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51 Terms

1

Is Python free to use?

Yes, Python is open-source software and free to use.

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2

Is Python 2 still in use?

No, Python 2 has been officially retired. Most environments now use Python 3.

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3

Can Python run on different operating systems?

Yes, Python is cross-platform and can run on Windows, macOS, and Linux.

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4

What is the smallest increment that a regular Python float can represent?

Approximately 2.2 x 10^-16, known as machine epsilon for floating point arithmetic.

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5

What is round-off error?

The error caused by approximating a number due to the finite precision of its representation.

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6

What happens when you add 1.0 and 1e-17 in Python?

It returns 1.0 due to the precision limits of floating point arithmetic.

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7

Name two root finding algorithms.

Newton Raphson Method and Bisection Method.

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8

Are NumPy arrays the same as Python lists?

No, NumPy arrays are homogeneous and provide more functionality for numerical operations.

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9

Can you store different data types within a NumPy array?

No, NumPy arrays are homogeneous; all elements must be of the same data type.

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10

How to write ln(2^4) in Python NumPy syntax?

import numpy as np; result = np.log(2**4).

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11

How to write log(10^-3) in Python NumPy syntax?

import numpy as np; result = np.log10(10**-3).

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12

What do the three arguments of np.arange() represent?

Start value, stop value (exclusive), and step size.

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13

What do the three arguments of np.linspace() represent?

Start value, stop value (inclusive), and number of samples.

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14

What is the first element of np.arange(11,2,-2)?

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15

What is the last element of np.arange(11,2,-2)?

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16

What is the first element of np.linspace(2,15,20)?

2.0.

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17

What is the last element of np.linspace(2,15,20)?

15.0.

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18

Name three numerical differentiation methods.

Forward difference, Backward difference, Central difference.

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19

What is the forward difference approximation?

(f(x + h) - f(x)) / h.

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20

What is the backward difference approximation?

(f(x) - f(x - h)) / h.

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21

What is the central difference approximation?

(f(x + h) - f(x - h)) / (2 * h).

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22

What point do you lose with forward difference?

The last point.

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23

What point do you lose with backward difference?

The first point.

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24

What points do you lose with central difference?

The first and last points.

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25

Name four numerical methods for solving ODEs.

Euler’s method, Runge-kutta methods, Adams-Bashforth methods, Finite difference methods.

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26

What is the difference between numerical differentiation methods and numerical methods for solving ODEs?

Numerical differentiation approximates derivatives, while numerical ODE methods find the solution function.

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27

When solving ODEs numerically, are we given the ODEs?

Yes, the ODE must be known to solve it numerically.

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28

Are initial values typically given when solving ODEs numerically?

Yes.

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29

What is the goal of solving an ODE numerically?

To approximate the function that solves the ODE.

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30

Define forward Euler’s method.

Explicitly updates the dependent variable using the function value at the current point.

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31

Define backward Euler’s method.

Requires solving an implicit equation for the dependent variable at the next step.

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32

What is the advantage of forward Euler’s method?

It provides an explicit update equation, making it easier to implement.

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33

What is the disadvantage of backward Euler’s method?

It is more computationally expensive due to solving an implicit equation.

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34

What is the difference between explicit and implicit methods?

Explicit methods use known quantities for the next value, while implicit methods involve solving an equation.

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35

Which methods are explicit and which are implicit?

Forward Euler: explicit; Backward Euler: implicit; Midpoint: explicit; RK4: explicit.

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36

What are the steps to rewrite a higher-order ODE into a system of first-order ODEs?

  1. Define new variables for each derivative. 2. Rewrite the original ODE. 3. Write the system in matrix form. 4. Solve using numerical methods.

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37

How many auxiliary variables are needed to rewrite a nth order ODE?

n-1 auxiliary variables.

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38

Can you use forward Euler’s method with a system of first-order ODEs?

Yes.

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39

What are the four key parameters of scipy.integrate.solve_ivp()?

Fun: function to evaluate the derivative; t_span: interval of integration; y0: initial conditions; method: integration method.

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40

Define absolute error.

The absolute difference between the true value and the approximate value.

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41

Define relative error.

The absolute error divided by the absolute value of the true value.

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42

Define sum of squared error.

The sum of the squares of the differences between true values and approximations.

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43

Define mean squared error.

The average of the sum of squared errors.

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44

Define root mean squared error.

The square root of the mean squared error.

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45

What are the order of the forward, backward, and central difference methods?

Forward difference: First order; Backward difference: First order; Central difference: Second order.

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46

Define global and local errors.

Global error: Accumulated error over all steps; Local error: Error at a single step.

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47

What is the local error order of forward Euler’s method?

First order.

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48

What is the global error of a numerical method with nth order local error?

Typically one order less than the local error.

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49

Define phase portrait.

A graphical representation of trajectories in the phase plane.

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50

What is a trajectory in a phase space?

The path that a system follows through the phase space.

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51

What is a slope field?

A graphical representation showing the slope of a differential equation at various points.

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