<p>

<div align="right">
Massimo Nocentini<br>
<small>
<br>February 4, 2018: exponential P
</small>
</div>
</p>
<br>
<br>
<div align="center">
<b>Abstract</b><br>
Exponential $\mathcal{P}$, according to Paul Barry's book.
</div>


```python
from sympy import *
from sympy.abc import n, i, N, x, lamda, phi, z, j, r, k, a, alpha

from commons import *
from matrix_functions import *
from sequences import *
import functions_catalog

init_printing()
```

---


```python
m = 10
```


```python
eP = Matrix(m, m, lambda n,k: factorial(n)*binomial(n,k)/factorial(k))
eP
```


```python
inspect(eP)
```




    nature(is_ordinary=False, is_exponential=True)




```python
eP_pm = production_matrix(eP)
eP_epm = production_matrix(eP, exp=True)

eP_pm, eP_epm
```


```python
F = Matrix(m, m, diagonal_func_matrix(factorial))
F_inv = F**(-1)

F, F_inv
```

In order to factorize `eP` as `F U F^{-1}`, for some matrix `U`


```python
B = F_inv * eP * F
B
```


```python
U = Matrix(m, m, rows_shift_matrix(by=1))
U
```


```python
F_inv * U * F
```


```python
F_inv * U * F * B
```


```python
B**(-1) * F_inv * U * F * B
```


```python
F * B**(-1) * F_inv * U * F * B * F_inv
```

---


```python
P = Matrix(m, m, binomial)
```


```python
P_bar = Matrix(m, m, lambda i, j: binomial(i, j) if j < i else 0)
P_bar
```


```python
production_matrix(P_bar[1:,:-1], exp=False), production_matrix(P_bar[1:,:-1], exp=True)
```


```python
j=3
(P_bar**j).applyfunc(lambda i: i/factorial(j))
```

---
<a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/"></a><br />This work is licensed under a <a rel="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License</a>.

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notes/matrices-functions/exp-Pascal.ipynb
massimo-nocentini/simulation-methods
f7578a9719b1a22e5a25a8de85cc229aef4c259d
MIT
notes/matrices-functions/exp-Pascal.ipynb
massimo-nocentini/simulation-methods
f7578a9719b1a22e5a25a8de85cc229aef4c259d
MIT
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