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fix on several equations

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Lieuwe Leene
2025-01-22 16:17:09 +01:00
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15
content/posts/2024/generator-functions.md
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@ -71,7 +71,7 @@ $$ P(x) = (x-p_1) (x-p_2) (x-p_3) ... (x-p_n) $$
$$ P(x) = a_n x^n + ... + a_2 x^2 + a_1 x + a_0 $$
We can back calculate the corresponding initial conditions for our cascade
of integrators by considering the super-position of each component $a_n$
of integrators by considering the super-position of each component \\(a_n\\)
seperately. Lets denote our initial conditions as \\( c_n + ... + c_2 + c_1 + c_0 \\)
for our Nᵗʰ order polynomial. As shown in the diagram above the coefficient
\\(c_n\\) is directly accumulated on the left most integrator. It should be obvious
@ -82,19 +82,20 @@ contribution from \\(a_2\\) on the 2nd order derivative is calculated as taking
the 2nd derivative of \\(P(x)\\) and evaluating its value with x=0 which gives
us \\(c_2 = 2 * a_2\\). Now equating the 1st derivative from \\(a_2\\)
similarly gives \\(c_1 = a_2\\) and finally \\(c_0 = 0\\). If there were lower
order terms, the contribution from a_1 for example would be calculated
order terms, the contribution from \\(a_1\\) for example would be calculated
independently and added together.
This was a simpler example but one can reason that if the mapping for a particular \\( a_n \\) is
\\( m_n, ... , m_1, m_0 \\) such that for all \\( n \\) the initial conditions are
\\(c_n = m_n a_n\\). Then for some given mapping of a Nᵗʰ order polynomial
\\( m(n,i), ... , m(n,1), m(n,0) \\) such that for all \\( i \\) the initial conditions are
\\(c_n = \sum^n_{i=0} m(n,i) * a_i\\). The final value for the initial condition \\(c_n\\) is then
sum of all mapping terms from each \\(a_n\\). Then for some given mapping of a Nᵗʰ order polynomial
we can add one more integration stage to the far right integrating the output
to realize a N+1ᵗʰ order polynomial. This is equivalent to multiplying the
response with \\( x + 1 \\). Now it should be clear that when we equate the
derivative terms the N+1ᵗʰ order terms can be derived from the Nᵗʰ order terms
simply by adding the appropriate contributions after the aforementioned
multiplication. That is \\( k_n = m_n + m_{n-1} \\) where \\( k_n \\) are the
mapping terms for the N+1ᵗʰ order polynomial.
multiplication. That is \\( m(n+1,i) = m(n,i) + m(n,i-1) \\) where \\( m(n+1,i) \\)
are the mapping terms for the N+1ᵗʰ order polynomial.
Interestingly the mapping here generates a set of basis coefficients related to
the sterling-numbers of the second kind. More specifically the
[A019538](https://oeis.org/A019538) sequence. Using python and numpy as np
@ -122,7 +123,7 @@ def mapping_coefficients(order: int) -> np.array:
```
This function will derive the \\( m_n \\) mapping values based on our recursive
This function will derive the \\( m(n,i) \\) mapping values based on our recursive
derivation above. In order to then determine the initial conditions we similarly
iterate over the characteristic coefficients \\( a_n \\) and accumulate all
contributions to resolve the initial conditions \\( c_n \\).