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51 lines
1.1 KiB
Markdown
51 lines
1.1 KiB
Markdown
---
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title: "Real Analysis Sequences in Haskell"
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date: 2019-05-21T22:18:21-04:00
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draft: false
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math: true
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tags: ["Math", "Haskell"]
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---
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In Real Analysis it is useful to look at terms of a sequence. One of the best ways I've found to do this is in believe it or not Haskell. This is mainly for these two reasons
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- Support for infinite data structures
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- Built-in Data Type to keep fractional precision
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## Code
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Let's get started, first let us define a sequence by the following:
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$$
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f(1) = 1, f(2) = 2, f(n) = \frac{1}{2}(f(n - 2) + f(n - 1))
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$$
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That is equivalent to the following haskell code:
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```haskell
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f :: Integral a => a -> Ratio a
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f 1 = 1
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f 2 = 2
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f n = 0.5 * (f (n - 2) + f (n - 1))
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```
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Now to generate the sequence we just need to map $f$ onto the natural numbers.
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```haskell
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nsequence = map f [1..]
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```
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If you want to look at specific subsequences, such as even or odd:
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```haskell
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odd_generator n = 2 * n - 1
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odds = map odd_generator [1..]
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even_generator n = 2 * n
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evens = map odd_generator [1..]
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```
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To look at the differences between each term:
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```haskell
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diff x = map (\(a, b) -> a - b) $ zip (tail x) (init x)
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```
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