CoCalc Public FilesTA Sandbox / KatieG Sandbox / Lab5_2G_Demo.sagews
Author: Katie Gostic
Views : 50
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

array([[ 5.00000000e+00], [ 4.05292127e+00], [ 3.28523415e+00], [ 2.66295906e+00], [ 2.15855276e+00], [ 1.74968883e+00], [ 1.41827019e+00], [ 1.14962747e+00], [ 9.31869897e-01], [ 7.55359039e-01], [ 6.12282120e-01], [ 4.96306230e-01], [ 4.02297999e-01], [ 3.26096414e-01], [ 2.64328608e-01], [ 2.14260601e-01], [ 1.73676262e-01], [ 1.40779241e-01], [ 1.14113441e-01], [ 9.24985590e-02], [ 7.49778749e-02], [ 6.07758843e-02], [ 4.92639748e-02], [ 3.99326036e-02], [ 3.23687402e-02], [ 2.62375915e-02], [ 2.12677789e-02], [ 1.72393268e-02], [ 1.39739271e-02], [ 1.13270454e-02], [ 9.18152475e-03], [ 7.44239953e-03], [ 6.03269193e-03], [ 4.89000518e-03], [ 3.96376125e-03], [ 3.21296251e-03], [ 2.60437686e-03], [ 2.11106691e-03], [ 1.71119787e-03], [ 1.38707024e-03], [ 1.12433697e-03], [ 9.11369142e-04], [ 7.38741089e-04], [ 5.98811603e-04], [ 4.85387059e-04], [ 3.93447046e-04], [ 3.18921914e-04], [ 2.58513004e-04], [ 2.09546505e-04], [ 1.69855018e-04], [ 1.37681233e-04], [ 1.11602828e-04], [ 9.04641528e-05], [ 7.33293882e-05], [ 5.94398572e-05], [ 4.81810626e-05], [ 3.90548621e-05], [ 3.16573712e-05], [ 2.56617791e-05], [ 2.08010546e-05], [ 1.68609783e-05], [ 1.36668051e-05], [ 1.10779503e-05], [ 8.97967137e-06], [ 7.27864377e-06], [ 5.89986754e-06], [ 4.78218093e-06], [ 3.87634247e-06], [ 3.14232703e-06], [ 2.54714854e-06], [ 2.06386835e-06], [ 1.67203296e-06], [ 1.35513524e-06], [ 1.09835446e-06], [ 8.90895362e-07], [ 7.22554993e-07], [ 5.85680759e-07], [ 4.75037894e-07], [ 3.85196470e-07], [ 3.12547077e-07], [ 2.53786617e-07], [ 2.06203092e-07], [ 1.67709140e-07], [ 1.35905534e-07], [ 1.09901178e-07], [ 8.87855067e-08], [ 7.06306304e-08], [ 5.75378163e-08], [ 4.64983282e-08], [ 3.73428460e-08], [ 2.99370698e-08], [ 2.42378929e-08], [ 1.96487081e-08], [ 1.59787464e-08], [ 1.31446345e-08], [ 1.07832733e-08], [ 8.85239351e-09], [ 7.28453313e-09], [ 5.96087603e-09], [ 4.87866360e-09]])