This tutorial is intended for someone who may need to quickly make a calculator to evaluate definite integrals, or perhaps needs the algorithm for use in a larger scale program. This algorithm is useful if you need to evaluate a definite integral in a larger program specified toward something else, or if you want to check your answer for any definite integrals done by hand. Last Post; Nov 29 . Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715.. What is numerical integration method? Fill them out appropriately just as in number 3 above. efficiently odeint can solve the system. The code is released under the MIT license. A.5 Numerical Python with numpy. See the help function for romberg for further details. The copyright of the book belongs to Elsevier. The trapezoidal rule tends to overestimate the value . Zack Fizell 1.2K Followers In this section we show how Scientific Python can help through its high level mathematical algorithms. * & * & 0 & * & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ integration (and therefore the bounds) is from the innermost integral to the romberg -- Integrate func using Romberg integration. How many transistors at minimum do you need to build a general-purpose computer? If you already have an editor/compiler, you can skip these steps. implements the system of differential equations. 0 & 0 & 0 & 0 & 0 & * & * & * & 0 & * & 0 & 0 & 0 & 0 \\ The example we shall use in this tutorial is the dynamics of a spring-mass system in the presence of a drag force. How to Compute Numerical integration in Numpy (Python)? The functionality of the function Now, let's take a function from the scipy.misc library and calculate the value of the derivative at the point x = 1. 0 & 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 0 & 0 & * & * & * \\ functions dblquad and tplquad. You evaluated the function over the range [1,6]; the poster needs the integral over the range [0,6]. I don't see anywhere that you've dealt with the definite integral of a given function. parabola. my go-to programming language is Python, so I created an empty python file expecting this to take only 10 to 15 minutes. \(\{u_0, u_1, \ldots, u_{N-1}, v_0, v_1, \ldots, v_{N-1}\}\) simpson -- Use Simpson's rule to compute integral from samples. s += h * (f (a + i*h) + f (a + (i-1)*h))/2.0. When the variables are interleaved, \(\{u_k\}\) and \(\{v_k\}\) into a single vector of length \(2N\). suppose you wish to integrate a bessel function jv(2.5, x) along are the nonzero diagonals. Part 1:Getting Started with Numerical Simulation. That function is f(x) = Sin[x]. How does the Chameleon's Arcane/Divine focus interact with magic item crafting? An The next two arguments are the Numerical Integration - Definite Integration. More than 1500 numerical integration schemes for line segments, circles, disks, triangles, quadrilaterals, spheres, balls, tetrahedra, hexahedra, wedges, pyramids, n-spheres, n-balls, n-cubes, n-simplices, the 1D half-space with weight functions exp(-r), the 2D space with weight functions . Does Python have a ternary conditional operator? Specifically, f is a function that is passed to integrate. derivatives into the \(\mathbf{y}\) vector. The efficiency of the numerical solution increases compared to other methods when the size of the problem increases. The two obvious choices are 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & * & 0 & * & * \\ and At this point you are done: you have a working definite integral algorithm coded up in Python that runs smoothly and gives very accurate answers. if the function is a polynomial of order 3 or less. You do have to choose your own integration method though. A Simple Method for Numerical Integration in Python | by Harrison Hoffman | Towards Data Science Write Sign up Sign In 500 Apologies, but something went wrong on our end. 3.) The solution of solve_ivp with its standard parameters shows a big deviation I can't install anything new I need to use the default python library and I have to integrate a function. Non-constant integration bounds can be treated in a similar manner; the When I run the program with the values N = 10000, a = 0, and b = 10, I get the answer 333.33333249999964. If we let b-a/n be dx (the 'width' of our sample) then we can write this in python as such: Note that we make use of higher-order functions here. WARNING: This section is dense, and I want to clear some things up that could potentially be confusing. Part 1:Getting Started with Numerical Simulation; Free Chapter. We also assume we have a function, \(f(x)\), that can be computed for any of the grid points, or that we have been given the function implicitly as \(f(x_i)\). 0 & * & * & * & 0 & 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 0 \\ as \(\{u_0, u_1, \ldots, u_{N-1}, v_0, v_1, \ldots, v_{N-1}\}\), Here, a Python function is defined that carries out the algorithm of numerical integration using the midpoint rule. As you notice, we get both the integral value and the error estimate in only three lines of code, without bothering about the number of trapeziums or the accuracy. This is seen in the piece of code that shows up as: value += f(a+((n-(1/2))*((b-a)/N))). This last example shows that multiple integration can be handled using Our simple integration program will divide the interval 0 to 2 in equally spaced slices and spend the same time calculating the integrand in each of these slices. In the supporting picture, imagine that each rectangle in the designated area is of equal width. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. For Python, a block is distinguished by indentations: each task-performing-section has its own indent, and there can be indented blocks within other indented blocks. At this point it is worth commenting on how Python segregates blocks of code. Our complete system of \(2N\) ordinary differential equations is (1) The "*" just means "all." Then, we will take an array as y. The function is linearly approximated at each iteration to obtain a better estimate of the zero point.. Methods for Integrating Functions given function object. an object whose y field is an array with consecutive solution values as Before running the program again, under the comment "#type your function after return," type: sin(x) where x**2 is currently located. 0.10097672 1.04643602 1.91060117 2.49872472 3.08684827, sol1.y[1]: [0.35502805 0.328952 0.12801343 0.04008508 0.01601291 0.00623879, airy(sol.t)[0]: [0.35502805 0.328952 0.12804768 0.03995804 0.01575943 0.00562799, sol2.y[1][::6]: [0.35502805 0.19145234 0.06368989 0.0205917 0.00554734 0.00106409], airy(sol2.t)[0][::6]: [0.35502805 0.19145234 0.06368989 0.0205917 0.00554733 0.00106406], \(\left\{x_0, x_1, \ldots, x_{N-1}\right\}\), \(\{u_0, u_1, \ldots, u_{N-1}, v_0, v_1, \ldots, v_{N-1}\}\), \(\{u_0, v_0, u_1, v_1, \ldots, u_{N-1}, v_{N-1}\}\). As an interesting reminder, if \(\mathbf{A}\left(t\right)\) From there, the next piece of the code utilizes the variable called "value2" which is then assigned to be the sum of all of the heights of each rectangle multiplied by the standardized width of each rectangle--this is our final answer that we want displayed by our program, and is thus the return value of the "Integrate" Python function. Asking for help, clarification, or responding to other answers. is \(2^{k}+1\) for some integer \(k\), then Romberg romb Ordinary Differential Equation - Initial Value Problems, Predictor-Corrector and Runge Kutta Methods, Chapter 23. For example, suppose it is desired to find the solution to the and lower bandwidths are \(N\). . \(x \in [0, L]\) are, where \(D_u\) and \(D_v\) are the diffusion coefficients of the This is just a matter of putting the values that were input by the user ("N," "a," and "b") into the "Integrate" Python function and printing it on the screen. This integral can be evaluated using the expression below (Note the use of the . For a definite integral, we seek the area between two points (labeled a and b respectively). trapezoid rule at step-sizes related by a power of two and then Who invented integration by parts? On the same graph, compare the plot of the sin function with the plot of the integral of the cos function in the range [-, ].This can be done as follows: Many physical phenomena are modeled by differential equations: oscillations of simple systems (spring-mass, pendulum, etc. Numerical integration, 3. To learn more, see our tips on writing great answers. The user must link any math libraries, orders until the difference in the integral estimate is beneath some Write a program to use the above integral to calculate the ratio T/T0 for integral amplitudes 0 90. Chapter 1: Introducing Simulation Models. \(x\) is on the interval \([0, \infty]\). 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(both in the C function and ctypes argtypes) if not needed. sage: numerical_integral(x^2, 0, 1, max_points=100) (0.3333333333333333, 3.700743415417188e-15) is banded: Lets ensure that they have computed the same result: https://en.wikipedia.org/wiki/Rombergs_method, https://en.wikipedia.org/wiki/Method_of_lines. python machine-learning integration gpu automatic-differentiation pytorch high-performance-computing numerical-integration monte-carlo-integration vegas torchquad multidimensional-integration vegas-enhanced. This allows you to use more complicated mathematical functions that can be integrated. when the variables are interleaved, the bandwidth is much Your one-stop shop for numerical integration in Python. Given an integer array A and a range, find a subarray within the range with a minimum sum. The copyright of the book belongs to Elsevier. Creating Local Server From Public Address Professional Gaming Can Build Career CSS Properties You Should Know The Psychology Price How Design for Printing Key Expect Future. A basic definite integral represents the area under a curve defined by a function e.g. For example, Python (v5.11.0) R Julia Javascript (v2.16.1) ggplot2 . November 9, 2014 3 Comments code, math, python The definite integral over a range (a, b) can be considered as the signed area of X-Y plane along the X-axis. This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods. Most such algorithms are based on first order differential equations, so it will probably not be a bad idea to start by putting our second-order equation in the form of a system of two first-order differential equations: To write the numerical integration program, we shall use odeint, which is part of scipy.integrate. Scipy uses three methods to integrate a one-dimensional function: trapezoidal (integrate.trapz), Simpson (integrate.simps) and Romberg (integrate.romb). following second-order differential equation: with initial conditions \(w\left(0\right)=\frac{1}{\sqrt[3]{3^{2}}\Gamma\left(\frac{2}{3}\right)}\) and \(\left.\frac{dw}{dz}\right|_{z=0}=-\frac{1}{\sqrt[3]{3}\Gamma\left(\frac{1}{3}\right)}.\) It is known that the solution to this differential equation with these These functions both use the module performs Richardson extrapolation on these estimates to approximate These function can be mathematically integrated within the code. odeint -- General integration of ordinary differential equations. The text in quotes, after the input command, shows up when you run the program telling you what type of value to type in. Then, let's set the function value in the form of pairs x, y with a step of 0.01 for the range of x from 0 to 4. The copyright of the book belongs to Elsevier. The correct analytical answer, is 333.333. Numerical Integration using Python A simple method to numerically integrate equations and visualize results in Python Photo by Arnold Franciscaon Unsplash Whether you are an engineer, physicist, or just a dynamics enthusiast, chances are you've had to work with equations of motion (EOMs) in the form of ordinary differential equations (ODEs). To accomplish this goal, we assume that the interval has been discretized into a numeral grid, \(x\), consisting of \(n+1\) points with spacing, \(h = \frac{b - a}{n}\). Note: There are \(n+1\) grid points because the count starts at \(x_0\). inf as one of the arguments. neural networks), and I've discovered as I try to read through the algorithms that my calculus has gotten a bit rusty. SUBSCRIBE https://bit.ly/drmanabIn this video, you are going to learn how to perform numerical integration in python using trapezoidal and Simpson's rul. within \(1.04\times10^{-11}\) of the exact result well below the dblquad -- General purpose double integration. In the case of this program I have named the variables "N," "a," and "b." A user desiring reduced integration times may pass a C function In this day and age, an editor and compiler are usually integrated, however that is not always the case. 0 & 0 & * & 0 & * & * & * & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ themselves are available as special functions returning instances of The book "Numerical Recipes" explains how to handle this type of improper integral. * & * & * & 0 & 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 0 & 0 \\ Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide, To avoid the question being closed without being answered, please refer to. Load shared library into Python using ctypes and set restypes and for the functions \(u(x, t)\) and \(v(x, t)\) on the interval As it can be seen solve_ivp determines its time steps automatically if not 1) Solve by the method you prefer (Riemann, Trapezoid or Simpson) 2) Defend the method you use or why it was better to use in this case. The Gray-Scott equations and the initial conditions vector, y0, as input arguments and returns We define. However, this program can be improved. This represents tasks within tasks, and essentially tells the order in which the code needs to be executed. I can get the value for any f(x) and I need to integrate from 0 to 6 for my function f(x). There is an equal amount of positive area as there is negative area in this interval, so if you add up the total area, you should get zero. It goes as follows: a command (task) is set forth, a colon follows the command, and what the command does is indented underneath. That is, the main diagonal and the two diagonals immediately Any text present after a "#" is a comment that allows the programmer to follow the code in a humanistic way; I have made certain comments in my code which you will be copying, but feel free to add any comments that help you specifically. to achieve better results. https://www.enthought.com/product/canopy/, Make Your Own Customisable Desktop LED Neon Signs / Lights, Smart Light Conversion Using ESP8266 and a Relay, Wi-Fi Control of a Motor With Quadrature Feedback, A personal computer with access to the internet, Click the download button corresponding to your operating system, Follow the instillation instructions after starting the execution file, Click "Editor" from the program main menu, Click "create a new file" on the center of the screen. In both cases, there are just five nontrivial diagonals, but The performance increase here arises from two factors. No credit card required. pointer through scipy.LowLevelCallable to quad, dblquad, See the earlier answer for a good presentation. It should be clear why these are designated as such. Lets see how Scipy could simplify our work: The output will be (8.153364119811167, 9.0520525739669813e-014). is a mapping from \(\mathcal{R}^{N}\) to \(\mathcal{R}^{N}.\) First, recall that the area of a trapezoid with a height of h and bases of length b1 and b2 is given by Area = 1 2h(b1 + b2). The integrate.quad() routine takes the function and the integration limits as input arguments.An overview of scipy.integrate modules can be accessed by typing in the shell window: The period of a pendulum of length l oscillating at a large angle is given by. An alternative solution is to pass a function reference to the integration function. to write a little C to reduce computation time significantly. Python Methods for Numerical Differentiation. In contrast, the integrate.quad() routine from Scipy is arbitrary callable (adaptive), in the sense that it can adjust the function evaluations to concentrate on the more important regions (quad is short for quadrature, an older name for integration). odeint -- General integration of ordinary differential equations. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Explain the result when = 0. A variable is a name given to a value that will be operated on and that can change. If the samples are not The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb.. Quadrature is a historical mathematical term that means calculating area. function the time vector created by solve_ivp is passed to the airy function. This summation requires a couple more variables, one of which will act as the return value for the entire "Integrate" Python function. 1 I know how to perform a double integral in python import numpy as np import scipy.integrate as integrate integrate.dblquad (x*y, 0, 1, lambda x: -np.sqrt (1-x**2), lambda x: np.sqrt (1-x**2)) where x and y are, say, (200,) numpy arrays. Recall from step 5 you imported the math library array of Python functions into the program. Of course we have to estimate the number of trapeziums to use; the accuracy of our method depends on this number. We introduced the concept of Monte Carlo integration and illustrated how it differs from the conventional numerical integration methods. To work with the Python code in this chapter, you'll need the following files . Quadrature () uses the adaptive GaussKronrod method. with this as it is OS-dependent). The base implementation of Python includes the basic programming language, the tools to write loops, check conditions, build and manipulate lists, and all of the other things that we saw in the previous section. to the airy function. the uniformly spaced grid of \(N\) points \(\left\{x_0, x_1, \ldots, x_{N-1}\right\}\), with You can name these anything you want, but to match the formulas given in "Understanding the Algorithm Part 3: The Midpoint Rule," it is best to keep them the same. Notice the use of a lambda- The copyright of the book belongs to Elsevier. The end points and the interior points, Faster integration using low-level callback functions, Solving a system with a banded Jacobian matrix, http://groups.csail.mit.edu/mac/projects/amorphous/GrayScott/. array containing the point the function f is evaluated at, and user_data The rigid properties of rectangles will leave some of the total area untouched, hence why this is considered an approximation; however, the more rectangles you can cram in between the boundaries (a and b), the more accurate the approximation will be since the untouched regions become more sparse. Differential equations for the 1-D Gray-Scott equations. We . 0 & 0 & 0 & 0 & * & 0 & 0 & 0 & 0 & 0 & * & * & * & 0 \\ We also have this interactive book online for a better learning experience. In the case of the defined Python function "Integrate," everything within that function is indented out one block thus distinguishing the tasks that will be executed within that function. A simple analogy . You will learn how to develop you own numerical integration method and how to get a specified accuracy. Here well show you how to numerically solve these equations. Jacobian matrix. Simpsons rule approximates the function between three adjacent points as a import numpy import quadpy a = 0.0 b = 1.0 n = 100 start_points = numpy.linspace (a, b, n, endpoint=False) h = (b-a) / n end_points = start_points + h intervals . To do a numerical integration with python, a solution is to use the trapezoidal rule from numpy numpy.trapz or the Simpson's rule from scipy scipy.integrate.simps: Note: to do an integration from a known function see the scipy method called quad. Building a Numerical Integration Tool in Python From Scratch. It requires the derivative, fprime, the time span [t_start, t_end] We can now starting implementing this system in code. as \(\{u_0, v_0, u_1, v_1, \ldots, u_{N-1}, v_{N-1}\}\) is. of the Jacobian matrix: Now set ml=2 and mu=2, so odeint knows that the Jacobian matrix The following sections give some of the most common methods of approximating \(\int_a^b f(x) dx\). Great tutorial. 1 2x (f(x0) + f(x1)). We define \(u_j(t) \equiv u(x_k, t)\) and \(v_j(t) \equiv v(x_k, t)\), and For a large 1 Answer. We also have this interactive book online for a better learning experience. Methods for Integrating function given function object: quad - General Purpose Integration dblquad - General Purpose Double Integration nquad - General Purpose n- fold Integration \(\{u_0, v_0, u_1, v_1, \ldots, u_{N-1}, v_{N-1}\}\). Basic calculus knowledge is expected, but relevant mathematical information is reviewed. "def Integrate(N, a, b)" reads as: define a function called "Integrate" that accepts the variables "N," "a," and "b," and returns the area underneath the curve (the mathematical function) which is also defined within the "Integrate" Python function. < CHAPTER 21. Hence, you can see the output is in float value. You can call this Python function anything when you do the coding, but it makes sense to call it integrate since it is a function that indeed integrates a mathematical function. Created February 09, 2019 | Viewed 32157 | by Benjamin Edit. Body of the package implementing numerical integration: to be chosen. Each of us optimizes an itinerary to reach a destination, organize MOSFET is getting very hot at high frequency PWM. Example, let's try to integrate the function. It will involve implementing numerical methods to solve the following problems: 1. Each method approximates the area under \(f(x)\) for each subinterval by a shape for which it is easy to compute the exact area, and then sums the area contributions of every subinterval. system, this improves the performance significantly, as demonstrated in the Find centralized, trusted content and collaborate around the technologies you use most. Updated 16 days ago. If you coded the program as shown in the preceding pictures, you have just integrated f(x) = x^2 over some bounds. Unlimited access to the entire Packt Library from 2nd - 5th December. . in the args argument. \(I_{n}\) is shown below: As example for non-constant limits consider the integral. 0 (Leetcode) Brick wall. Blog:https://www.halvorsen.blogPython Resources:https://www.halvorsen.blog/documents/programming/python/Python Programming Videos:https://www.youtube.com/pla. If the jacobian matrix of function is known, it can be passed to the solve_ivp gfg = lambda x: x**2. is the period of the same pendulum at small amplitudes. It works on the average of a function denoted by <f . or a list of functions for the non-constant integration bounds. If you're not using Canopy, then you probably don't even need to follow this step at all and running the program may require different procedures. Before the for-loop, I have designated these variables as "value," and "value2." Immediately after defining the "integrate" Python function, you will define another Python function called f(x). The total online course (discounted): https://www.udemy.com/programming-numerical-methods-in-python/?couponCode=PNMP19 Python Programming And Numerical Methods: A Guide For Engineers And Scientists, Chapter 2. The limits of all Getting Started with Python on Windows, Python Programming and Numerical Methods - A Guide for Engineers and Scientists. Numerical Integration in Python | Speculum Minor Numerical Integration in Python last updated Jan 5, 2017 Numerical integration aims to find the area under a curve without using analytical methods. The trapezium (trapezoidal) method is the most straightforward of the three. * & * & * & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ If the function to integrate takes additional parameters, they can be provided That is not the only mathematical function this program can integrate. . Variables and Basic Data Structures, Chapter 7. integration bounds are an iterable object: either a list of constant bounds, An "int" is an integer, and a "float" is a floating point value (i.e a decimal). the integral with a higher degree of accuracy. Copy the code in the picture below the previous code. One could also make each integration function generic, instead of making the whole package generic. Help us identify new roles for community members, Proposing a Community-Specific Closure Reason for non-English content. What is this fallacy: Perfection is impossible, therefore imperfection should be overlooked, I want to be able to quit Finder but can't edit Finder's Info.plist after disabling SIP. If x is None, the sample points are assumed to be evenly . Connect and share knowledge within a single location that is structured and easy to search. (refer to picture). quadrature, which performs Gaussian quadrature of multiple Python Basics Learning with examples Numerical Integration Numerical Integration You will probably encounter many situations in which analytical integration of a function or a differential equation is difficult or impossible. weights of a large variety of orthogonal polynomials (the polynomials # dydt is the return value of this function. Monte Carlo Integration is a process of solving integrals having numerous values to integrate upon. error bound may underestimate the error due to possible numerical error in the Refresh the page, check Medium 's site status, or find something interesting to read. The return value is a tuple, with the first Here, we denote each point in \(x\) by \(x_i\), where \(x_0 = a\) and \(x_n = b\). In this section we show how Scientific Python can help through its high level mathematical algorithms. The points can be ( inf) to indicate infinite limits. \(x_0 = 0\) and \(x_{N-1} = L\). We must combine Suggested for: Numerical Integration in Python Python Numerical integration over a disk with polar coordinates. So I leave this step as an example of the flaws this program has and perhaps an exercise to anyone who wants to improve the program further. where n is the number of samples. quad -- General purpose integration. The trapezoidal rule In this case the thrown-in value is "x," and this "x" term will take the value of what ever you throw it--it is a temporary value. The illustrative problem is simple. For instance, let's take the function y = f (x), y = x2. I've divided the steps into 3 sections: understanding the algorithm that will be used to make the program, coding the program using the Python programming language, and running the program. rev2022.12.9.43105. integrand from the use of quad ). 0 & 0 & 0 & 0 & 0 & 0 & * & 0 & * & * & * & 0 & 0 & 0 \\ (refer to picture). 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & * & 0 & * & * & * & 0 \\ Using the Newton-Raphson algorithm for root finding Instead of specifying the step size ##du## in the numerical integration, I can specify the number of areas/points ##N##, and then calculate ##du=y/N##, and then pass it as an argument to the recursive function. In the picture, the turquoise region is the area I'm referring to, and the equation for determining this is also shown in that region. element holding the estimated value of the integral and the second element holding an upper bound on the error. cumulative_trapezoid -- Use trapezoidal rule to cumulatively compute integral. Next, a for-loop acts as the summation defined in the formulas in the "Understanding the Algorithm" section of this tutorial. This mathematical function is displayed in the first accompanying picture, plotted from 0 to 2, and the area of interest is shaded in turquoise. How can I use a VPN to access a Russian website that is banned in the EU? We only need to translate the last equation into a Python set of instructions. option to the function call solve_ivp returns the solutions of these time points The Of course, it's entirely possible to read through the descriptions of the algorithms involved without . This allows the underlying From this point you should see a blank white window with a cursor resembling a basic word processing document. Run the program by hitting the green play button again. First, we define the functions for the source and reaction The copyright of the book belongs to Elsevier. Visit Stack Exchange Tour Start here for quick overview the site Help Center Detailed answers. did anything serious ever run on the speccy? You'll get a detailed solution from a subject matter expert that helps you learn core concepts. This evaluates the function, it doesn't do any integration. The simple trapezium formula calculates the integral of a function f(x) as the area under the curve representing f(x) by approximating it with the sum of trapeziums: The area of each trapezium is calculated as width times the average height.Example: Evaluate the integral: using the basic trapezium rule.We shall write a small program to evaluate the integral. 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