Matrixexponential and Normaldistribution/Gauss' function (II)
Gottfried Helms
Annette.Warlich at t-online.de
Sat Mar 4 21:54:47 CET 2006
With another important known function the matrix-exponential
can be associated to provide an extremely simple structure.
My original problem was to find derivatives and integrals
of the Gauss'-error-function for smoothing of statistical
data, so the connection here is somehow surprising...
Gauss' function may be written as:
NV(z) = 1 / sqrt(2*pi) * exp ( - z²/2)
denote the left constant term as "c" and the exponentialterm
as E(z) such that:
f:= NV(z) = cE(z)
and write the derivatives in a table, one can see, how the
derivatives are just polynomial multiples of the original
function itself (which is not so surprising, since it is
an exponential function)
f = cE(z) *( 1 )
f' = cE(z) *( -1 z)
f" = cE(z) *( -1 + 1 z^2)
f"' = cE(z) *( 3 z - 1 z^3)
f"" = cE(z) *( 3 - 6 z^2 + 1 z^4)
f5 = cE(z) *( -15 z +10 - 1 z^5)
f6 = cE(z) *(-15 +45 z^2 -15 z^4 +1 z^6)
... usw
The triangle of the coefficients of the polynomials has a nice
structure (I used "o" for zeroes):
Row: coefficients
0 1
1 o -1
2 - 1 o 1
3 o 3 o -1
4 3 o -6 o 1
5 o -15 o 10 o -1
6 -15 o 45 o -15 o 1
7 o 105 o -105 o 21 o -1
8 105 o -420 o 210 o -28 o 1
which has a simple recursive structure by the rules of
computing of derivatives and this similarity to the
binomial-triangle.
One nice thing, by the way, one can apply the recursion-
rule inversely to find the coefficients for the *integrals*
too; I write them here as rows with negative row-indexes
(row index 0 denotes the original function)
The triangle will be continued upwards and *rightwards*,
where now the polynoms hafe infinitely many coefficients.
(The symbol "!²" means the two-step-factorial-function
n*(n-2)*(n-4)*(n-6)*...*1 )
-5 o 1 o 3/3 o 6/5!² o 10/7!² o ... *1/3 (to multiply all entries by 1/3)
-4 1 o 2 o 3/5!² o 4/7!² o 5/9!² ... *1/3 (to multiply all entries by 1/3)
-3 o 1 o 2/3 o 3/5!² o 4/7!² o ... ^
-2 1 o 1/3 o 1/5!² o 1/7!² o ... |
-1 o 1 o 1/3 o 1/5!² o 1/7!² o ... integrals
0 1 ------------------------------------------------ original function ---------
1 o -1 derivatives
2 - 1 o 1 |
3 o 3 o -1 V
4 3 o -6 o 1
5 o -15 o 10 o -1
6 -15 o 45 o -15 o 1
7 o 105 o -105 o 21 o -1
8 105 o -420 o 210 o -28 o 1
The coefficients for the integrals have again ovious patterns;
but I don't want to go into that here.
-----------------------------------
Since the lower triangle of coefficients reminds to the binomial-
triangle, I applied the same idea to that triangle as I did
with the matrix-logarithm/exponential to the binomial-triangle:
"Gauß-Triangle"
| 1 0 0 0 0 0 0 0 |
| 0 -1 0 0 0 0 0 0 |
| -1 0 1 0 0 0 0 0 |
| 0 3 0 -1 0 0 0 0 |
| 3 0 -6 0 1 0 0 0 |
| 0 -15 0 10 0 -1 0 0 |
| -15 0 45 0 -15 0 1 0 |
| 0 105 0 -105 0 21 0 -1 |
inverted:
| 1 0 0 0 0 0 0 0 |
| 0 -1 0 0 0 0 0 0 |
| 1 0 1 0 0 0 0 0 |
| 0 -3 0 -1 0 0 0 0 |
| 3 0 6 0 1 0 0 0 |
| 0 -15 0 -10 0 -1 0 0 |
| 15 0 45 0 15 0 1 0 |
| 0 -105 0 -105 0 -21 0 -1 |
and - surprise, surprise - the matrix-logarithm is
A= ln(inv("Gauß-Triangle"))
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 1 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 6 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 15 0 0 0 |
| 0 0 0 0 0 0 0 0 |
While the matrix-logarith of the binomial-triangle has the sequence
of natural numbers n in the *first* subdiagonal, we have here the triangular
numbers (2n-1)*2n / 2 in the *second* subdiagonal (but only each second
entry is different from zero.
I would call things like that maximal compression of information...
... and a strong surprise.
But one can even smooth that thing and proceed;
use A as the "Gauss-triangle", and compute its square:
A1 = A*A
| 1 0 0 0 0 0 0 0 |
| 0 1 0 0 0 0 0 0 |
| -2 0 1 0 0 0 0 0 |
| 0 -6 0 1 0 0 0 0 |
| 12 0 -12 0 1 0 0 0 |
| 0 60 0 -20 0 1 0 0 |
| -120 0 180 0 -30 0 1 0 |
| 0 -840 0 420 0 -42 0 1 |
and now the matrix logarithm has no holes in the sequence:
lA1 = ln(A*A)
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 2 0 0 0 0 0 0 0 |
| 0 6 0 0 0 0 0 0 |
| 0 0 12 0 0 0 0 0 |
| 0 0 0 20 0 0 0 0 |
| 0 0 0 0 30 0 0 0 |
| 0 0 0 0 0 42 0 0 |
and even better: ---------------------------------------------
A2 = sqrt(A*A)
| 1 0 0 0 0 0 0 0 |
| 0 1 0 0 0 0 0 0 |
| -1 0 1 0 0 0 0 0 |
| 0 -3 0 1 0 0 0 0 |
| 3 0 -6 0 1 0 0 0 |
| 0 15 0 -10 0 1 0 0 |
| -15 0 45 0 -15 0 1 0 |
| 0 -105 0 105 0 -21 0 1 |
then the matrix-logarithm has the -say- most simple representation (second only
to that for the binomial-matrix):
LA2 = ln(A2)
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 1 0 0 0 0 0 0 0 |
| 0 3 0 0 0 0 0 0 |
| 0 0 6 0 0 0 0 0 |
| 0 0 0 10 0 0 0 0 |
| 0 0 0 0 15 0 0 0 |
| 0 0 0 0 0 21 0 0 |
which is n*(n+1)/2 in its *second* subdiagonal.
so that ( note that using matrix-squaring means A*A, not A*transp(A) ! )
exp("natural numbers") = "binomial-triangle"
_________________
exp("triangular-numbers") = \/"Gauß-triangle" ²
This seems to be the start of an interesting family of numbers/functions...
Gottfried
Note that the coefficients of the "Gauss-triangle" are known in the OEIS
as
> 1,-1,-1,1,3,-1,3,-6,1,-15,10,-1,-15,45,-15,1 ,10,1 id:A096713
>
> Triangle of nonzero coefficients of matching polynomial of complete graph of order n.
http://www.research.att.com/~njas/sequences/?q=id%3AA096713
and in mathworld in context to the "hermitean polynomials",
http://mathworld.wolfram.com/HermitePolynomial.html
(around "eq (60)")
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