Numerical methods

PHY224 
Introduction to Scientific Computing
In this assignment we focus on coding practices more common in science, like numerical methods, calculating
statistics, and vectorization. You can use any Python libraries that are suitable for answering the question
unless excluded in the question, but you are responsible for their correctness.
1 numpy operations
Consider the following definitions of variables
import numpy as np
a = 1
b = np.array([3.0,2.3,1.0])
c = np.array([3,.3,.03])
d = np.array([[2,4],[4,6],[7,8]])
f = [9,90,900]
Write Python code to calculate and print the following calculations
1. The sum of a and b
2. The sum of c and b
3. The sum of c and d (adding c to each column of d)
4. The sum of b and f
5. The type variable stored in d, and in f
6. The len of d, and the len of f.
2 numpy functions
Consider the following definitions of variables (the array word shows the type of these variables, a numpy
array).
t= array([0. , 0.1155, 0.2287, 0.3404, 0.4475,
0.5546, 0.6607, 0.7753, 0.8871, 1. ])
y= array([ 0. , 0.1655, 0.2009, 0.1124,
-0.0873, -0.3996, -0.8197, -1.3977,
-2.0856, -2.905 ])
Write Python code to calculate and print the following calculations
1. The average of all the values in y
2. The sample standard deviation of values in y
3. The differential of y with respect to t. i.e. dy
dt .
4. The integral of energy R
E dt, where E =
1
2
y
2
.
One way of writing the differential dy
dt numerically is to approximate the dy with δyi = yi+1 − yi where yi
is
the ith element of the array y, and similarly for δti
. Then use dt
dy ≈
δy
δt . You might need to use a for loop to
calculate the value at each point in time and position.
One way of writing the integral is to calculate the value of E(yi), then multiply by δti and sum all of the
elements. This method is known as the Riemann sum.
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PHY224 University of Toronto
In both cases, numpy provides functions that can assist you in the calculation. Consult the numpy documentation for help.
3 Implementing equations in Python
A common task in scientific computing is to implement an equation in Python code. Consider the equation
for the period of a simple pendulum.
T = 2π
s
l
g
where l is the pendulum length, g is the acceleration due to gravity, and T is the period of oscillation.
1. Write a function that calculates and returns a value for the acceleration due to gravity given a
pendulum length and period of oscillation. Test your code for a pendulum of length 2.50m that oscillates
with a period of 5.16s.
2. Real measurements have uncertainties. Write a second function to propagate uncertainties in pendulum
length and oscillation period through to the uncertainty in the predicted gravitational accelerations.
i.e. calculate δg given l ± δl and T ± δT. Test your code and print the uncertainty assuming l =
2.40 ± 0.01m and T = 5.0 ± 0.01s. Do not use the uncertainties package for this exercise.
4 Conditionals
Conditionals are used to perform different calculations based on the input to the code. For example, a grading
scheme that is Pass/Fail at 70% could be written as
if grade > 70 :
mark = "pass"
else:
mark = "fail"
According to the University of Toronto grading scheme the following table provides the conversion from
numerical mark to letter grade and GPA:
Numerical Scale of Mark Letter Grade Grade Point Value
90-100 A+ 4.0
85-89 A 4.0
80-84 A- 3.7
77-79 B+ 3.3
73-76 B 3.0
70-72 B- 2.7
67-69 C+ 2.3
63-66 C 2.0
60-62 C- 1.7
57-59 D+ 1.3
53-56 D 1.0
50-52 D- 0.7
0-49 F 0
Write the function called gpa that takes a value corresponding to “Numerical Scale of Mark” and returns
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PHY224 University of Toronto
“Grade Point Value”.
5 Loops
Loops are used to perform the same calculation on difference elements of a collection (such as a list). The
marks for forty (entirely fictional) students from PHY224 in 2019 are given in the marks list below. Calculate
the average GPA for the class.
marks= array([72, 82, 72, 72, 79, 57, 59, 71, 66, 80,
67, 62, 91, 74, 77, 62, 71, 78, 65, 80,
70, 74, 70, 95, 76, 66, 85, 64, 79, 57,
63, 78, 84, 78, 75, 73, 62, 69, 72, 87])
6 Data comparison
One common question when you make a new measurement of a known value is whether your measurement
“agrees” with the known value. If we assume the known value has no uncertainty (or much smaller uncertainty)
the question can be answered by comparing the difference between your measurement and the known value,
relative the the uncertainty in your measurement.
For example, if you measure the mass of the electron to be me ± se, and the known value is mke you should
compare the |mke − me| to the uncertainty se. If the difference is smaller than your uncertainty you can
reasonably state that the measurement agrees with the known value.
Write a function that takes your measurement with uncertainty, and the known value of a parameter, and
returns True if they agree, and False if they don’t.
Test your function on the following values, printing the result of each call to your function.
Measurement Uncertainty Known value
19.2 0.1 19.41
19.5 0.8 19.41
19.5 0.1 19.41
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