Previously we looked at using Dual numbers get the value of the first derivative of a function. As useful as this is there is more potential if we can also obtain the second derivative. My initial, naive, approach to this was to extend the idea of a Dual to that of a Triple like this.
data Triple a = T a a a deriving (Show). Creating Triple somehow seemed ‘wrong’, or if not wrong then certainly clumsy as can be seen in some of the code below.
data Triple a = T a a a deriving (Show)
instance Fractional a => Fractional (Triple a ) where
fromRational n = T (fromRational n) 0 0
(T g g' g'') / (T h h' h'') = T (g / h) ((g * h' - h * g')/ h * h) secDiff where
secDiff = ( 2*h'*(g*h' - h*g') - h*(g*h'' - h*g'')) / (h * h * h)
Note how messy the code is! It’s the result of apply the quotient rule to the result of applying the quotient Read More
Just recently I came across ‘Kaprekar’s Constant‘ and maybe Mr Kaprekar had too much spare time… but still, it is quite interesting. The idea is to take a 4 digit number where the digits are not all the same then make the largest and smallest numbers possible with these digits, subtract the smaller from the larger then rinse and repeat with the result of the subtraction. e.g start with 4123 4321 – 1234 = 3087 8730 – 0378 = 8352 8532 – 2358 = 6174 7641 – 1467 = 6174 … repeats… and in fact all 4 digits ‘converge’ to Read More
At the end of the previous post I had intended this posting to be an exploration of a recursive definition of Dual that will give an infinite (lazy) list of derivatives. However, there’s still a lot to play with using our simple
data Dual a = Dual a a Let’s try a simple function of two variables… and at we have Now we can evaluate at using dual numbers with a subscript of x or y to ‘remember’ where it came from…i.e we want but really are the ‘same thing’. Notice that the coefficients of are the same as the Read More
Overview Just recently I came across the interesting and, at first viewing, the rather abstract idea of dual numbers. I suppose they are no more or less abstract than other numbers… anyway the idea is similar to that of complex numbers where we have Dual numbers are quite similar, we have the dual number d as So now lets take this idea of a dual number and explore how adding, multiplying and dividing them might be defined. Addition and subtraction are simple – we just add the corresponding components – in much the same way that Read More
I think this post will wrap up the series on Vectors and Simple Mechanics and we’ll look at Simple Harmonic Motion (SHM) and compare the numerical solutions to SHM using the naive step function from the previous post – aka the ‘Euler‘ step and the more accurate ‘Euler-Cromer‘ method. Here’s the Euler Step from last time.
step :: Accnf -> Float -> State -> State
step f dt st@(t, r, v) = (t', r', v') where
t' = t + dt
r' = r ^+^ v ^* dt
v' = v ^+^ f st ^* dt
a very simple change to the above yields the ‘Euler-Cromer‘ step where the ‘new velocity‘ rather than the ‘old‘ is used to determine the ‘new‘ position.
ecStep :: Accnf -> Float -> State -> State
ecStep f dt st@(t, r, v) = (t', r', v') where
t' = t + dt
r' = r ^+^ v' ^* dt
v' = v ^+^ f st ^* dt
These two functions have the same signature,
Accnf -> Float -> State -> State which allows us to generalise a solution based Read More
In this post we’ll continue the previous one about vectors and take a look at calculating the path of a projectile and rendering that path to the screen. Imagine a single particle in three dimensional space, we can characterise a state for it as its position and velocity at some instant in time.
type State = (Time, Displacement, Velocity) From basic mechanics and Newtons laws we know that if no forces act on it then not a lot happens really! Time will increase and depending on your point of view, not much else will change. However, if some forces are acting then things become more Read More
Here we will continue the previous ideas about vectors and take a look at basic rendering of Vectors – for which we will use the Haskell Gloss package at https://hackage.haskell.org/package/gloss Gloss claims that “Gloss hides the pain of drawing simple vector graphics behind a nice data type and a few display functions. Gloss uses OpenGL under the hood, but you won’t need to worry about any of that. Get something cool on the screen in under 10 minutes.” and to be fair I found it very easy to use but not without problems when installing. However I believe these problems Read More
I like vectors! A long time ago my maths teacher introduced me to them and I just like the way three numbers (typically three) can express the notion of a position in space and combining these under different operations can produce other interesting properties. Just recently I found this paper Learn Physics by Programming in Haskell which gives a very interesting Haskell oriented discussion on vectors and their use in mechanical problems. Reading it has inspired me to create my own implementation of vectors, bore you all senseless with it and see where it goes. What I’d like to do Read More
This is the third part of parsing in Haskell. The previous posts are available at Haskell Parsers Part 1 and Haskell Parsers Part 2. In part one we defined a parser as a type and derived a very simple parser, a parser of char. Part two extended the ideas and created monadic, functorial and applicative instances of the parser type. This allowed us to combine parsers monadically – using ‘do’ or in an applicative style – e.g. using ‘<*>’ etc. Which to use is often a stylistic choice. In this post we’ll look at creating a few extra functions to Read More
This post continues from Haskell Parsers Part 1 and here we look at elevating the parser first to a Functor then an Applicative and finally to a Monad. Creating Functor, Applicative and Monad instances for the parser will provide useful operations and allow parsers to be combined in interesting and useful ways. For a Functor we need to implement the fmap function and in the context of a Parser the signature of fmap is
fmap :: (a -> b) -> Parser a -> Parser b
in other words “A function a -> b applied to the result of a parser of ‘a’ gives a parser of ‘b’” and the implementation Read More