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Chapter 5: Problem 5

Simulate by hand a vectorized expression. Suppose \(\mathrm{x}\) and \(\mathrm{t}\) are two arrays of the same length, entering a vectorized expression, $$ y=\cos (\sin (x))+\exp (1 / t) $$ If \(\mathrm{x}\) holds two elements, 0 and 2, and \(\mathrm{t}\) holds the elements 1 and 1.5, calculate by hand (using a calculator) the y array. Thereafter, write a program that mimics the series of computations you did by hand (typically a sequence of operations of the kind we listed on page 182 - use explicit loops, but at the end you can use Numerical Python functionality to check the results). Name of program file: simulate_vector_computing.py.

Short Answer

Expert verified
The \(y\) array is \([3.718, 2.564]\).

Step by step solution

01

Initialize Arrays

First, initialize the arrays. Let \(x = [0, 2]\) and \(t = [1, 1.5]\). These arrays will be our input for the vectorized expression.
02

Calculate Inner Functions

Calculate the inner functions for each element in the arrays:- For \(x_1 = 0\), calculate \(\sin(x_1) = \sin(0) = 0\).- For \(x_2 = 2\), calculate \(\sin(x_2) = \sin(2) \approx 0.909\).- For \(t_1 = 1\), calculate \(1/t_1 = 1/1 = 1\).- For \(t_2 = 1.5\), calculate \(1/t_2 = 1/1.5 \approx 0.667\).
03

Apply Cosine and Exponential

Apply the cosine and exponential functions on the results from Step 2:- For \(x_1\), calculate \(\cos(0) = 1\) and \(\exp(1) \approx 2.718\).- For \(x_2\), calculate \(\cos(0.909) \approx 0.615\) and \(\exp(0.667) \approx 1.949\).
04

Calculate Final y Array

Calculate the elements of the \(y\) array using the results from Step 3:- \(y_1 = 1 + 2.718 = 3.718\).- \(y_2 = 0.615 + 1.949 \approx 2.564\). So the \(y\) array becomes \([3.718, 2.564]\).
05

Write the Python Program

Create a Python file named `simulate_vector_computing.py` and write a program that implements the same calculations using explicit loops: ```python import numpy as np x = np.array([0, 2]) t = np.array([1, 1.5]) y = np.zeros_like(x, dtype=float) for i in range(len(x)): sin_x = np.sin(x[i]) cos_sin_x = np.cos(sin_x) exp_t = np.exp(1/t[i]) y[i] = cos_sin_x + exp_t # Check results with NumPy operations y_check = np.cos(np.sin(x)) + np.exp(1/t) print('Manual:', y) print('NumPy:', y_check) ``` This program mirrors the calculations done by hand and verifies them using NumPy.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Python
Numerical Python, often shortened to NumPy, is an essential library for anyone dealing with scientific programming in Python. It provides support for arrays, which are powerful tools to store and manipulate data efficiently. This makes it possible to perform mathematical operations on entire arrays without writing detailed loops. This
  • improves the efficiency of the code,
  • makes the code more concise, and
  • helps minimize the chance of errors.
Working with NumPy in our example, we're simulating how mathematical functions such as sine, cosine, and exponential are applied to arrays. It shows the advantage of vectorized operations, where we compute everything element-wise simultaneously, rather than iterating through each item individually. This capability illustrates NumPy’s ability to make tasks that involve large data sets more manageable and quicker compared to traditional looping methods. As seen in the Python code provided, NumPy's function calls replace manual calculations effectively.
Array Manipulation
Array manipulation in Python, particularly using NumPy, is a fundamental skill in working with large datasets efficiently. Arrays are versatile data structures that can hold multiple numbers organized in rows and columns. With their capability to handle entire datasets with single operations, arrays reduce the need for complex nested loops. In the given exercise, we are mainly using NumPy arrays to perform vectorized computing which allows simultaneous calculations on multiple data points.
The operations demonstrated include:
  • Creation of arrays using `np.array`.
  • Initialization with zeros using `np.zeros_like`, to match the dimensions of existing arrays, ensuring that the output matches the input size.
  • Element-wise mathematical function applications like sine, cosine, and exponential operations, perfectly illustrating their manipulation capacity.
These manipulations make the use of arrays indispensable in scientific computing scenarios, where the volume of data might be overwhelming if processed one element at a time.
Mathematical Functions
Mathematical functions in a programming context allow us to perform complex calculations swiftly and accurately. Python, with libraries like NumPy, supplies a robust set of functions that can operate over entire arrays seamlessly. These include trigonometric functions like sine and cosine, exponential functions, and many others, which can be applied directly to array elements.
In the exercise, the use of functions such as `np.sin`, `np.cos`, and `np.exp` is highlighted. These functions apply:
  • Sine to each element of the array `x`.
  • Cosine to the result of the sine function, again element-wise.
  • Exponential to the inverse of each element in array `t`.
By using these functions, we simplify the computational process since we're working with entire datasets as entities, instead of individual elements. This not only speeds up the manipulation and calculation processes but also keeps the code neat. Understanding how to utilize these mathematical functions efficiently is crucial for anyone looking to work in data science or scientific programming.
Scientific Programming
Scientific programming is vital when you need to solve real-world scientific problems through computers. By using robust programming techniques, alongside numerical libraries like NumPy, it is possible to handle complex computations involving large datasets encountered in scientific research. In our example, we have demonstrated how a vectorized expression can simplify calculations taken out on arrays. The code provided shows three key points for effective scientific programming:
  • Utilizing loops for manual verification of computations ensures accuracy.
  • Implementing vectorized expressions saves time and computational power.
  • Using libraries like NumPy, which are well-tested and optimized, ensures reliability in results.
Through these practices, scientific programming in Python facilitates efficient computation and allows for the analysis of large-scale data, helping researchers gain sharper insights into the data. This promotes the productive analysis required in various scientific fields, from physics to biology.

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Most popular questions from this chapter

Plot a formula for several parameters. Make a program that reads a set of \(v_{0}\) values from the command line and plots the corresponding curves \(y(t)=v_{0} t-0.5 g t^{2}\) in the same figure (set \(g=9.81\) ). Let \(t \in\left[0,2 v_{0} / g\right]\) for each curve, which implies that you need a different vector of \(t\) coordinates for each curve. Name of program file: plot_bal12.py.

Plot a formula. Make a plot of the function \(y(t)=v_{0} t-0.5 g t^{2}\) for \(v_{0}=10, g=9.81\) and \(t \in\left[0,2 v_{0} / g\right] .\) The label on the \(x\) axis should be 'time (s)' and the label on the \(y\) axis should be 'height \((\mathrm{m}) '\). Name of program file: plot_ball1.py.

Plot Taylor polynomial approximations to \(\sin x\). The sine function can be approximated by a polynomial according to the following formula: $$ \sin x \approx S(x ; n)=\sum_{j=0}^{n}(-1)^{j} \frac{x^{2 j+1}}{(2 j+1) !} $$ The expression \((2 j+1) !\) is the factorial (see Exercise 3.14). The error in the approximation \(S(x ; n)\) decreases as \(n\) increases and in the limit we have that \(\lim _{n \rightarrow \infty} S(x ; n)=\sin x\). The purpose of this exercise is to visualize the quality of various approximations \(S(x ; n)\) as \(n\) increases. The first part of the exercise is to write a Python function \(\mathrm{S}(\mathrm{x}\), (n) that computes \(S(x ; n)\). Use a straightforward approach where you compute each term as it stands in the formula, i.e., \((-1)^{j} x^{2 j+1}\) divided by the factorial \((2 j+1) !\). (We remark that Exercise A.16 outlines a much more efficient computation of the terms in the series.) The next part of the exercise is to plot \(\sin x\) on \([0,4 \pi]\) together with the approximations \(S(x ; 1), S(x ; 2), S(x ; 3), S(x ; 6)\), and \(S(x ; 12)\) Name of program file: plot_Taylor_sin.py.

Plot the velocity profile for pipeflow. A fluid that flows through a (very long) pipe has zero velocity on the pipe wall and a maximum velocity along the centerline of the pipe. The velocity \(v\) varies through the pipe cross section according to the following formula: $$ v(r)=\left(\frac{\beta}{2 \mu_{0}}\right)^{1 / n} \frac{n}{n+1}\left(R^{1+1 / n}-r^{1+1 / n}\right) $$ where \(R\) is the radius of the pipe, \(\beta\) is the pressure gradient (the force that drives the flow through the pipe), \(\mu_{0}\) is a viscosity coefficient (small for air, larger for water and even larger for toothpaste), \(n\) is a real number reflecting the viscous properties of the fluid ( \(n=1\) for water and air, \(n<1\) for many modern plastic materials), and \(r\) is a radial coordinate that measures the distance from the centerline \((r=0\) is the centerline, \(r=R\) is the pipe wall). Make a function that evaluates \(v(r) .\) Plot \(v(r)\) as a function of \(r \in[0, R]\), with \(R=1, \beta=0.02, \mu=0.02\), and \(n=0.1 .\) Thereafter, make an animation of how the \(v(r)\) curves varies as \(n\) goes from 1 and down to \(0.01\). Because the maximum value of \(v(r)\) decreases rapidly as \(n\) decreases, each curve can be normalized by its \(v(0)\) value such that the maximum value is always unity. Name of program file: plot_velocity_pipeflow.py.

Plot a smoothed "hat" function. The "hat" function \(N(x)\) defined by (3.5) on page 109 has a discontinuity in the derivative at \(x=1\). Suppose we want to "round" this function such that it looks smooth around \(x=1\). To this end, replace the straight lines in the vicinity of \(x=1\) by a (small) cubic curve $$ y=a(x-1)^{3}+b(x-1)^{2}+c(x-1)+d $$ for \(x \in[1-\epsilon, 1+\epsilon]\), where \(a, b, c\), and \(d\) are parameters that must be adjusted in order for the cubic curve to match the value and the derivative of the function \(N(x)\). The new rounded functions has the specification $$ \tilde{N}(x)= \begin{cases}0, & x<0 \\ x, & 0 \leq x<1-\epsilon \\\ a_{1}(x-1)^{3}+b(x-1)+c(x-1)+d_{1}, & 1-\epsilon \leq x<1 \\\ a_{2}(x-1)^{3}+b(x-1)+c(x-1)+d_{2}, & 1 \leq x<1+\epsilon \\ 2-x, & 1+\epsilon \leq x<2 \\ 0, & x \geq 2\end{cases} $$ with \(a_{1}=\frac{1}{3} \epsilon^{-2}, a_{2}=-a_{1}, d_{1}=1-\epsilon+a_{1} \epsilon^{3}, d_{2}=1-\epsilon-a_{2} \epsilon^{3}\), and \(b=c=0 .\) Plot this function. (Hint: Be careful with the choice of \(x\) coordinates!) Name of program file: plot_hat.py.
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