I knew it!

I have in mind a game of dice that reminds me of my time as the Russian military attaché to the city state of Coruscant and its territories during the traitorous popular uprising fomented by the blasphemous teachings of a fundamentalist religious sect known as the Jedi. ]]>

An easy way to create rotationally symmetric functions, known as radial basis functions, is to apply univariate functions that are symmetric about zero to the distance between the interpolation's argument and their associated nodes. PDFs are a rich source of such functions and, in fact, the second bell shaped curve that we considered is related to that of the Cauchy distribution, which has some rather interesting properties. ]]>

An alternative approach is to construct a single function that passes through all of the points and, given that

and together are known as the spectral decomposition of

In this post, we shall add it to the

`ak`

library using the `householder`

and `givens`

functions that we have put so much effort into optimising.]]>

Splendid!

I propose a game that is popular amongst Antipodean opal scavengers as a means to improve their skill at guesswork.

Opals, as any reputable botanist will confirm, are the seeds of the majestic opal tree which grows in some abundance atop the vast monoliths of that region. Its mouth-watering fruits are greatly enjoyed by the Titans on those occasions when, attracted by its entirely confused seasons, they choose to winter thereabouts. ]]>

The columns of the matrix of transformations

and, since the product of

which is known as the spectral decomposition of

Last time we saw how we could efficiently apply the Householder transformations in-place, replacing the elements of

When I heard these rules I was reminded of the game of Cram and could see that, just like it, the key to figuring the outcome is to recognise that the Baron could always have kept the remaining draughts in a state of symmetry, thereby ensuring that however Sir R----- had chosen he shall subsequently have been free to make a symmetrically opposing choice. ]]>

implying that the columns of

where

From a mathematical perspective the combination of Householder transformations and shifted Givens rotations is particularly appealing, converging on the spectral decomposition after relatively few matrix multiplications, but from an implementation perspective using

`ak.matrix`

multiplication operations is less than satisfactory since it wastefully creates new `ak.matrix`

objects at each step and so in this post we shall start to see how we can do better.
]]>
We have thus far employed it to model the solar system itself, uniformly distributed bodies of matter and the accretion of bodies that are close to Earth's orbit about the Sun. Whilst we were most satisfied by its behaviour, I should now like to report upon an altogether more surprising consequence of its engine's action. ]]>

and, since the transpose of

which is known as the spectral decomposition of

Unfortunately, the way that we used Givens rotations to diagonalise tridiagonal symmetric matrices wasn't particularly efficient and I concluded by stating that it could be significantly improved with a relatively minor change. In this post we shall see what it is and why it works. ]]>

Good man! Good man!

I must say that the contrast between the warmth of this fire and the frost outside brings most vividly to my mind an occasion during my tenure as the Empress's ambassador to the land of Oz; specifically the time that I attended King Quadling Rex's winter masked ball during which his southern palace was overrun by an infestation of Snobbles! ]]>

known as the eigenvectors and the eigenvalues respectively, with the vectors typically restricted to those of unit length in which case we can define its spectral decomposition as the product

where the columns of

You may recall that this is a particularly convenient representation of the matrix since we can use it to generalise any scalar function to it with

where

You may also recall that I suggested that there's a more efficient way to find eigensystems and I think that it's high time that we took a look at it. ]]>