daviddoji/CalculusWithJuliaNotes.jl
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jverzani
2023-06-27 18:39:13 -04:00
parent f0f81e86d1
commit c8342b85a8
3 changed files with 31 additions and 23 deletions
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14
quarto/derivatives/symbolic_derivatives.qmd
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@@ -104,7 +104,7 @@ function D(::Val{:+}, ::Val{:nary}, args, var)
end
```
The `args` are always held in a container, so the unary method must pull out the first one. The binary case should read as: apply `D` to each of the two arguments, and then create a quoted expression containing the sum of the results. The dollar signs interpolate into the quoting. (The "primes" are unicode notation achieved through `\prime[tab]` and not operations.) The *nary* case does something similar, only uses splatting to produce the sum.
The `args` are always held in a container, so the unary method must pull out the first one. The binary case should read as: apply `D` to each of the two arguments, and then create a quoted expression containing the sum of the results. The dollar signs interpolate into the quoting. (The "primes" are unicode notation achieved through `\prime[tab]` and not operations.) The *nary* case does something similar, only using splatting to produce the sum.
Subtraction must also be implemented in a similar manner, but not for the *nary* case:
@@ -195,7 +195,15 @@ function D(::Val{:cos}, ::Val{:unary}, args, var)
end
```
The pattern is similar for each. The `$a′` factor is needed due to the *chain rule*. The above illustrates the simple pattern necessary to add a derivative rule for a function. More could be, but for this example the above will suffice, as now the system is ready to be put to work.
The pattern is similar for each. The `$a′` factor is needed due to the *chain rule*. The above illustrates the simple pattern necessary to add a derivative rule for a function.
:::{.callout-note}
Several automatic differentiation packages use a set of rules defined following an interface spelled out in the package `ChainRules.jl`. Leveraging multi-dimensional derivatives, the chain rule is the only rule needed of the sum, product, quotient and chain rules.
:::
More functions could be included, but for this example the above will suffice, as now the system is ready to be put to work.
```{julia}
@@ -223,5 +231,3 @@ D(D(ex₃, :x), :x)
```
The length of the expression should lead to further appreciation for simplification steps taken when doing such a computation by hand.