| brain activity log
02.04.2006 Sunday - diary and some goodies
Yesterday we have been in Florence
and returned really late. Couldn't sleep this morning, so woken up early.
In the first minutes thought that because of the missing hours of sleep
and because of recent developments it would be a bad day. ...but after opening
the window discovered that there was a fantastic Sun outside:
been waiting for this since several months.
Never undestimate the energy emitted by the Sun:
a ball of plasma with a diameter of 1.392 million km and a mass of about 2.0x10^30 kg. Approx 5500 K at surface...
... man ... that's HOT :D
Etherea sent me an english translation of my CV.
Thnx Ethy :******
In her blog I have also found this funny image of Iakko :D
(Hum.. Cle... there is no way to link your blog directly without loosing the external frame)
While getting my coffee I've added a couple of comparison operators to the SSMatrix
class. The scalar ops are still missing, but the sun is calling me thus I'll add them the next time :).
I've also upped the first SSComplex template that can be used with SSMatrix. Enjoy :)
It seems that there is some new movement around KVirc.
For chronicle purposes I also to need to record the following (important) note.
Segmentation fault. Core dumped.
"Hum... still need some debugging"
01.04.2006 Saturday - April Fools
1st April. In many countries this is the day of "Jokes".
"Prima Aprilis: uwazaj bo sie pomylisz".. which is in Polish and means
"1 April, be careful because you might be wrong". "April Fools" and "Pesce d'Aprile".
The (net-)jokes already started.
Heh.. after you read some of them, all the of the web starts to look like a joke :D
An Insightful post on slashdot stated:
Try date -u and you will see that it is in fact April Fool's Day.
Note that April Fool's Day, as defined by the International April Pranksters Association,
goes by UTC, not by local time zones. Because IAPA is not widely recognized as an international
standards body and many people's problems to understand time zones, this has led to some problems
since its introduction in 2002. It is especially uncertain for regional publications, which are
reluctant to adopt the new standard because they fear to irritate their audience. One example
is the Hubsborough Gazette, which famously spread confusion on the evening of March 31st, 2004 (EST)
when an article claiming that aliens have attacked the Whitehouse appeared on their website.
Despite the seemingly obvious nature of the hoax, many believed it and called the authorities or
local clergery for guidance. One family even is reported to have spent two weeks in their backyard
bunker. Since then, the editor has announced that they will only publish April Fool's articles during
the hours when April 1st of their local time zone and UTC overlap, and take down articles afterwards.
Many publications have followed their example in the following years.
The guys from Centrica fooled me about bex2 emitting all zero size output... that would be a problem.
It is a 1st April joke..isn't it ? :D
31.03.2006 Friday - diary and other goodies
Hum.. a lot of images (in random order):
- The Diophantine Equation
- A nearly black cat
- The Youla-Kucera theorem (kinda hard proof for the multivariable case)
- A pair of sweet lips :)
- The missed Frantz sister's party: mea culpa :(
- Francesca in the lecture room
- Silvia pressing for messaging
- Samuel spending up to the last cent for a drum set
- The two girls near Skile tripping three times around the block
- A sensation close to deja-vu
- Iakko laughing hard after drinking 4 Hoegarden glasses :D
- ...
Nice four days :)
The promised LU decomposition code is here. Pretty straightforward
once the formula is known. The file contains also the algorithms for matrix determinant (computed
either by LU or by a recursive adaptation of the Liebnitz fomula) and for matrix inversion. Enjoy :)
"I have a monster in me and if I wake him up he is probably going to kill me"
27.03.2006 Monday - The LU decomposition of a matrix
Back to my fast determinant computed by using the LU decomposition of a matrix.
The idea is to decompose the matrix A in a product of two matrixes L and U in that
L is lower triangular and U is upper triangular. We then know that det(A) = det(L) * det(U)
and the determinants of triangular matrixes are trivial to compute.
The LU decomposition is not unique so we can finally choose some arbitrary elements
of L or U. We choose the diagonal of L to be made of all ones so the determinant of L is directly
computed as 1.
Determining L and U such that A = LU is a system of m^2 equations (where m is the rank of A)
in 2 * m^2 unknown variables. The bounds of L being lower triangular with a diagonal of all ones
and U being upper triangular reduce the number of unknown variables to m^2. The system looks
like the following one (3 x 3 case):
Which can be expanded to:
This shows us that we can compute each row of L and U independently by solving a
m-equantion in m-variables system. In the 3x3 case the three systems look like:
Which can be solved by inverting the coefficient matrix or
by Gauss-Jordan manipulation. The Gauss-Jordan solution for the 3x3 case looks like:
There's obviously a pattern in the solution vector and it can be written down as follows:
Now we have a LU decomposition and can compute the determinant just by multiplying together
the diagonal elements of U. Nice eh ?
This is pretty fast. While the Liebnitz method costs like n!, the LU method costs like n^3
which is faaaaaar bettter :)
Now one should prove that the LU decomposition exists for every non-singular matrix
and verify the numeric stability of the method... I'll do it in the next days
while implementing this stuff and report you later.
26.03.2006 Sunday - 1 is a number
- 1 is a number
- Every number has a successor (next one) that itself is a number
- The number "before" the successor is the predecessor (of the successor)
This allows us to define the set N of natural numbers.
We give each number a "name" (one, two, three...) that is defined by rules beyond the scope of this document.
- the predecessor of 1 is 0
We include 0 in the set N to allow the following definitions.
- We define the operation of sum as follows:
The sum of number a and number b is computed by applying the "successor" rule
to a and the "predecessor" rule to b until b reaches 0.
For example summing four to three leads to the following steps:
- successor of four is five, predecessor of three is two
- successor of five is six, predecessor of two is one
- successor of six is seven, predecessor of one is zero
- the result is seven
- We define the operation of subtraction as the inverse of sum. Being pedantic one
can define it in terms of successor and predecessor itself: subtracting b from a means
applying the predecessor rule to both a and b until b reaches zero.
Oops... we can't subtract b from a when b is in the successor chain of a: we haven't defined
the predecessor of zero. We define it now as well as all the other predecessors.
This leads us to the definition of the set Z of relative integers. Z obviously includes N.
- We define multiplication of two numbers a and b as summing b to zero a times. Being pedantic:
we start with an accumulator of 0 and sum b to it while applying the predecessor rule
to a until it reaches zero.
- We define division as the inverse operation of multiplication. If a multiplied b given c then
c divided b gives a and c divided a gives b.
Oops... we can now attempt to divide b by a when there is no number in Z the multiplied a gives b.
- Such a "thing" is still a number and is exactly b/a: a rational number.
This leads us to the definition of the set Q of rational numbers: the ones that can be rappresented
by a fraction (ratio). Q obviously includes Z and thus N.
- We define the "power" operation of numbers a and b as muliplying 1 for a b times.
Thus a to the power of b is computed as multiplying 1 for a and applying the predecessor rule
to b until zero is reached.
- We define the extracion of the N-th root as the inverse of the power operation. If a to the power of
b given c then the b-th root of c gives a.
Oops... the N-th root of certain numbers is not inside Q: you can attempt to extract the N-th root
of a number x when there is no number y in Q that leads y^n=x.
The square root of 2 is such a number. This leads us to the definition of the set R of real numbers.
Obviously R contains Q and thus Z and N. R is dense: between two numbers in R there is always another number.
Oops... this still does not allow the extraction of certain N-th roots of negative numbers.
The square root of -1 doesn't lie in R. In other words: there is no such number x in R that
x^2=-1. We define such a number as the immaginary unity j (engineers use j :) and extend
the set R with it. This leads us to the definition of the set C of complex numbers: the
ones that have a real part (that lies in R) and an immaginary part (that doesn't lie in R).
We rappresent such numbers as a+jb.
...
Just to make sure you know :)
Now go and conquer the world!
want more ?
... really ? :D
Browse around then.
You're viewing 5 posts per page: you can view more or less, if you want.
The entries marked in red are the ones you're viewing now.
| 
