The Cooper Union for the Advancement of Science and Art 
ChE352 
Numerical Techniques for Chemical Engineers 
Professor Stevenson 
Lecture 17 The Cooper Union for the Advancement of Science and Art 
Exams = Monte Carlo integration 
•I can't ask you about every topic 
•I can't use a predictable sequence 
•Therefore, I must use a pseudo-random 
subset... 
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Midterm rules 
•You will have 90 minutes. 
•You may use only these resources: the two 
course textbooks (F&B and PNM), my slides, 
your own notes, your group's HWs, and Colab 
•You may use laptops and/or tablets, but not 
phones. 
•The exam will be graded based on your blue 
book. Show all your work clearly in your blue 
book and draw a box around each answer. 
•If you have a question, raise your hand. The Cooper Union for the Advancement of Science and Art 
Midterm rules 
•You will have 90 minutes. 
•You may use only these resources: the two 
course textbooks (F&B and PNM), my slides, 
your own notes, your group's HWs, and Colab 
–You can use my graded pdfs of your 
group's HWs, too 
–You can be on the Wifi in order to get to 
these resources like PNM and Colab, but 
no general internet usage 
–No using the AI features in Colab The Cooper Union for the Advancement of Science and Art 
Linear algebra 
5. (10 points) Define the following in one sentence (or less) each: 
A.Linear operator 
B.Non-singular matrix 
C.Positive definite matrix 
D.Dot product 
E.Eigenvector 
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Linear algebra 
5. (10 points) Define the following in one sentence (or less) each: 
A.Linear operator 
An operation transforming one vector into another which can be 
expressed as matrix multiplication by some matrix 
B.Non-singular matrix 
A matrix which has an inverse, aka can be solved in a linear system 
Ax = b. Also valid: full-rank and square, determinant != 0. 
C.Positive definite matrix 
A matrix for which the expression xTPx always gives a positive scalar 
for any nonzero column vector x of the correct length 
D.Dot product 
The operation of multiplying all corresponding entries of two vectors 
x1, x2 and summing all the results to produce a single scalar 
E.Eigenvector 
A characteristic vector x  of a given matrix H satisfying the expression  
Hx = ax  for some scalar a (the corresponding eigenvalue), meaning 
that the vector x does not change direction when multiplied by the 
matrix H, it is only scaled by a constant (the eigenvalue). Common mistake: not saying eigenvector depends on the matrix The Cooper Union for the Advancement of Science and Art 
Optimization formalism 
Or abbreviated as: 
Optimal value Objective 
function 
Constraint set 
Problem 
name 
●Which values are changing over the course of 
optimization? How? 
●Which values are not changing? The Cooper Union for the Advancement of Science and Art 
Interpolation/regression Methods 
•Linear regression 
•Polynomial regression 
•Lagrange polynomial interpolation 
•Piecewise linear interpolation 
•Cubic spline interpolation 
← Which method is this? 
When would you use 
each of these methods? The Cooper Union for the Advancement of Science and Art 
Numerical derivatives & integrals 
come from approximation methods 
•You have learned some methods for 
approximating a function f(x) based on 
individual data points ( Examples? )
•How can you use these approximation 
methods to get numerical derivatives and 
integrals? The Cooper Union for the Advancement of Science and Art 
Numerical derivatives & integrals 
come from approximation methods 
•You have learned some methods for 
approximating a function f(x) based on 
individual data points ( Examples? )
•We used functions like polynomials which 
have easy analytic derivatives & integrals 
•We can also use these to estimate the 
derivative & integral of the true function f(x) The Cooper Union for the Advancement of Science and Art 
Bisection Newton 
Convergence? 
Always 
converges? 
Special 
conditions? 
Good when? Root-finding methods: what & why? The Cooper Union for the Advancement of Science and Art 
Root-finding methods: what & why? 
Bisection Newton 
Convergence? Linear Quadratic 
Always 
converges? Yes No, if bad p0 or if 
we hit f’(pn) ≈ 0 
Special 
conditions? Need the 
bounds a, b Need a guess p0, 
need to have f’(x) 
Good when? We need 
stability We need 
accuracy & speed The Cooper Union for the Advancement of Science and Art 
Sensitivity analysis 
•How can you tell if an approximation is stable ?
Ariane 5 algorithm was 
conditionally stable The Cooper Union for the Advancement of Science and Art 
•How can you tell if an approximation is stable ?
•Try small  perturbations in input (“small” 
depends on the problem at hand) 
•If the output changes significantly  (as defined 
by the problem at hand), you have instability 
Ariane 5 algorithm was 
conditionally stable Sensitivity analysis The Cooper Union for the Advancement of Science and Art 
What is floating point? 
•Not every real number can be represented 
What numbers can’t be represented? How many decimal digits can we store in 23 bits? •Computer math is almost always floating point  
•Like scientific notation with powers of 2 only The Cooper Union for the Advancement of Science and Art 
•np.float32 holds ~7 decimal digits 
•np.float64 holds ~16 decimal digits 
•Not every real number can be represented 
•Too big = overflow, too small = underflow 
•Only binary fractions (no exact 1/3, 1/5, etc) 
•Computer math is almost always floating point  
•Like scientific notation with powers of 2 only What is floating point? The Cooper Union for the Advancement of Science and Art 
Review: IVPs 
•When is an IVP like an integral? 
○Only when y' = f(t) , no y 
○Then y = ∫ f(t), can solve with quadrature 
•In general, y' = f(t, y) 
○We know t exactly, but y (after y0) is an 
estimate, so y' = f(t, y) is also an estimate 
○Subject to accumulating errors, so be careful 
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Review: IVP systems 
•Same idea, but u' = f(t, u) where u is a vector 
•Solving is similar to 1-D IVPs, but more values 
to keep track of 
•Vector math rules apply: vector + vector = 
vector, scalar * vector = vector, etc 
Euler for systems (w = our estimate for u): 
Which index i, j is the timestep? What is the other index? The Cooper Union for the Advancement of Science and Art 
Setting up IVP systems 
•Every ui corresponds to an element ui' = f(t, u) 
Vector function Vector function The Cooper Union for the Advancement of Science and Art 
Review: Euler for IVP systems 
What is the Euler 
step for this system? The Cooper Union for the Advancement of Science and Art 
Review: Euler for IVP systems 
f(t, u) for this IVP: 
Euler step definition: 
Euler step for this IVP: The Cooper Union for the Advancement of Science and Art 
Review: higher-order IVPs 
•Often we know only a higher-order derivative 
of our desired function y, like y'' = f(t, y, y') 
•Treat every derivative of y as an element of u 
in an IVP system: u1 = y, u2 = y'
•du2/dt = f(t, u1, u2)      du1/dt = u2•And initial conditions y0 & y'0