Re-wrote Thomas Algorithm chapter to include more details #329
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jiegillet
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the
Chapter Edit
leios
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Aug 5, 2018
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| # Thomas Algorithm | |||
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| As alluded to in the [Gaussian Elimination chapter](../gaussian_elimination/gaussian_elimination.md), the Thomas Algorithm (or TDMA -- Tri-Diagonal Matrix Algorithm) allows for programmers to **massively** cut the computational cost of their code from $$\sim O(n^3) \rightarrow \sim O(n)$$! This is done by exploiting a particular case of Gaussian Elimination, particularly the case where our matrix looks like: | |||
| As alluded to in the [Gaussian Elimination chapter](../gaussian_elimination/gaussian_elimination.md), the Thomas Algorithm (or TDMA, Tri-Diagonal Matrix Algorithm) allows for programmers to **massively** cut the computational cost of their code from $$ O(n^3)$$ to $$O(n)$$ in certain cases! This is done by exploiting a particular case of Gaussian Elimination, in the case where our matrix looks like: | |||
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Last sentence:
This is done by exploiting a particular case of Gaussian Elimination where the matrix looks like this:
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| By this, I mean that our matrix is *Tri-Diagonal* (excluding the right-hand side of our system of equations, of course!). Now, at first, it might not be obvious how this helps; however, we may divide this array into separate vectors corresponding to $$a$$, $$b$$, $$c$$, and $$d$$ and then solve for $$x$$ with back-substitution, like before. | |||
| This matrix shape is called *Tri-Diagonal* (excluding the right-hand side of our system of equations, of course!). | |||
| Now, at first, it might not be obvious how this helps. Well, first of all, the system easier to encode: we may divide it into four separate vectors corresponding to $$a$$, $$b$$, $$c$$, and $$d$$ (in some implementation, you will see the missing $$a_0$$ and $$c_n$$ set to zero to get four vectors of the same size). | |||
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"well, first of all" -> "Well, firstly, it makes..."
There was no verb in that sentence and also switching "first of all" to "firstly" to match "secondly" below
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"in some implementations" fix to plural
| By this, I mean that our matrix is *Tri-Diagonal* (excluding the right-hand side of our system of equations, of course!). Now, at first, it might not be obvious how this helps; however, we may divide this array into separate vectors corresponding to $$a$$, $$b$$, $$c$$, and $$d$$ and then solve for $$x$$ with back-substitution, like before. | ||
| This matrix shape is called *Tri-Diagonal* (excluding the right-hand side of our system of equations, of course!). | ||
| Now, at first, it might not be obvious how this helps. Well, first of all, the system easier to encode: we may divide it into four separate vectors corresponding to $$a$$, $$b$$, $$c$$, and $$d$$ (in some implementation, you will see the missing $$a_0$$ and $$c_n$$ set to zero to get four vectors of the same size). | ||
| Secondly, equations this short are easy to solve analytically, let's see how. |
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| In particular, we need to find an optimal scale factor for each row and use that. What is the scale factor? Well, it is the diagonal $$-$$ the multiplicative sum of the off-diagonal elements. | ||
| In the end, we will update $$c$$ and $$d$$ to be $$c'$$ and $$d'$$ like so: | ||
| We'll apply the mechanisms introduced in the Gaussian Elimination chapter. |
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What mechanisms, exactly? Maybe say "We'll start by applying mechanisms similar to those seen in the [Gaussian Elimination](../gaussian_elimination/gaussian_elimination.md) chapter."
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| Of course, the initial elements will need to be specifically defined as | ||
| Let's transform row $$(i)$$ in two steps. |
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Let me know what you think, maybe I went into too much detail?