advancement of Science and Art 
ChE352 
Numerical Techniques for Chemical Engineers 
Professor Stevenson 
Lecture 5 The Cooper Union for the Advancement of Science and Art 
Why is finding catalysts difficult? 
●The Schrödinger equation lets us calculate energy 
barriers for any reaction 
●But catalysts are still an open question - Why?  The Cooper Union for the Advancement of Science and Art 
How can we reason about speed? 
●Each computer's 
speed is different 
●Even on the same 
computer, timing 
will vary randomly 
●There is a way to 
describe speed that 
avoids all of this... 
The Cooper Union for the Advancement of Science and Art 
How can we reason about speed? 
●Each computer's 
speed is different 
●Even on the same 
computer, timing 
will vary randomly 
●There is a way to 
describe speed that 
avoids all of this... 
Big-O Notation The Cooper Union for the Advancement of Science and Art 
Big-O notation 
Big-O simplifies an expression by discarding 
everything but the largest term in the limit of 
large N, also dropping constant prefactors 
Run time Big-O Description 
   t = N/2 + 999 ? ?
   t = 3N2 + 5N ? ?
   t = 7N3 + 8N2 ? ?
   t = 2N + N9999 ? ?
This gets rid of computer speed & noise The Cooper Union for the Advancement of Science and Art 
Big-O notation 
Big-O simplifies an expression by discarding 
everything but the largest term in the limit of 
large N, also dropping constant prefactors 
Run time Big-O Description 
   t = N/2 + 999 O(N) Linear 
   t = 3N2 + 5N O(N2) Quadratic 
   t = 7N3 + 8N2 O(N3) Cubic 
   t = 2N + N9999 O(2N) Exponential 
This gets rid of computer speed & noise The Cooper Union for the Advancement of Science and Art 
Big-O notation examples 
O(N) algorithms act on each input a 
constant number of times 
How might an O(N2) algorithm arise? The Cooper Union for the Advancement of Science and Art 
Big-O notation examples 
O(N) algorithms act on each input a 
constant number of times 
How might an O(N2) algorithm arise? 
Doing N*N things, like building an NxN matrix The Cooper Union for the Advancement of Science and Art 
Big-O notation examples 
Doing N*N things, like building an NxN matrix The Cooper Union for the Advancement of Science and Art 
Big-O notation examples 
How might an O(2N), or 
worse, algorithm arise? The Cooper Union for the Advancement of Science and Art 
Big-O notation examples 
Trying every permutation (Traveling Salesman, 
Knapsack Problem, Schrödinger equation )How might an O(2N), or 
worse, algorithm arise? 
The Cooper Union for the Advancement of Science and Art 
Constant factors 
Big-O notation ignores constant factors: 
If you wait an hour at every step, 
your big-O stays the same! 
(You should still try not to do this) Run time Big-O 
 t ∝ 1,000,000  N O(N) 
 t ∝ 1,000,000  N2 O(N2)The Cooper Union for the Advancement of Science and Art 
What is fast and what is slow? 
Modern computer architecture from a speed 
perspective ( really fast, fast, medium,  slow)Memory  
(“RAM”) CPU 
(a “core”) GPU 
(“graphics card”) 
CPU 
(a “core”) CPU 
(a “core”) CPU 
(a “core”) CPU 
(a “core”) CPU 
(“cores”) Storage 
SSDDisk
Cache Internet The Cooper Union for the Advancement of Science and Art 
Why is code slow? 
•Big-O is too high (bad algorithm) 
–How would you find big-O? 
•Too much input/output (especially files) 
–Why is I/O slower than calculations? 
•Too much task switching 
–Not vectorized  (needs more numpy) 
• Convergence  is poor (algorithm issue) The Cooper Union for the Advancement of Science and Art 
Why is code slow? 
import timeit
def arithmetic_answer ():
   return 2 + 2
def file_answer ():
   with open('data.txt' , 'w') as f:
       f.write(f'{4}')
   with open('data.txt' , 'r') as f:
       answer = int(f.read())
   return answerWhat is "timeit"? 
What do these 
two functions do? 
Which one is 
faster? The Cooper Union for the Advancement of Science and Art 
Why is code slow? 
import timeit
def arithmetic_answer ():
   return 2 + 2
def file_answer ():
   with open('data.txt' , 'w') as f:
       f.write(f'{4}')
   with open('data.txt' , 'r') as f:
       answer = int(f.read())
   return answer
arithmetic_time  = timeit.timeit(arithmetic_answer , number=1000)
file_time  = timeit.timeit(file_answer , number=1000)
print(f'{arithmetic_time =:.1e}, {file_time =:.1e}')Try this code in 
Colab 
What % of 
file_time is the 
file writing vs the 
file reading? The Cooper Union for the Advancement of Science and Art 
Big-O for Getting Smaller 
•Big-O also works for numbers getting smaller, 
such as error α - αn shrinking as n grows 
○O(1/n + 1/n2) = ?The Cooper Union for the Advancement of Science and Art 
Big-O for Getting Smaller 
•Big-O also works for numbers getting smaller, 
such as error α - αn shrinking as n grows 
○O(1/n + 1/n2) = O(1/n) as n ➔ ∞
•Same idea if real-valued input h goes to zero: 
○O(h2 + h4) = ?For some K The Cooper Union for the Advancement of Science and Art 
Big-O for Getting Smaller 
•Big-O also works for numbers getting smaller, 
such as error α - αn shrinking as n grows 
○O(1/n + 1/n2) = O(1/n) as n ➔ ∞
•Same idea if real-valued input h goes to zero: 
○O(h2 + h4) = O(h2) as h ➔ 0For some K 
For some K The Cooper Union for the Advancement of Science and Art 
●Open Python Numerical Methods, go to 
Chapter 8.4: Summary and Problems 
https://pythonnumericalmethods.berkeley.edu
/notebooks/chapter08.04-Summary-and-Prob
lems.html 
●Do the first three problems, starting with: Summary and Problems 
○How would you define the size of the following tasks? 
■Solving a jigsaw puzzle. 
■Passing a handout to a class. 
■Walking to class. 
■Finding a name in dictionary.