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This course is about Numerical Methods and covers some of the popular methods and approaches being used daily by mathematicians and everyone involved in computation.
This course will teach you about
This course consists of the following materials:
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Section 1: Introduction | |||
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Lecture 1 | 04:20 | ||
Some motivating examples of what the course can help you do. |
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Section 2: Machine Representation of Numbers | |||
Lecture 2 | 04:29 | ||
How do you store numbers using only 0s and 1s? |
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Lecture 3 | 13:49 | ||
How computers actually store numbers. A discussion of mantissa, exponent and bias. |
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Lecture 4 | 03:39 | ||
We talk about the notion of the machine epsilon and the fact that it's not really that useful when specifying calculation tolerances. |
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Lecture 5 | 06:02 | ||
The two types of errors that reduce the accuracy of numerical methods. |
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Quiz 1 | 2 questions | ||
Use MATLAB to calculate round-off and truncation errors. |
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Section 3: Basic Concepts | |||
Lecture 6 | 07:50 | ||
What makes a function continuous? |
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Lecture 7 | 03:41 | ||
The claim that a differentiable function that has equal states at two distinct point has a stationary point somewhere in between. |
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Lecture 8 | 02:25 | ||
Another theorem you need to be aware of. |
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Lecture 9 | 02:49 | ||
If a<b and f(a)f(b)<0 then the root of f(x) = 0 lies between a and b. |
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Section 4: One-Point Iterative Methods | |||
Lecture 10 | 08:59 | ||
What are one-point iterative methods and what are they used for? |
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Lecture 11 | 07:20 | ||
The notion of convergence (and divergence, too). |
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Lecture 12 | 06:02 | ||
A look at Aitken's Δ² process and Steffensen's method. |
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Quiz 2 | 1 question | ||
Use MATLAB to fast solve the root-finding problem. |
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Lecture 13 | 03:29 | ||
What is the order of convergence and why do we care? |
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Lecture 14 | 08:08 | ||
An extremely efficient and popular root finding method. Quadratic convergence, woo-hoo! |
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Quiz 3 | 1 question | ||
Use MATLAB to solve the root-finding problem. |
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Lecture 15 | 12:13 | ||
Newton's method applied in many dimensions. Useful for solving systems of non-linear equations! |
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Quiz 4 | 1 question | ||
Use MATLAB to solve the root-finding problem. |
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Lecture 16 | 13:56 | ||
What if you cannot get the Jacobian matrix in analytic form? Use finite differences! (Note: finite differences are actually discussed in a later section, so you can come back to this clip later.) |
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Quiz 5 | 2 questions | ||
Use MATLAB to solve a root-finding problem. |
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Lecture 17 | 08:11 | ||
A way of speeding up polynomial evaluations. |
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Quiz 6 | 1 question | ||
Use MATLAB to evaluate polynomial. |
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Section 5: Two-Point Iterative Methods | |||
Lecture 18 | 06:06 | ||
A very simple method that leverages Bolzano's theorem. |
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Quiz 7 | 2 questions | ||
Use MATLAB to solve the root-finding problem. |
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Lecture 19 | 05:10 | ||
Similar to the Bisection method, Regula Falsi can, in most cases, provide faster convergence than the Bisection method. |
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Quiz 8 | 3 questions | ||
Use MATLAB to solve the root-finding problem. |
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Lecture 20 | 05:20 | ||
Yet another single-point iteration method. |
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Quiz 9 | 2 questions | ||
Use MATLAB to solve the root-finding problem. |
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Section 6: Interpolation and Curve Fitting | |||
Lecture 21 | 06:57 | ||
An introduction to the concept of interpolation, with a simple example. |
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Lecture 22 | 06:52 | ||
A better way of defining the interpolating polynomial. |
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Lecture 23 | 15:10 | ||
Did you think the Newton basis was cool? With divided difference, you don't even have to solve the triangular set of equations! |
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Lecture 24 | 06:11 | ||
The derivations of divided differences took too much time, so the examples get their own separate lesson. |
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Lecture 25 | 13:49 | ||
What kind of simplicifications can be made to divided differences if we assume the points are equally spaced? |
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Lecture 26 | 13:26 | ||
An interpolation formula for Lagrange polynomial. |
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Quiz 10 | 3 questions | ||
Use MATLAB to determite the interpolated value of a point. |
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Lecture 27 | 18:37 | ||
An improvement of Lagrangian interpolation. |
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Quiz 11 | 2 questions | ||
Use MATLAB to determite the interpolated value of a point. |
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Section 7: Differentiation | |||
Lecture 28 | 10:21 | ||
Finite difference approximations of derivatives - forward, backward and central differences. |
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Lecture 29 | 04:38 | ||
Now a formula for the 2nd derivative approximation. |
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Lecture 30 | 14:02 | ||
A look that the error terms in first and second derivatives that arise from using finite difference methods. |
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Lecture 31 | 13:24 | ||
A method of combining approximations for improving accuracy. |
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Section 8: Integration (Quadrature) | |||
Lecture 32 | 02:22 | ||
Why would we want to integrate things numerically? |
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Lecture 33 | 08:30 | ||
The simplest way of estimating the value of an integral. |
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Lecture 34 | 06:44 | ||
Subdivide an integral into several strips, evaluate functions as midpoints, treat strips as rectangles. Profit! |
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Lecture 35 | 09:16 | ||
A way of numerically calculating a specific type of integral. |
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Lecture 36 | 06:11 | ||
Another numeric procedure for a very specific integral. Usable for calculating the Gamma function! |
Dmitri Nesteruk is a developer, speaker and podcaster. His interests lie in software development and integration practices in the areas of computation, quantitative finance and algorithmic trading. His technological interests include C#, F# and C++ programming as well high-performance computing using technologies such as CUDA. He has been a C# MVP since 2009.
Dmitri is a graduate of University of Southampton (B.Sc. Computer Science) where he currently holds a position as a Visiting Researcher. He is also an instructor on an online intro-level Quantitative Finance course, and has also made online video courses on CUDA, MATLAB, D, the Boost libraries and other topics.
I am an engineer, mathematician and web developer. I make web applications using C# and .NET technology stack, and have 10 published scientific articles. I graduated from a university in 2011. At that time I engaged in research into matrix algebra and conducted work related to request processing optimization in a learning management system. Right now, I am interested in participating in research into financial engineering.