Lagrange Point Calculations and Methods
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Questions and Answers

What is the distance from Earth to the Lagrange Point calculated in the code?

  • 1,506,117,016 meters
  • 58,354,928 meters
  • 326,045,071 meters (correct)
  • 147,993,882,983 meters
  • Which mathematical approach is utilized in the provided code for finding roots?

  • fsolve (correct)
  • Gradient Descent
  • Integration
  • Differentiation
  • In the plot, which marker represents the distance to the Moon?

  • x marker
  • red star
  • black circle (correct)
  • blue triangle
  • What is the range of values used to create the plot for the Lagrange Point?

    <p>0.5e8 to 3.8e8</p> Signup and view all the answers

    What was the distance from the Moon to the Lagrange Point according to the output?

    <p>58,354,928 meters</p> Signup and view all the answers

    What will the 'relax' function return for an input value of 2?

    <p>-2.5</p> Signup and view all the answers

    What does the variable 'TOO_BIG' likely represent in the context of this code?

    <p>1000</p> Signup and view all the answers

    How often does the program report values after the first few iterations?

    <p>3000</p> Signup and view all the answers

    What is the primary function of 'scipy.optimize.fsolve'?

    <p>To find the roots of a function</p> Signup and view all the answers

    If the function's value at x3 is negative, which value is reassigned to x1?

    <p>x3</p> Signup and view all the answers

    What is indicated by the statement 'Not Converging!' in the output?

    <p>The maximum iterations were reached without convergence.</p> Signup and view all the answers

    In the binary search method described, what happens if f(x3) is greater than zero?

    <p>x2 is replaced with x3.</p> Signup and view all the answers

    What is the maximum number of iterations set for the relaxation method?

    <p>20000</p> Signup and view all the answers

    What happens when f(T2) is greater than f(T3)?

    <p>Slide T3 and T4 down</p> Signup and view all the answers

    What is the main purpose of the goldsearch function?

    <p>To implement a search algorithm for maximum value</p> Signup and view all the answers

    Which condition leads to sliding T1 and T2 up in the search process?

    <p>When f(T2) &lt; f(T3)</p> Signup and view all the answers

    What is the purpose of the tol parameter in the goldsearch function?

    <p>To stop the search when x1 and x4 are close enough</p> Signup and view all the answers

    In the goldsearch function, when is x3 updated to x2?

    <p>When func(x2) is greater than func(x3)</p> Signup and view all the answers

    What occurs after each adjustment in the values during the goldsearch routine?

    <p>The search continues recursively with the new intervals</p> Signup and view all the answers

    What is the initial requirement for the range parameters x1, x2, x3, and x4 in goldsearch?

    <p>They must be provided in ascending order</p> Signup and view all the answers

    In the context of the goldsearch implementation, what does the variable x2 represent?

    <p>The midpoint of the first comparison</p> Signup and view all the answers

    What is the function returned by the func definition?

    <p>10 - (x - 2)²</p> Signup and view all the answers

    In the context of gold search, what is being maximized?

    <p>The output of the function</p> Signup and view all the answers

    What value of x from goldsearch yields the maximum output in this function?

    <p>2.0</p> Signup and view all the answers

    In the equation for visible light maximization, which variable is associated with the temperature?

    <p>T</p> Signup and view all the answers

    When simplifying the denominator in the light bulb temperature equation, what does it become?

    <p>e - 1</p> Signup and view all the answers

    What is the primary purpose of the goldsearch function in the provided code?

    <p>To find the maximum of the function</p> Signup and view all the answers

    In the context of light maximization, what is represented by 'I λ' in the equation?

    <p>Intensity of light</p> Signup and view all the answers

    What effect does increasing 'x' have on the output of the func function?

    <p>Output decreases until x equals 2</p> Signup and view all the answers

    What is the primary goal of the Six-Hump Camelback function in optimization?

    <p>Minimize the function value to find optimization points</p> Signup and view all the answers

    Which method is NOT mentioned as an optimization technique used for the Six-Hump problem?

    <p>Gradient descent</p> Signup and view all the answers

    In the visualization of the Six-Hump problem, which programming construct is used to create a grid for plotting?

    <p>meshgrid</p> Signup and view all the answers

    What is the output format of the optimization results from the BFGS method?

    <p>An array of coordinates</p> Signup and view all the answers

    What is a key requirement for a function used in least squares fitting?

    <p>It should return 0 when the fit is perfect</p> Signup and view all the answers

    What visualization type is used to display the Six-Hump Camelback function in a 3D plot?

    <p>Surface plot</p> Signup and view all the answers

    What does the term 'residual' refer to in the context of nonlinear curve fitting?

    <p>The difference between observed and predicted values</p> Signup and view all the answers

    Which library function is mentioned for performing the least squares fitting in the example?

    <p>optimize.leastsq</p> Signup and view all the answers

    What does the function effincandescent_bulb(temp) return?

    <p>The maximum efficiency of the bulb at a specified temperature</p> Signup and view all the answers

    Which constant is used to calculate COEFF3?

    <p>A ratio involving $ rac{15.0}{ ext{pi}^4}$</p> Signup and view all the answers

    What is the lower limit of integration in the effincandescent_bulb function for a given temperature?

    <p>COEFF2 / temp</p> Signup and view all the answers

    What integration method is used in the effincandescent_bulb function?

    <p>Numerical quadrature</p> Signup and view all the answers

    At what initial temperature is the efficiency first calculated in the program?

    <p>300 K</p> Signup and view all the answers

    What variable represents the peak temperature found by the search?

    <p>peak_temp</p> Signup and view all the answers

    What does the goldsearch function accomplish in the program?

    <p>Finds the peak efficiency temperature</p> Signup and view all the answers

    Which of the following statements about the filament temperature range is correct?

    <p>It ranges from 2000 K to 3300 K.</p> Signup and view all the answers

    What libraries does the script import for mathematical computations?

    <p>numpy and scipy</p> Signup and view all the answers

    What kind of data is visualized using the matplotlib.pyplot library?

    <p>Temperature vs Efficiency</p> Signup and view all the answers

    Study Notes

    Introduction to Python for Rapid Engineering Solutions

    • Course offered by the Ira A. Fulton Schools of Engineering at Arizona State University
    • Instructor: Steve Millman
    • Topic: Finding Roots

    Finding Roots of Nonlinear Equations

    • Objectives include examining numerical methods, using fsolve, and looking at examples
    • Topics include Newton's method, Relaxation method, Binary Search
    • Example Code: Available

    Newton's Method

    • Advantage: Converges quickly.
    • Disadvantages: Needs the derivative (f'(x)), multiple roots exist, can jump too far if f'(x) is small.
    • Possible Solution: Introduce a factor (r) to limit Δx.
      • Formula: xnew = x - r * f(x) / f'(x)

    Relaxation Method

    • Goal: Rearrange the equation to the form x = f(x).
    • Steps:
      1. Start with an initial guess for x.
      2. Plug the guess into f(x) to get a new value of x.
      3. Repeat steps 1 and 2 until Δx < ε.
    • Disadvantages: Can have multiple roots, may not converge, not suitable for a general form of x = f(x).
    • Example: x² + 2x + 1 = 0, rewritten as x = -(x² + 1)/2
    • Goal: Find an x₁ such that f(x₁) < 0 and an x₂ such that f(x₂) > 0.
    • Steps:
      1. Set x₃ = (x₁ + x₂)/2
      2. If f(x₃) < 0, set x₁ = x₃, else set x₂ = x₃
      3. Repeat until satisfactory accuracy.
    • Purpose: Efficiently finding a root based on finding values of either side of the root.

    fsolve (scipy.optimize)

    • Purpose: Finds roots of functions
    • Usage: fsolve(function, initial_guess, args=(arg_list))
      • function: The target function.
      • initial_guess: Initial approximation of roots (can be an array).
      • arg_list (Optional): Additional arguments passed to the target function.
    • Important Note: Don't confuse with numpy.linalg.solve as fsolve deals with nonlinear equations.

    Lagrange Point (Earth-Moon/Sun-Earth)

    • Goal: Identify the point where gravitational forces of celestial bodies balance.
    • Methods: Mathematical equations (differential equations) describing the forces are solved using fsolve (scipy routine).
    • Use Cases: Placing satellites in stable orbit.

    Optimization Methods

    • Goal: Find the minimum or maximum of a function
    • Methods: Previous lectures introduced simplified methods. Several Python optimization functions in scipy are more sophisticated.
    • Considerations: Trappable in local minima during the optimization process.

    Six-Hump Function

    • Purpose: Used for testing optimization algorithms, demonstrating a complex function with multiple local minima.
    • Methods: Utilized fmin_bfgs and basinhopping`` methods from scipy` to find the minimum in 3D.

    Residuals

    • Goal: Fit a curve to data.
    • Method: Use a least squares fit
    • Requirement: Function that returns 0 for an exact match.
    • Implementation: leastsq function in scipy calls the residual function which in turn calls the target function.
      • Used with known parameter(s) and sampled values.

    Diode Example

    • Goal:Find circuit voltages V₁ and V₂.
    • Methods: Uses fsolve to solve simultaneous equations.
    • Goal: Find the maximum value of a function.
    • Method: Systematically narrows the search range by splitting the range using a golden ratio.
    • Purpose: More reliable optimization algorithm than a simple range-reduction approach.
    • Usage: Often preferred when precise optima are needed.

    Bulb Problem (Lightbulb Efficiency)

    • Goal: Determine the ideal temperature for maximizing visible light from a lightbulb.
    • Method: Uses numerical integration with pre-computed parameters to optimize the efficiency equation via the golden search algorithm.
      • Calculates and plots efficiency over a range of temperatures.

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    Description

    Test your understanding of the various mathematical methods and approaches used to find the Lagrange Point and root-finding techniques in the provided code. This quiz covers distances, plotting ranges, and function behaviors related to Lagrange Point calculations.

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