CoCalc Public FilesTA Sandbox / KatieG Sandbox / Lab5_2G_Demo.sagewsOpen in with one click!
Author: Katie Gostic
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Description: Eulers_method_partial_solution
# Iterate Euler's method for 1000 steps (time 0 to 100 in steps of 0.1) # Use the change equation 0.2*x*(1-x/100) # Store N values in a list called "N" # Initial value is 10 counts = srange(0, 1000) # tell the for loop to run for 1000 steps changevec(x) = 0.2*x*(1-x/100) # defined the change vector as a function of the current state N = [10] # Define initial value for i in counts: # x_new = x_old + changevec(x_old)*delta_t # N[-1] Pulls out the last entry from the list N. (The last entry = N_current) # changevec(N[-1]) takes the last entry of the list N (N_current) as an input into the changevec equation. This outputs the change vector # Finally, we multiply the change vector by 0.1 and add to x_current. N_new = N[-1] + changevec(N[-1])*0.1 N.append(N_new) # The final line takes the newly calculated N value and appends it (glues it on) to the end of the list # So, e.g. if N was 4 elements long when you stared the loop [?, ?, ?, ?], it's now 5 elements long: [?, ?, ?, ?, N_new] # Next time we start the loop, N_new (this step's output) will become N_current (next step's input)
N # Show final list of N values
[10, 10.1800000000000, 10.3628735200000, 10.5486531608817, 10.7373714073976, 10.9290606065975, 11.1237529455809, 11.3214804285737, 11.5222748533262, 11.7261677868336, 11.9331905403773, 12.1433741438903, 12.3567493196484, 12.5733464552916, 12.7931955761805, 13.0163263170940, 13.2427678932773, 13.4725490708478, 13.7056981365714, 13.9422428670207, 14.1822104971285, 14.4256276881541, 14.6725204950778, 14.9229143334436, 15.1768339456718, 15.4343033668624, 15.6953458901155, 15.9599840313958, 16.2282394939672, 16.5001331324319, 16.7756849164029, 17.0549138938481, 17.3378381541397, 17.6244747908507, 17.9148398643370, 18.2089483641508, 18.5068141713282, 18.8084500206003, 19.1138674625768, 19.4230768259530, 19.7360871797947, 20.0529062959569, 20.3735406116931, 20.6979951925157, 21.0262736953681, 21.3583783321730, 21.6943098338204, 22.0340674146637, 22.3776487375901, 22.7250498797374, 23.0762652989248, 23.4312878008740, 23.7901085072900, 24.1527168248781, 24.5191004153711, 24.8892451666427, 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## BEST PRACTICE: Write your scripts first. Then turn them into functions once you know they work. ## Turn this script into a function by "wrapping" it in a def statement and a return statement. ## Inputs: none (this tells us what we need to put in the parentheses in the def statement) ## Outputs: N list (this tells us what we need to put in the return statement) def iterateEulers(initial): counts = srange(0, 1000) # tell the for loop to run for 1000 steps changevec(x) = 0.2*x*(1-x/100) # defined the change vector as a function of the current state N = [initial] # Define initial value for i in counts: N_new = N[-1] + changevec(N[-1])*0.1 N.append(N_new) return N
iterateEulers()
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## Add step_size variable as an input (replace 0.1) def iterateEulers(step_size): counts = srange(0, 100/step_size) # tell the for loop to run for 1000 steps changevec(x) = 0.2*x*(1-x/100) # defined the change vector as a function of the current state N = [10] # Define initial value for i in counts: N_new = N[-1] + changevec(N[-1])*step_size N.append(N_new) return N
iterateEulers(step_size = 0.01)
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# Hints for # 15 # Replace x limits with a variable! plot(16*x^2, (x, -5, 5))+point([1,16], color = "red", size = 50)
## Hints for #16 # 16 ## Eventually, you will have to define an interactive function that inputs the x2 value of interest ## For now, pick a single x_2 value and get your plot working for that value only ## Then, once your code works for one x_2 value, you can get the rest working. ## ## SUGGESTED STEPS: ## 1. Define the function you want to plot, f(x)=16*x^2, in your code ## 2. Plot the function ## 3. The point you're focused on (x_1, y_1) is [1,16]. Add that point to the plot and color it red. ## 4. Next, you need to add the secant line to the plot. Let's think about how to do this. ## BEFORE YOU WRITE ANY MORE CODE, figure out the equation for the secant line and write it in a comment. ## --> What is the slope of the secant line (hint - slope = average rate of change from x1 to x2) ## --> Once you are able to calculate the slope, you can write down the secant line equation in point-slope form ## 5. Above your plot code, add a line of code that calculates the slope, given your current value of x2. Store this to a variable called "slope" or "mm" or something like that. ## 6. Use your calculated slope to add the secant line to your existing plot. Choose a color other than blue. Look at your comments above if you forget the equation. ## 7. Add the point [x2, f(x2)] to your plot. Choose the same color as your secant line. ## 8. FINALLY: Now that your script works, turn it into a function that takes "x2" as an input (aka, turn x2 into a variable, and add that variable name as an input into the function. ## 9. Turn this function into an interact ## desolve_odeint t = srange(0, 10, 0.1) f(x) = -2.1*x sol = desolve_odeint(f, ics = 5, times = t, dvars = x) sol
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