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Add log for 2024-02-02 (#339)
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45 changes: 45 additions & 0 deletions _posts/2024-02-02-discussion-about-functions.md
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---
layout: post
title: "Functions, docstrings and a mistake by me at the end"
tags:
- functions-and-data-structures
---

In class today we spoke about group coursework a bit but mainly spent time
looking at docstrings.

You can see a recording of this here: [https://cardiff.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=7c020a93-9855-44f2-9f30-b1050118b4a9](https://cardiff.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=53ced2ac-b8f5-4b07-9b3a-b10201087ba1)

We spoke about the numpy docstring format, which is specified here: [https://vknight.org/pfm/building-tools/02-functions-and-data-structures/how/main.html#how-to-write-a-docstring](https://vknight.org/pfm/building-tools/02-functions-and-data-structures/how/main.html#how-to-write-a-docstring)

I used the example of verifying the following identity to demonstrate this:

$$(a + b) ^ n = \sum_{i=0}^n\binom{n}{i} a ^ i b ^ {n - i}$$

You can find a notebook with all this [here]({{site.baseurl}}/assets/nbs/2023-2024/functions-and-data-structures.ipynb)

**I'm really sorry about how confusing the ending of class today must have been
I made a mistake (my fault entirely)**.

The notebook linked to above does not contain the error.

When it came to fixing the floating point error I wrote:

```python
sum(int(scipy.special.binom(n, i) * a ** i * b ** (n - i)) for i in range(n + 1))
```

But the mistake I made here (and again I apologise for the crash and burn at the
end of the class) is that the closing bracket of `int` should have been after
the `binom` and not after the entire calculation.

```python
sum(int(scipy.special.binom(n, i)) * a ** i * b ** (n - i) for i in range(n + 1))
```

The difference is subtle but it is important as `scipy.special.binom(n, i)`
will always be a float with 0 fractional part. So taking the `int` of it will
give what we need.

I'll happily go over this again on Tuesday if I can clarify things. The
important thing to take from today is the approach and the docstrings.
305 changes: 305 additions & 0 deletions assets/nbs/2023-2024/functions-and-data-structures.ipynb
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"Verify the following identity for all integer values of $0 \\leq a \\leq\n",
"100$, $0 \\leq b \\leq 100$ and $1 \\leq n \\leq 10$:\n",
"\n",
"$$(a + b) ^ n = \\sum_{i=0}^n\\binom{n}{i} a ^ i b ^ {n - i}$$"
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"def get_lhs(a, b, n):\n",
" \"\"\"\n",
" This calculates the left hand side of the identity.\n",
" It does it directly, using python exponentiation.\n",
"\n",
" Parameters\n",
" ----------\n",
" a : int\n",
" This is the value of a in the equation.\n",
" b : int\n",
" This is the value of b in the equation.\n",
" n : int\n",
" This is the value of n in the equation. It is the value of the exponent.\n",
"\n",
"\n",
" Returns\n",
" -------\n",
" int\n",
" The value of the left hand side.\n",
" \"\"\"\n",
" return int((a + b ) ** n)"
]
},
{
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"id": "63635921-e137-40a5-b776-fe32d02d4e39",
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"id": "7054d82e-98d3-4810-8d9a-953b3911f5f3",
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"id": "6a12fc60-ce48-4786-9de3-4d2f4943df38",
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"import scipy.special\n",
"\n",
"def get_rhs(a, b, n):\n",
" \"\"\"\n",
" Obtain the right hand side of the equation.\n",
"\n",
" This uses scipy's special import for the binomial coefficient.\n",
"\n",
" Note that this also makes use of the fact that the rhs is an integer.\n",
" Thus we take `int` of the output of scipy's binom. This is important as otherwise\n",
" some numerical errors are not captured for the values for example of a=0, b=85 and n=10.\n",
" \n",
" Parameters\n",
" ----------\n",
" a : int\n",
" This is the value of a in the equation.\n",
" b : int\n",
" This is the value of b in the equation.\n",
" n : int\n",
" This is the value of n in the equation. It is the value of the exponent.\n",
"\n",
"\n",
" Returns\n",
" -------\n",
" int\n",
" The value of the left hand side.\n",
" \"\"\"\n",
" return sum(int(scipy.special.binom(n, i)) * a ** i * b ** (n - i) for i in range(n + 1))"
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"cell_type": "code",
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"id": "81202b71-0fd2-4e58-858e-043aae4b63cf",
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"def check_identity(a, b, n):\n",
" \"\"\"\n",
" This calculates the lhs and the rhs of the equation.\n",
"\n",
" It then compares them to check if the equality holds.\n",
"\n",
" Note this uses `get_lhs` and `get_rhs`.\n",
"\n",
" Parameters\n",
" ----------\n",
" a : int\n",
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" b : int\n",
" This is the value of b in the equation.\n",
" n : int\n",
" This is the value of n in the equation. It is the value of the exponent.\n",
"\n",
" Returns\n",
" -------\n",
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" Whether or not the lhs is equal to the rhs\n",
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"checks = [\n",
" check_identity(a=a_value, b=b_value, n=n_value)\n",
" for a_value in range(101)\n",
" for b_value in range(101)\n",
" for n_value in range(1, 11)\n",
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