Newton's Method, Euler's Method, and Modeling Dynamical Systems

Yadu Chittampalli

Improved Euler's Method

\[ x_{n+1}^{*}=x_n+hf\left( t_n,x_n,y_n\right) \]

\[ y_{n+1}^{*}=y_n+hg\left( t_n,x_n,y_n\right) \]

\[ t_{n+1}=t_n+h \]

\[ x_{n+1}=x_n+0.5h\left[f\left( t_n,x_n,y_n\right) +f\left( t_{n+1},x_{n+1}^{*},y_{n+1}^{*}\right) \right] \]

\[ y_{n+1}=y_n+0.5h\left[g\left( t_n,x_n,y_n\right) +g\left(t_{n+1},x_{n+1}^{*},y_{n+1}^{*}\right) \right] \]

\[ \frac{dx}{dt} = f\left( t_n,x_n,y_n\right) = x - xy - 0.75y \] \[ \frac{dy}{dt} = g\left( t_n,x_n,y_n\right) = xy - y - 0.75x \]

Improved Euler's Method

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All trajectories are diagonal straight lines.
Disregard the trajectory for step size 1.

Improved Euler's Method Exception (Step Size 1)

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x decreases as y increases.
The trajectory almost looks like that of the Euler's method. Only difference is that there is no steep drop to begin with.

Euler's Method

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Smoother trajectory for lower step sizes.

Improved Euler's Method vs. Euler's Method

The improved Euler's method displays divergence for step sizes less than 1.
The trajectories for the improved Euler's method are a straight diagonal line, whereas for the Euler's method, the trajectories are more curved.

Newton's Method

\[ x_{k+1}=x_k - \frac{f^\prime \left( x_k \right)}{f^{\prime \prime} \left( x_k \right) } \]

\[ \epsilon = tolerance \]

\[ \mid x_{k+1} - x_k \mid < \epsilon \]

\[ \mid f^\prime \left( x_k \right) \mid < \epsilon \]

Keep on reiterating until one of the two conditions listed above is met.

Two Different Functions (Piecewise Function and Quadratic Function)

\[ \left\{ \begin{array}{ll} 4x^{3} - 3x^{4} & x > 0 \\ 4x^{3} + 3x^{4} & x < 0 \\ \end{array} \right. \]

\[ \left\{ \begin{array}{ll} x^{2} + 2x & 3\leq x\leq 6 \\ \end{array} \right. \]

Newton's Method (How the Tolerance Affects Convergence of a Quadratic)

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For each tolerance convergence occurs at approximately -1.

Newton's Method (How the Tolerance Affects Convergence of a Piecewise at 0.4)

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The higher the tolerance, the lower the convergence value for x.

Newton's Method (How the Tolerance Affects Convergence of a Piecewise at 0.6)

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For x = 0.6, it does not matter what tolerance we pick because each of the tolerances yield the same trajectory of convergence.
For each tolerance convergence occurs at approximately 1.

Newton's Method (How the Tolerance Affects Number of Iterations Required)

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Newton's Method (How the Tolerance Affects Number of Iterations Required for Convergence)

The higher the tolerance, the lesser number of iterations required for convergence (shorter trajectory).
For the piecewise function where x1 = 0.6 and for the quadratic function it does not matter what tolerance we pick because convergence occurs at approximately the same number of iterations (7-9 for the piecewise function and 2-3 for the quadratic).

Newton's Method (How the Initial Value Affects Convergence of a Quadratic Function)

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For intial x-values less than -1, x increases and then converges at -1. But if x is initally greater than -1, x decreases and then converges at -1.

Newton's Method (How the Initial Value Affects Number of Iterations for Convergence of a Quadratic Function)

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Generally, as the initial value of x increases, the number of iterations required for convergence increases.
However, when x is initially -1 only one iteration is required for convergence.

Newton's Method (Convergence of a Piecewise Function (x > 1))

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For all initial values of x greater than 1, convergence occurs at approximately 1.
As the initial value of x is increased, the length of the trajectory increases and the slope becomes initially steeper.

Newton's Method (How x > 1 Affects Convergence of a Piecewise Function)

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As the initial value of x increases, the number of iterations required for convergence increases, thus yielding a trajectory that is of a longer length.

Newton's Method (Convergence of a Piecewise Function (0 < x < 1))

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Newton's Method (Convergence of a Piecewise Function (0 < x < 1))

There are uniquely different trajectories for many of these initial values of x.
When x is initially less than or equal to 0.6 convergence occurs when x reaches 0.
When x is initially greater than 0.6, convergence occurs when x reaches 1.

Newton's Method (How the Initial Value Affects Convergence)

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No particular trend shown in the graph.
However, if the number of iterations required for the initial value of 0.5 was 4 or greater, the histogram would look like a bell-shaped curve or in other words a normal distribution.

How the Half-life of Mercury Affects the Amount Accumulated in a Person's Lifetime

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As the half-life of mercury increases, the maximum amount of mercury accumulated in a person's body during a lifetime would indubitably increase due to the fact that it would take a longer time to decay.

How the Mass Affects the Amount of Mercury Accumulated in a Person's Lifetime

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As the mass of the consumed fish increases, mercury accumulated in a person's body during a lifetime would indubitably increase due to the fact that the intake of the amount of fish would be large.

Maximum Amount of Fish per Month

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As the mercury half-life increases, the maximum amount of fish that can be consumed per month decreases such that the concentration of mercury in a person's body does not exceed 50 mg/kg.

Maximum Amount of Fish per Month

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There is exponential decay shown in the graph. As the mass of the fish consumed is greater, the maximum amount of fish that can be consumed per month decreases.