I suppose since the advent of cheap calculators the task of working out the square root of a number has become simply a matter of pressing a few buttons and reading off the answer. But there was a time when we didn’t have calculators or slide rules or log tables – like these! Before all of these we used one of several ways to calculate a square root. One technique, which I’ll describe, is a bit like long division and works like this… First off group the numbers in pairs starting from the least significant digit before any decimal point Read More

# Ever Decreasing Circles.

A few days ago I came across this really interesting problem on the ‘dailyprogrammer‘. Here’s the question. Imagine you’ve got a square in 2D space, with axis values between 0 and 1, like this diagram. The challenge today is conceptually simple: can you place a circle within the square such that exactly half of the points in the square lie within the circle and half lie outside the circle, like here? You’re going to write a program which does this – but you also need to find the smallest circle which solves the challenge, ie. has the minimum area of Read More

# Dual Numbers and Differentiation. Part 3.

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.

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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

# Kaprekar’s Constant.

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

# Dual Numbers and Differentiation. Part 2.

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

# Dual Numbers and Differentiation. Part 1.

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

# Haskell, Vectors and Simple Mechanics – part 4.

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.

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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.

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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

# Haskell, Vectors and Simple Mechanics – part 3.

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

# Haskell, Vectors and Simple Mechanics – part 2.

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

# Haskell, Vectors and Simple Mechanics – part 1.

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