From e69dc734073818f29c87a16c775a503662c9e24c Mon Sep 17 00:00:00 2001 From: James Threlfall Date: Mon, 14 Aug 2023 17:08:49 +0100 Subject: [PATCH] Update documentation --- MAT301_DifferentialEquationSolving1.html | 61 +++++++++++++----- MAT301_DifferentialEquationSolving2.html | 58 ++++++++++------- MAT301_DifferentialEquationSolving3.html | 32 +++++++-- MAT301_Iterative_Techniques.html | 36 ++++++----- MAT301_Numerical_Integration.html | 6 +- MAT301_Python_Intro.html | 4 +- ...T301_DifferentialEquationSolving1_17_1.png | Bin 0 -> 37442 bytes ...T301_DifferentialEquationSolving1_19_1.png | Bin 0 -> 44963 bytes ...T301_DifferentialEquationSolving1_23_1.png | Bin 0 -> 41405 bytes ...T301_DifferentialEquationSolving1_27_1.png | Bin 0 -> 42287 bytes ...T301_DifferentialEquationSolving2_16_0.png | Bin 12907 -> 12907 bytes ...T301_DifferentialEquationSolving2_25_0.png | Bin 12374 -> 12374 bytes ...T301_DifferentialEquationSolving2_34_0.png | Bin 7431 -> 7431 bytes ...T301_DifferentialEquationSolving2_40_0.png | Bin 13234 -> 13234 bytes ...AT301_DifferentialEquationSolving2_8_0.png | Bin 7245 -> 7245 bytes ...T301_DifferentialEquationSolving3_12_1.png | Bin 15826 -> 53744 bytes _images/MAT301_Python_Intro_37_0.png | Bin 72029 -> 72029 bytes .../MAT301_DifferentialEquationSolving1.ipynb | 18 +++--- .../MAT301_DifferentialEquationSolving2.ipynb | 11 +++- .../MAT301_DifferentialEquationSolving3.ipynb | 11 ++-- _sources/MAT301_Iterative_Techniques.ipynb | 20 ++++-- _sources/MAT301_Numerical_Integration.ipynb | 6 +- searchindex.js | 2 +- 23 files changed, 174 insertions(+), 91 deletions(-) create mode 100644 _images/MAT301_DifferentialEquationSolving1_17_1.png create mode 100644 _images/MAT301_DifferentialEquationSolving1_19_1.png create mode 100644 _images/MAT301_DifferentialEquationSolving1_23_1.png create mode 100644 _images/MAT301_DifferentialEquationSolving1_27_1.png diff --git a/MAT301_DifferentialEquationSolving1.html b/MAT301_DifferentialEquationSolving1.html index 3529d78..b6a467a 100644 --- a/MAT301_DifferentialEquationSolving1.html +++ b/MAT301_DifferentialEquationSolving1.html @@ -262,6 +262,11 @@

MAT301 Numerical Methods

Euler’s method +
  • + + Example 9.1 + +
  • Midpoint method @@ -316,6 +321,11 @@

    Contents

    Euler’s method
  • +
  • + + Example 9.1 + +
  • Midpoint method @@ -344,7 +354,7 @@

    Contents

    Differential Equation Solving 1

    -

    Notebook to implement Euler, Midpoint and Euler Trapezium methods to solve differential equations. Designed to support Abertay University undergrad course MAT301 (JT, 2022).

    +

    Many physical laws governing natural behaviour take the form of differential equations. These are often complicated, coupled equations where the behaviour of quantities in space and time depend on others. While some have analytical solutions, often it is quicker and easier to apply numerical tools to obtain approximate solutions.

    Motivation

    A differential equation is a relationship between a function, \(f(x)\), its independent variable, \(x\), and any number of its derivatives. An ordinary differential equation or ODE is a differential equation where the independent variable, and therefore also the derivatives, is in one dimension. For our purposes, we can assume that an ODE can be written

    @@ -395,8 +405,11 @@

    Euler’s method\(S(t_{j+1})\) given the state at \(S(t_j)\). Starting from a given initial value of \(S_0 = S(t_0)\), we can use this formula to integrate the states up to \(S(t_f)\); these \(S(t)\) values are then an approximation for the solution of the differential equation. The Explicit Euler formula is the simplest and most intuitive method for solving initial value problems. At any state \((t_j, S(t_j))\) it uses \(F\) at that state to “point” toward the next state and then moves in that direction a distance of \(h\). Although there are more sophisticated and accurate methods for solving these problems, they all have the same fundamental structure. As such, we enumerate explicitly the steps for solving an initial value problem using the Explicit Euler formula.

    -explicit Euler

    Lets go ahead and see this in action, using the equation and question posed in from Example 9.1 in the lectures:

    +

    This formula is called the Explicit Euler Formula, and allows us to approximate the state at \(S(t_{j+1})\) given the state at \(S(t_j)\). Starting from a given initial value of \(S_0 = S(t_0)\), we can use this formula to integrate the states up to \(S(t_f)\); these \(S(t)\) values are then an approximation for the solution of the differential equation. The Euler formula is the simplest and most intuitive method for solving initial value problems. At any state \((t_j, S(t_j))\) it uses \(F\) at that state to “point” toward the next state and then moves in that direction a distance of \(h\). Although there are more sophisticated and accurate methods for solving these problems, they all have the same fundamental structure. As such, we enumerate explicitly the steps for solving an initial value problem using the Explicit Euler formula.

    +explicit Euler

    +

    So we see that our approximation displays similar behaviour, but doesn’t follow the trend exactly; in fact the graphs appear to be diverging as \(x\) increases. What if we make the step size ten times smaller?

    @@ -516,20 +533,25 @@

    Euler’s method -_images/MAT301_DifferentialEquationSolving1_19_0.png +
    /var/folders/9y/379gbxjx3255qz9rg64kh7mm0000gn/T/ipykernel_52116/4080072740.py:8: MatplotlibDeprecationWarning: The seaborn styles shipped by Matplotlib are deprecated since 3.6, as they no longer correspond to the styles shipped by seaborn. However, they will remain available as 'seaborn-v0_8-<style>'. Alternatively, directly use the seaborn API instead.
    +  plt.style.use('seaborn-poster')
    +
    +
    +_images/MAT301_DifferentialEquationSolving1_19_1.png

    As expected, cranking up the number of steps improves the accuracy of the solution. However, in practice, we often have a finite amount of numerical resources (e.g. CPU speed, number of nodes, memory size), so we can’t always rely on the ability to simply decrease \(h\). In those cases, we have to be smarter, and apply other techniques.

    Midpoint method

    -

    In the lectures, we were introduced to a more accurate scheme, the Midpoint method. This essentially estimates the gradient of the tangent at the midpoint between steps \(x_n+\frac{1}{2}h\), which lies halfway between \(x_n\) and \(x_{n+1}\). Using Euler’s method, we then determine \begin{equation}y_{n+1}=y_n+hk_2,\end{equation}where -$\( +

    In the lectures, we were introduced to a more accurate scheme, the Midpoint method. This essentially estimates the gradient of the tangent at the midpoint between steps \(x_n+\frac{1}{2}h\), which lies halfway between \(x_n\) and \(x_{n+1}\). Using Euler’s method, we then determine \(y_{n+1}=y_n+hk_2,\) where

    +
    +\[\begin{split} \begin{align} -k_1&=f(x_n,y_n) \\ +k_1&=f(x_n,y_n), \\ k_2&=f(x_n+\frac{1}{2}h, y_n+\frac{1}{2}hk_1). \end{align} -\)$

    +\end{split}\]

    This looks a bit weird, but all this means is that we start to evaluate the function not only at the grid point values, but also now at different locations in \(x\) and \(y\).

    Let’s go ahead and implement this, in exactly the same manner as the previous method:

    We quickly see that the midpoint method is more accurate than the Euler method for the same step size. The trade-off is the additional calculation required to evaluate \(k_1\) and \(k_2\).

    @@ -571,12 +597,13 @@

    Modified Euler (Euler-Trapezium Method)\(y_{n+1}^p\) is an initial prediction. We then use the derivative prediction, \(y_{n+1}'^p\), from

    \[ -y_{n+1}'^p=f\left(x_{n+1}, y_{n+1}^p\right) +y_{n+1}'^p=f\left(x_{n+1}, y_{n+1}^p\right), \]
    -

    to correct the value -$\( +

    to correct the value

    +
    +\[ y_{n+1}^c=y_n+\frac{h}{2}\left(y_n'+y_{n+1}'^p\right). -\)$

    +\]

    Once again, we can apply this solution to the ODE problem from earlier in the following way:

    Once again, the Modified Euler does a much better job than the original Euler method, and (for this ODE) is comparable to the Midpoint method in accuracy.

    diff --git a/MAT301_DifferentialEquationSolving2.html b/MAT301_DifferentialEquationSolving2.html index 5bf9db1..702b125 100644 --- a/MAT301_DifferentialEquationSolving2.html +++ b/MAT301_DifferentialEquationSolving2.html @@ -538,11 +538,12 @@

    4th Order Runge Kutta Algorithm

    1. Linear Population Equation

    -

    Our first ODE will be linear: -$\( +

    Our first ODE will be linear:

    +
    +\[ y^{'}=0.1y, \ \ (2000 \leq t \leq 2020), -\)$ -with the initial condition,

    +\]
    +

    with the initial condition,

    \[ y(2000)=6. @@ -600,19 +601,21 @@

    Discrete Interval\(2000\leq t \leq 2020,\) -$\( +

    Here, the interval is \(2000\leq t \leq 2020,\)

    +
    +\[ h=\frac{2020-2000}{200}=0.1. -\)$

    +\]

    This gives the 201 discrete points:

    \[ t_0=2000, \ t_1=2000.1, \ ... t_{200}=2020, \]
    -

    which we can generalise to -$\( +

    which we can generalise to

    +
    +\[ t_n=2000+0.1n, \ \ \ n=0,1,...,200. -\)$

    +\]

    Let’s go ahead and set this range of \(t\) values as a list in Python:

    @@ -668,14 +671,15 @@

    Tailoring our 4th Order Runge Kutta scheme\(k_2\): -$\( +

    Subsequent coefficients are more complicated to evaluate mathematically, e.g. \(k_2\):

    +
    +\[\begin{split} \begin{align} k_2&=f\left(x_n+\frac{h}{2},y_n+\frac{h}{2}k_1\right)=0.1\left(y_n+0.05k_1\right) \\ &=0.1\left(y_n+0.05(0.1y_n)\right)=0.1y_n(1+0.005) \\ &=0.1y_n(1.005)=0.1005y_n. \end{align} -\)$

    +\end{split}\]

    This is a lot of mathematical effort to then plug into our calculators!

    The beauty of this method relies on making the computer to do the hard work for us! Now that we’ve defined a function to spit out values of \(f(x,y)\), we can also evaluate shifted values of \(x\) and \(y\) as input into that function to evaluate \(k_1-k_4\), and hence the next iteration of the solution, \(y_{n+1}\):

    @@ -908,14 +912,16 @@

    Results and Graph\( +

    we can therefore find:

    +
    +\[ y(t)=\left({5e^{\dfrac{-(t+c)}{5}}+0.05}\right)^{-1}, -\)$

    -

    with a contant of integration -$\( +\]

    +

    with a contant of integration

    +
    +\[ c=-5\ln{\left(\frac{0.14\exp{(400)}}{6}\right)}. -\)$

    +\]

    The plot below again shows the RK4 numerical approximation, \(y_n\) (circles) and the analytical solution to the non-linear population equation:

    @@ -1058,16 +1064,18 @@

    3. Non-Linear Population Equation with an oscillation -

    with the initial condition, -$\( +

    with the initial condition,

    +
    +\[ y(2000)=6. -\)$

    +\]

    Implementing RK4

    -

    As with the previous cases, lets express the right hand side of the ODE as a function -$\( +

    As with the previous cases, lets express the right hand side of the ODE as a function

    +
    +\[ f_3(x,y)=0.2y-0.01y^2+\sin(2\pi x), -\)$

    +\]

    Once again, we could mathematically calculate expressions for \(k_1\rightarrow k_4\). This time though, you can see that \(f_3(x,y)\) now contains terms that depend on \(y\) and \(t\)!

    Now that we’re more familiar with how to implement RK4 numerically, we’ll simply move on to define our function representing \(f_3(x,y)\) and let the code evaluate the coefficients:

    diff --git a/MAT301_DifferentialEquationSolving3.html b/MAT301_DifferentialEquationSolving3.html index 4a8b598..7850b42 100644 --- a/MAT301_DifferentialEquationSolving3.html +++ b/MAT301_DifferentialEquationSolving3.html @@ -262,6 +262,11 @@

    MAT301 Numerical Methods

    Simple attempt: Euler

  • +
  • + + Example 9.4 + +
  • Further work @@ -304,6 +309,11 @@

    Contents

    Simple attempt: Euler
  • +
  • + + Example 9.4 + +
  • Further work @@ -375,7 +385,10 @@

    Simple attempt: EulerThis ensures that the error does not accumulate, since the velocity calculation is recalculated from a more accurate displacement calculation at each step.

    One additional complexity is that this model is not self-starting: each new displacement value requires two previous values, which is an issue for an initial value problem. One common approach is to use a single Euler step to calculate the first displacement, with subsequent values using the Verlet algorithm.

    A second complexity is that the velocity calculation requires knowledge of the next displacement in time, so calculation of this lags behind the displacement calculation. If the velocity is required after 5 steps, we would need to calculate the displacement over 6 steps to achieve the required solution.

    -

    Lets use the algorithm to tackle Example 9.4 from the Lecture notes. In this example, a mass of \(2\)kg is propelled along a rough horizontal table which exerts a resistance of \(0.4\)N. The initial velocity of the mass is \(8{\rm{ms}}^{-1}\). We are to use the Verlet algorithm with a step size of \(h=0.1\) to approximate the velocity and displacement of the mass after \(0.5\)s, assuming \(x\left(t=0\right)=0\).

    + +
    +

    Example 9.4

    +

    To see the algorithm in action, we will re-use Example 9.4 from the Lecture notes. In this example, a mass of \(2\)kg is propelled along a rough horizontal table which exerts a resistance of \(0.4\)N. The initial velocity of the mass is \(8{\rm{ms}}^{-1}\). We are to use the Verlet algorithm with a step size of \(h=0.1\) to approximate the velocity and displacement of the mass after \(0.5\)s, assuming \(x\left(t=0\right)=0\).

    My first step is to determine the acceleration at a given time. In this case the acceleration (force per unit mass) is constant, so we’ll store acceleration as a constant:

    @@ -426,13 +439,14 @@

    Simple attempt: Euler

    We can see that the desired solution at \(t=0.5\) is given by the final element in the list of \(x\) and \(v\), and matches the values we calculated in the lectures.

    -

    This equation of motion also has exact solutions (the acceleration is constant so we can use \(v=u+at\) and \(s=ut+\frac{1}{2}at^2\)). The exact solutions are: -$\( +

    This equation of motion also has exact solutions: the acceleration is constant so we can use \(v=u+at\) and \(s=ut+\frac{1}{2}at^2\). From these, we should recover:

    +
    +\[\begin{split} \begin{align} - v(t)&=&8-0.2t \\ - x(t)&=&8t-0.1t^2 + v(t)&=8-0.2t \\ + x(t)&=8t-0.1t^2 \end{align} -\)$

    +\end{split}\]

    Lets compare our numerical approximation with the true solutions:

    This is not a particularly exciting example, but it matches the mathematical steps you were shown in the lectures.

    diff --git a/MAT301_Iterative_Techniques.html b/MAT301_Iterative_Techniques.html index edcb2a3..2d7ba0e 100644 --- a/MAT301_Iterative_Techniques.html +++ b/MAT301_Iterative_Techniques.html @@ -263,8 +263,8 @@

    MAT301 Numerical Methods

  • - - Iteration Using Loops + + Looping Simple Iteration in Python
  • @@ -320,8 +320,8 @@

    Contents

  • - - Iteration Using Loops + + Looping Simple Iteration in Python
  • @@ -350,7 +350,7 @@

    Contents

    Iterative Solutions to Equations

    -

    Notebook to implement iterative techinques elements of Abertay University undergrad course MAT301 (JT, 2022).

    +

    The first numerical technique we will explore are algorithms which find zeros: these are commonly called root-finding algorithms.

    Motivation

    The roots of a function are locations/values which make that function equal zero. @@ -361,8 +361,8 @@

    Motivation\(x_r\), exactly (noting that there may be zero, one or two real solutions of this equation). The quadratic formula is a rare example of root-finding with a simple solution; most other types of function require novel ways to identify roots. As computational power has significantly grown in recent years, so too have iterative ways to identify roots which take advantage of computational resources and procedures.

    -

    We will here discuss several iterative techinques to identify roots of equations. When testing some of these tools, we may use quadratic functions, in order to compare exact and approximate solutions.

    +

    evaluates the root(s) of the function, \(x_r\), exactly (noting that there may be zero, one or two real solutions of this equation). The quadratic formula is a rare example of root-finding with a simple solution, and can only be applied to equations which are quadratic (i.e. whose highest power in \(x\) is two, \(x^2\)); most other types of function require novel ways to identify roots. As computational power has significantly grown in recent years, so too have iterative ways to identify roots which take advantage of computational resources and procedures.

    +

    We will here discuss several iterative techinques to identify roots of equations. We will often use quadratic functions to test these tools: we already have an exact analytical way to solve these problems, so this makes a good way to test our numerical approaches and examine how accurate and efficient they are.

    +

    (Note that the termcolor package, used to change the font color of some output, doesn’t appear to work in Colab. This doesn’t really matter, as I was just using it to highlight some specific values).

    -
    -

    Iteration Using Loops

    +
    +

    Looping Simple Iteration in Python

    So far, we’ve effectively only used Python as a simple calculator. Python (and other languages) are far more powerful and can potentially check the derivative for convergence/divergence each time an iteration occurs. We can also build in a test of the difference between the latest iteration and previous values in order to not perform the calculation too many times. One could, in principle, combine all of these steps together in the following way:

    @@ -602,9 +605,7 @@

    Iteration Using Loops
    divergent
    -
    -
    -
    no. of steps: 1      final x: 3.0
    +no. of steps: 1      final x: 3.0
     
    @@ -641,15 +642,16 @@

    Iteration Using Loops

    Newton-Raphson Iteration

    Newton-Raphson (also sometimes known as Newton’s method) is an iterative technique which rapidly approximates the roots of a real-valued function. It is fast and efficient, yet relatively easy to understand and is in common use in many different practical applications.

    -

    The function we want to work with, \(f(x)\), must be smooth and continuous; let us define \(x_r\) as an unknown root of \(f(x)\). We could make an initial guess \(x_0\) for the value of \(x_r\). Unless \(x_0\) is a very lucky guess, \(f(x_0)\) will not be a root. If we don’t find the root, we want our next guess, \(x_1\), to be an improvement on \(x_0\) (i.e., closer to \(x_r\) than \(x_0\)). If we assume that \(x_0\) is “close enough” to \(x_r\), then we can improve upon it by taking the linear approximation of \(f(x)\) around \(x_0\), which is a line, and finding the intersection of this line with the x-axis. Written out, the linear approximation of \(f(x)\) around \(x_0\) is \(f(x) \approx f(x_0) + f^{\prime}(x_0)(x-x_0)\). Using this approximation, we find \(x_1\) such that \(f(x_1) = 0\). Plugging these values into the linear approximation results in the equation

    +

    The function we want to work with, \(f(x)\), must be smooth and continuous; let us define \(x_r\) as an unknown root of \(f(x)\). We could make an initial guess \(x_0\) for the value of \(x_r\). Unless we make a very lucky guess, \(f(x_0)\) will not be a root. If we don’t find the root, we want our next guess, \(x_1\), to be an improvement on \(x_0\) (i.e., closer to \(x_r\) than \(x_0\)). If we assume that \(x_0\) is “close enough” to \(x_r\), then we can improve upon it by taking the linear approximation of \(f(x)\) around \(x_0\), which is a line, and finding the intersection of this line with the x-axis. Written out, the linear approximation of \(f(x)\) around \(x_0\) is \(f(x) \approx f(x_0) + f^{\prime}(x_0)(x-x_0)\). Using this approximation, we find \(x_1\) such that \(f(x_1) = 0\). Plugging these values into the linear approximation results in the equation

    \[ 0 = f(x_0) + f^{\prime}(x_0)(x_1-x_0), \]
    -

    which when solved for \(x_1\) is -$\( +

    which when solved for \(x_1\) is

    +
    +\[ x_1 = x_0 - \frac{f(x_0)}{f^{\prime}(x_0)}. -\)$

    +\]

    An illustration of how this linear approximation improves an initial guess is shown in the following figure.

    Newton Step

    Written generally, a Newton step computes an improved guess, \(x_i\), using a previous guess \(x_{i-1}\), and is given by the equation

    diff --git a/MAT301_Numerical_Integration.html b/MAT301_Numerical_Integration.html index 4bae8d0..21ea32b 100644 --- a/MAT301_Numerical_Integration.html +++ b/MAT301_Numerical_Integration.html @@ -330,12 +330,12 @@

    Contents

    Numerical Integration

    -

    Notebook to implement Euler, Midpoint and Euler Trapezium methods to solve differential equations. Designed to support Abertay University undergrad course MAT301 (JT, 2022).

    +

    The next useful mathematical approach that can be applied numerically is numerical integration.

    Motivation

    Many laws of motion (for example Newton’s Laws) require differentiation or integration in order to extract different information. For example: integrating a displacement in time yields a velocity. The ability to integrate functions between two points is extremely useful in video games programming.

    Given a function \(f(x)\), we want to approximate the integral of \(f(x)\) over the total interval, \([a,b]\). The following figure illustrates the area this generates.

    -Illustration integral +Illustration integral

    To achieve this, we assume that the interval is broken up (“discretised”). Our independent variable, \(x\), will comprise \(N+1\) points with spacing, \(h = \dfrac{b - a}{n}\).

    We denote each point in \(x\) using the subscript “n”, i.e. \(x_n\) (which is common for this type of algorithm), beginning with 0 (\(x_0 = a\)) and ending with n (\(x_n = b\)). Note: There are \(N+1\) grid points because the count starts at \(x_0\). We also assume we have a function, \(f(x)\), that can be computed for any of the grid points, or that we have been given the function implicitly as \(f(x_n)\). The interval \([x_n, x_{n+1}]\) is referred to as a subinterval.

    Several methods exist to approximate the area under the function or \(\int_a^b f(x) dx\). Each method approximates the area under \(f(x)\) for each subinterval by a shape for which it is easy to compute the exact area, and then sums the area contributions of every subinterval. In MAT301 we learned about Simpson’s Rule and Trapezium Rule, which we will detail below.

    @@ -485,7 +485,7 @@

    Simpson’s Rule\(f_0\) and \(f_n\) (used elsewhere in the formula.

    +

    No warning appeared so we’re good to go. All we need to do is sum the elements that we’ve calculated according to the Simpson’s rule formula. We can be a bit clever here, and let Python decide which numbers are odd and which are even (by picking every second element in cleverly chosen ranges which omit \(f_0\) and \(f_n\)) at add on the remaining elements according to the formula.

    total_simp_area = h / 3.0 * ( f[0] + f[N] + 4.0 * sum(f[1:N:2]) + 2.0 * sum(f[2:N-1:2] ))
    diff --git a/MAT301_Python_Intro.html b/MAT301_Python_Intro.html
    index 65210f9..08d5e20 100644
    --- a/MAT301_Python_Intro.html
    +++ b/MAT301_Python_Intro.html
    @@ -437,7 +437,9 @@ 

    Symbolic Computation
    sin(x)**2 + cos(x)**2
    -1
    +
    +
    +
    1
     
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--git a/_images/MAT301_Python_Intro_37_0.png b/_images/MAT301_Python_Intro_37_0.png index be7315f53f0d5d3f6f534254264fb2c75d4101d9..b0d6c8a9e97fe39308805e7281e624f7d78baa35 100644 GIT binary patch delta 48 zcmcb+iRJDlmI-bO=6Z%Y3K=CO1;tkS`nicE1v&X8Ihjd%`9" ] @@ -123,7 +121,9 @@ "id": "LvEn08T2tqjg" }, "source": [ - "Lets go ahead and see this in action, using the equation and question posed in from Example 9.1 in the lectures:\n", + "## Example 9.1 ##\n", + "\n", + "To see this in action, we will use the equation and question posed in from Example 9.1 in the lectures:\n", "\n", "$$\n", "\\frac{{\\rm{d}}y}{{\\rm{d}}t}=xy,\n", @@ -472,10 +472,11 @@ "source": [ "## Midpoint method\n", "\n", - "In the lectures, we were introduced to a more accurate scheme, the Midpoint method. This essentially estimates the gradient of the tangent at the midpoint between steps $x_n+\\frac{1}{2}h$, which lies halfway between $x_n$ and $x_{n+1}$. Using Euler's method, we then determine \\begin{equation}y_{n+1}=y_n+hk_2,\\end{equation}where\n", + "In the lectures, we were introduced to a more accurate scheme, the Midpoint method. This essentially estimates the gradient of the tangent at the midpoint between steps $x_n+\\frac{1}{2}h$, which lies halfway between $x_n$ and $x_{n+1}$. Using Euler's method, we then determine $y_{n+1}=y_n+hk_2,$ where\n", + "\n", "$$\n", "\\begin{align}\n", - "k_1&=f(x_n,y_n) \\\\\n", + "k_1&=f(x_n,y_n), \\\\\n", "k_2&=f(x_n+\\frac{1}{2}h, y_n+\\frac{1}{2}hk_1).\n", "\\end{align}\n", "$$\n", @@ -578,10 +579,11 @@ "where $y_{n+1}^p$ is an initial prediction. We then use the derivative prediction, $y_{n+1}'^p$, from\n", "\n", "$$\n", - "y_{n+1}'^p=f\\left(x_{n+1}, y_{n+1}^p\\right)\n", + "y_{n+1}'^p=f\\left(x_{n+1}, y_{n+1}^p\\right),\n", "$$\n", "\n", "to correct the value\n", + "\n", "$$\n", "y_{n+1}^c=y_n+\\frac{h}{2}\\left(y_n'+y_{n+1}'^p\\right).\n", "$$" diff --git a/_sources/MAT301_DifferentialEquationSolving2.ipynb b/_sources/MAT301_DifferentialEquationSolving2.ipynb index ac3c87a..2c30ca3 100644 --- a/_sources/MAT301_DifferentialEquationSolving2.ipynb +++ b/_sources/MAT301_DifferentialEquationSolving2.ipynb @@ -61,9 +61,11 @@ "To illustrate how this method can be implemented numerically, we will apply it to three initial value problems. These happen to be models of population growth over time, but could in reality be for any type of ordinary differential equation (ODE, or indeed for a system of several coupled ODEs). I have selected these equations because they are broadly related, but gradually increase in complexity. The equations which we will investigate are the following:\n", "### 1. Linear Population Equation\n", "Our first ODE will be linear:\n", + "\n", "$$\n", "y^{'}=0.1y, \\ \\ (2000 \\leq t \\leq 2020), \n", "$$\n", + "\n", "with the initial condition,\n", "\n", "$$\n", @@ -140,8 +142,8 @@ "h=\\frac{b-a}{N}.\n", "$$\n", "\n", - "\n", "Here, the interval is $2000\\leq t \\leq 2020,$ \n", + "\n", "$$\n", "h=\\frac{2020-2000}{200}=0.1.\n", "$$\n", @@ -153,6 +155,7 @@ "$$\n", "\n", "which we can generalise to\n", + "\n", "$$ \n", "t_n=2000+0.1n, \\ \\ \\ n=0,1,...,200.\n", "$$\n", @@ -285,6 +288,7 @@ "$$\n", "\n", "Subsequent coefficients are more complicated to evaluate mathematically, e.g. $k_2$:\n", + "\n", "$$\n", "\\begin{align}\n", "k_2&=f\\left(x_n+\\frac{h}{2},y_n+\\frac{h}{2}k_1\\right)=0.1\\left(y_n+0.05k_1\\right) \\\\\n", @@ -671,11 +675,13 @@ "$$\n", "\n", "we can therefore find:\n", + "\n", "$$\n", "y(t)=\\left({5e^{\\dfrac{-(t+c)}{5}}+0.05}\\right)^{-1}, \n", "$$\n", "\n", "with a contant of integration \n", + "\n", "$$\n", "c=-5\\ln{\\left(\\frac{0.14\\exp{(400)}}{6}\\right)}.\n", "$$\n", @@ -922,13 +928,16 @@ "$$ \n", "y^{'}=0.2y-0.01y^2+\\sin(2\\pi t), \\ \\ (2000 \\leq t \\leq 2020), \n", "$$\n", + "\n", "with the initial condition,\n", + "\n", "$$\n", "y(2000)=6.\n", "$$\n", "\n", "### Implementing RK4\n", "As with the previous cases, lets express the right hand side of the ODE as a function\n", + "\n", "$$\n", "f_3(x,y)=0.2y-0.01y^2+\\sin(2\\pi x),\n", "$$\n", diff --git a/_sources/MAT301_DifferentialEquationSolving3.ipynb b/_sources/MAT301_DifferentialEquationSolving3.ipynb index 8c30a79..6390370 100644 --- a/_sources/MAT301_DifferentialEquationSolving3.ipynb +++ b/_sources/MAT301_DifferentialEquationSolving3.ipynb @@ -113,7 +113,9 @@ "id": "aDUnAhD884SL" }, "source": [ - "Lets use the algorithm to tackle Example 9.4 from the Lecture notes. In this example, a mass of $2$kg is propelled along a rough horizontal table which exerts a resistance of $0.4$N. The initial velocity of the mass is $8{\\rm{ms}}^{-1}$. We are to use the Verlet algorithm with a step size of $h=0.1$ to approximate the velocity and displacement of the mass after $0.5$s, assuming $x\\left(t=0\\right)=0$.\n", + "## Example 9.4 ##\n", + "\n", + "To see the algorithm in action, we will re-use Example 9.4 from the Lecture notes. In this example, a mass of $2$kg is propelled along a rough horizontal table which exerts a resistance of $0.4$N. The initial velocity of the mass is $8{\\rm{ms}}^{-1}$. We are to use the Verlet algorithm with a step size of $h=0.1$ to approximate the velocity and displacement of the mass after $0.5$s, assuming $x\\left(t=0\\right)=0$.\n", "\n", "My first step is to determine the acceleration at a given time. In this case the acceleration (force per unit mass) is constant, so we'll store acceleration as a constant:" ] @@ -252,11 +254,12 @@ "source": [ "We can see that the desired solution at $t=0.5$ is given by the final element in the list of $x$ and $v$, and matches the values we calculated in the lectures.\n", "\n", - "This equation of motion also has exact solutions (the acceleration is constant so we can use $v=u+at$ and $s=ut+\\frac{1}{2}at^2$). The exact solutions are:\n", + "This equation of motion also has exact solutions: the acceleration is constant so we can use $v=u+at$ and $s=ut+\\frac{1}{2}at^2$. From these, we should recover:\n", + "\n", "$$\n", "\\begin{align}\n", - " v(t)&=&8-0.2t \\\\\n", - " x(t)&=&8t-0.1t^2\n", + " v(t)&=8-0.2t \\\\\n", + " x(t)&=8t-0.1t^2\n", "\\end{align}\n", "$$\n", "\n", diff --git a/_sources/MAT301_Iterative_Techniques.ipynb b/_sources/MAT301_Iterative_Techniques.ipynb index fabb909..50c4954 100644 --- a/_sources/MAT301_Iterative_Techniques.ipynb +++ b/_sources/MAT301_Iterative_Techniques.ipynb @@ -8,7 +8,7 @@ "source": [ "# Iterative Solutions to Equations\n", "\n", - "Notebook to implement iterative techinques elements of Abertay University undergrad course MAT301 (JT, 2022)." + "The first numerical technique we will explore are algorithms which find zeros: these are commonly called **root-finding** algorithms. " ] }, { @@ -30,10 +30,9 @@ "x_r=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a},\n", "$$\n", "\n", - "evaluates the root(s) of the function, $x_r$, exactly (noting that there may be zero, one or two real solutions of this equation). The quadratic formula is a rare example of root-finding with a simple solution; most other types of function require novel ways to identify roots. As computational power has significantly grown in recent years, so too have iterative ways to identify roots which take advantage of computational resources and procedures.\n", + "evaluates the root(s) of the function, $x_r$, exactly (noting that there may be zero, one or two real solutions of this equation). The quadratic formula is a rare example of root-finding with a simple solution, and can only be applied to equations which are *quadratic* (i.e. whose highest power in $x$ is two, $x^2$); most other types of function require novel ways to identify roots. As computational power has significantly grown in recent years, so too have iterative ways to identify roots which take advantage of computational resources and procedures.\n", "\n", - "We will here discuss several iterative techinques to identify roots of equations. When testing some of these tools, we may use quadratic functions, in order to compare exact and approximate solutions.\n", - "\n" + "We will here discuss several iterative techinques to identify roots of equations. We will often use quadratic functions to test these tools: we already have an exact analytical way to solve these problems, so this makes a good way to test our numerical approaches and examine how accurate and efficient they are." ] }, { @@ -86,6 +85,13 @@ "from termcolor import colored" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "(Note that the termcolor package, used to change the font color of some output, doesn't appear to work in Colab. This doesn't really matter, as I was just using it to highlight some specific values)." + ] + }, { "cell_type": "markdown", "metadata": { @@ -460,7 +466,7 @@ "source": [ "We can see from this that we would only expect convergence using $F_1$ when $x=1$ and $F_2$ when $x=3$.\n", "\n", - "## Iteration Using Loops\n", + "## Looping Simple Iteration in Python\n", "\n", "So far, we've effectively only used Python as a simple calculator. Python (and other languages) are far more powerful and can potentially check the derivative for convergence/divergence each time an iteration occurs. We can also build in a test of the difference between the latest iteration and previous values in order to not perform the calculation too many times. One could, in principle, combine all of these steps together in the following way:" ] @@ -647,12 +653,14 @@ "\n", "Newton-Raphson (also sometimes known as Newton's method) is an iterative technique which rapidly approximates the roots of a real-valued function. It is fast and efficient, yet relatively easy to understand and is in common use in many different practical applications.\n", "\n", - "The function we want to work with, $f(x)$, must be smooth and continuous; let us define $x_r$ as an unknown root of $f(x)$. We could make an initial guess $x_0$ for the value of $x_r$. Unless $x_0$ is a very lucky guess, $f(x_0)$ will not be a root. If we don't find the root, we want our next guess, $x_1$, to be an improvement on $x_0$ (i.e., closer to $x_r$ than $x_0$). If we assume that $x_0$ is \"close enough\" to $x_r$, then we can improve upon it by taking the linear approximation of $f(x)$ around $x_0$, which is a line, and finding the intersection of this line with the x-axis. Written out, the linear approximation of $f(x)$ around $x_0$ is $f(x) \\approx f(x_0) + f^{\\prime}(x_0)(x-x_0)$. Using this approximation, we find $x_1$ such that $f(x_1) = 0$. Plugging these values into the linear approximation results in the equation\n", + "The function we want to work with, $f(x)$, must be smooth and continuous; let us define $x_r$ as an unknown root of $f(x)$. We could make an initial guess $x_0$ for the value of $x_r$. Unless we make a very lucky guess, $f(x_0)$ will not be a root. If we don't find the root, we want our next guess, $x_1$, to be an improvement on $x_0$ (i.e., closer to $x_r$ than $x_0$). If we assume that $x_0$ is \"close enough\" to $x_r$, then we can improve upon it by taking the linear approximation of $f(x)$ around $x_0$, which is a line, and finding the intersection of this line with the x-axis. Written out, the linear approximation of $f(x)$ around $x_0$ is $f(x) \\approx f(x_0) + f^{\\prime}(x_0)(x-x_0)$. Using this approximation, we find $x_1$ such that $f(x_1) = 0$. Plugging these values into the linear approximation results in the equation\n", "\n", "$$\n", "0 = f(x_0) + f^{\\prime}(x_0)(x_1-x_0),\n", "$$\n", + "\n", "which when solved for $x_1$ is\n", + "\n", "$$\n", "x_1 = x_0 - \\frac{f(x_0)}{f^{\\prime}(x_0)}.\n", "$$\n", diff --git a/_sources/MAT301_Numerical_Integration.ipynb b/_sources/MAT301_Numerical_Integration.ipynb index 99b42ef..dc19b9d 100644 --- a/_sources/MAT301_Numerical_Integration.ipynb +++ b/_sources/MAT301_Numerical_Integration.ipynb @@ -7,7 +7,7 @@ }, "source": [ "# Numerical Integration\n", - "Notebook to implement Euler, Midpoint and Euler Trapezium methods to solve differential equations. Designed to support Abertay University undergrad course MAT301 (JT, 2022)." + "The next useful mathematical approach that can be applied numerically is **numerical integration**." ] }, { @@ -21,7 +21,7 @@ "\n", "Given a function $f(x)$, we want to approximate the integral of $f(x)$ over the total **interval**, $[a,b]$. The following figure illustrates the area this generates. \n", "\n", - "\"Illustration\n", + "\"Illustration\n", "\n", "To achieve this, we assume that the interval is broken up (\"discretised\"). Our independent variable, $x$, will comprise $N+1$ points with spacing, $h = \\dfrac{b - a}{n}$. \n", "\n", @@ -335,7 +335,7 @@ "id": "f5ewoO7jpq2s" }, "source": [ - "No warning appeared so we're good to go. All we need to do is sum the elements that we've calculated according to the Simpson's rule formula. We can be a bit clever here, and let Python decide which numbers are odd and which are even (by picking every second element in cleverly chosen ranges which omit $f_0$ and $f_n$ (used elsewhere in the formula." + "No warning appeared so we're good to go. All we need to do is sum the elements that we've calculated according to the Simpson's rule formula. We can be a bit clever here, and let Python decide which numbers are odd and which are even (by picking every second element in cleverly chosen ranges which omit $f_0$ and $f_n$) at add on the remaining elements according to the formula." ] }, { diff --git a/searchindex.js b/searchindex.js index 907222e..f04987d 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ 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