[samuelson] Fixing the lecture

[bishmaybarik]

This PR brings the following changes to the lecture:

• Fixes [samuelson] Incorrect mathematical equation #530 : I have rectified the error in the mathematical equation
• Fixes [samuelson] Inconsistent usage of ($a$, $b$) and ($\alpha$, $\beta$) #456 : I have updated the lectures to use $a$ and $b$ throughout; and this is consistent with the explanation on text as well
• Fixes [samuelson] Does not follow Quantecon style guidelines #531 : Making some more changes to follow John's suggestions
  • large code cells with lots of plotting code are typically hidden now -- see, e.g., def param_plot()
  • we might want to truncate the number of decimal points in output such as (1.5371322893124, -0.9024999999999999)
  • we shouldn't have comments in cells, such as ### Test the categorize_solution function -- if necessary, those comments should be ordinary markdown above the cells.
  • I don't think we should have plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
  • the align syntax after "Notice that" and in other places seems nonstandard

[github-actions]

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[bishmaybarik]

I am not sure if I should completely delete the following three lines of code:

### code to reverse-engineer a cycle
### y_t = r^t (c_1 cos(ϕ t) + c2 sin(ϕ t))
###

from this entire code chunk (it can be found here):

### code to reverse-engineer a cycle
### y_t = r^t (c_1 cos(ϕ t) + c2 sin(ϕ t))
###

def f(r, ϕ):
    """
    Takes modulus r and angle ϕ of complex number r exp(j ϕ)
    and creates ρ1 and ρ2 of characteristic polynomial for which
    r exp(j ϕ) and r exp(- j ϕ) are complex roots.

    Returns the multiplier coefficient a and the accelerator coefficient b
    that verifies those roots.
    """
    g1 = cmath.rect(r, ϕ)  # Generate two complex roots
    g2 = cmath.rect(r, -ϕ)
    ρ1 = g1 + g2           # Implied ρ1, ρ2
    ρ2 = -g1 * g2
    b = -ρ2                # Reverse-engineer a and b that validate these
    a = ρ1 - b
    return ρ1, ρ2, a, b

## Now let's use the function in an example
## Here are the example parameters

r = .95
period = 10                # Length of cycle in units of time
ϕ = 2 * math.pi/period

## Apply the function

ρ1, ρ2, a, b = f(r, ϕ)

print(f"a, b = {a}, {b}")
print(f"ρ1, ρ2 = {ρ1}, {ρ2}")

Also, there are comments inside the code cell, like:

## Now let's use the function in an example
## Here are the example parameters

which I think might be a little confusing if people miss out reading each line of the code.

---

I plan to rewrite the entire thing like this:

def f(r, ϕ):
    """
    Takes modulus r and angle ϕ of complex number r exp(j ϕ)
    and creates ρ1 and ρ2 of characteristic polynomial for which
    r exp(j ϕ) and r exp(- j ϕ) are complex roots.

    Returns the multiplier coefficient a and the accelerator coefficient b
    that verifies those roots.
    """
    g1 = cmath.rect(r, ϕ)  # Generate two complex roots
    g2 = cmath.rect(r, -ϕ)
    ρ1 = g1 + g2           # Implied ρ1, ρ2
    ρ2 = -g1 * g2
    b = -ρ2                # Reverse-engineer a and b that validate these
    a = ρ1 - b
    return ρ1, ρ2, a, b

Now let's use the function in an example. Here are the example parameters:

r = .95
period = 10                # Length of cycle in units of time
ϕ = 2 * math.pi/period

## Apply the function
ρ1, ρ2, a, b = f(r, ϕ)

print(f"a, b = {a}, {b}")
print(f"ρ1, ρ2 = {ρ1}, {ρ2}")

---

Please let me know your thoughts on this @HumphreyYang and @mmcky. Once you give your inputs, I can make the changes accordingly.

[HumphreyYang]

I am not sure if I should completely delete the following three lines of code:

### code to reverse-engineer a cycle
### y_t = r^t (c_1 cos(ϕ t) + c2 sin(ϕ t))
###

from this entire code chunk (it can be found here):

### code to reverse-engineer a cycle
### y_t = r^t (c_1 cos(ϕ t) + c2 sin(ϕ t))
###

def f(r, ϕ):
    """
    Takes modulus r and angle ϕ of complex number r exp(j ϕ)
    and creates ρ1 and ρ2 of characteristic polynomial for which
    r exp(j ϕ) and r exp(- j ϕ) are complex roots.

    Returns the multiplier coefficient a and the accelerator coefficient b
    that verifies those roots.
    """
    g1 = cmath.rect(r, ϕ)  # Generate two complex roots
    g2 = cmath.rect(r, -ϕ)
    ρ1 = g1 + g2           # Implied ρ1, ρ2
    ρ2 = -g1 * g2
    b = -ρ2                # Reverse-engineer a and b that validate these
    a = ρ1 - b
    return ρ1, ρ2, a, b

## Now let's use the function in an example
## Here are the example parameters

r = .95
period = 10                # Length of cycle in units of time
ϕ = 2 * math.pi/period

## Apply the function

ρ1, ρ2, a, b = f(r, ϕ)

print(f"a, b = {a}, {b}")
print(f"ρ1, ρ2 = {ρ1}, {ρ2}")

Also, there are comments inside the code cell, like:

## Now let's use the function in an example
## Here are the example parameters

which I think might be a little confusing if people miss out reading each line of the code.

I plan to rewrite the entire thing like this:

def f(r, ϕ):
    """
    Takes modulus r and angle ϕ of complex number r exp(j ϕ)
    and creates ρ1 and ρ2 of characteristic polynomial for which
    r exp(j ϕ) and r exp(- j ϕ) are complex roots.

    Returns the multiplier coefficient a and the accelerator coefficient b
    that verifies those roots.
    """
    g1 = cmath.rect(r, ϕ)  # Generate two complex roots
    g2 = cmath.rect(r, -ϕ)
    ρ1 = g1 + g2           # Implied ρ1, ρ2
    ρ2 = -g1 * g2
    b = -ρ2                # Reverse-engineer a and b that validate these
    a = ρ1 - b
    return ρ1, ρ2, a, b

Now let's use the function in an example. Here are the example parameters:

r = .95
period = 10                # Length of cycle in units of time
ϕ = 2 * math.pi/period

## Apply the function
ρ1, ρ2, a, b = f(r, ϕ)

print(f"a, b = {a}, {b}")
print(f"ρ1, ρ2 = {ρ1}, {ρ2}")

Please let me know your thoughts on this @HumphreyYang and @mmcky. Once you give your inputs, I can make the changes accordingly.

Looks great to me! Thanks so much!

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