python in learning math

limit behavior of transition probability of a given markov chain

P.S. Note that below code use @ operator which requires python 3.5 or above, see matrix_operator.

>>> import numpy as np
>>> np.set_printoptions(precision=4)
>>> g = lambda trans_mat, cnt=20: reduce(lambda x1, x2: print(x1) or x1@x2, (trans_mat for _ in range(cnt)), np.eye(len(trans_mat)))
>>> t, t1 = np.array([[.5,.4,.1],[.3,.4,.3],[.2,.3,.5]]), \
np.array([[.7,0,.3,0],[.5,0,.5,0],[0,.4,0,.6],[0,.2,0,.8]])
>>> g(t,10)
[[ 1.  0.  0.]
[ 0.  1.  0.]
[ 0.  0.  1.]]
[[ 0.5  0.4  0.1]
[ 0.3  0.4  0.3]
[ 0.2  0.3  0.5]]
[[ 0.39  0.39  0.22]
[ 0.33  0.37  0.3 ]
[ 0.29  0.35  0.36]]
[[ 0.356  0.378  0.266]
[ 0.336  0.37   0.294]
[ 0.322  0.364  0.314]]
[[ 0.3446  0.3734  0.282 ]
[ 0.3378  0.3706  0.2916]
[ 0.333   0.3686  0.2984]]
[[ 0.3407  0.3718  0.2875]
[ 0.3384  0.3708  0.2908]
[ 0.3368  0.3702  0.2931]]
[[ 0.3394  0.3713  0.2894]
[ 0.3386  0.3709  0.2905]
[ 0.338   0.3707  0.2913]]
[[ 0.3389  0.3711  0.29  ]
[ 0.3387  0.371   0.2904]
[ 0.3385  0.3709  0.2906]]
[[ 0.3388  0.371   0.2902]
[ 0.3387  0.371   0.2903]
[ 0.3386  0.3709  0.2904]]
[[ 0.3387  0.371   0.2903]
[ 0.3387  0.371   0.2903]
[ 0.3387  0.371   0.2904]]
Out[20]:
array([[ 0.3387,  0.371 ,  0.2903],
[ 0.3387,  0.371 ,  0.2903],
[ 0.3387,  0.371 ,  0.2903]])

plot a sequence of function with different hyperparameter

Given f(x)=xa−1e−x/b (which is integrand of gamma function when b=1, see Gamma Function, we can plot a series of plots of functions.

import numpy as np
from sympy import *
%matplotlib inline
x = Symbol('x')
def g(a=2.,b=1.):
return x**(a-1) * exp(-x/b)
for v in np.linspace(0.5,2,6):
plot(g(v),(x,0,v-1+4),title = 'a=%s, b=1.0' % v)

the output is show as