Update 023_dysolve_propagator.md

Author: MathiB123

tutorials-v5/time-evolution/023_dysolve_propagator.md

@@ -22,7 +22,7 @@ This notebook shows how to compute time propagators with Dysolve using QuTiP. Dy
 
 For the moment, Dysolve can be used with the class `DysolvePropagator` and the function `dysolve_propagator` from QuTiP's solvers. They follow a similar structure to the class `Propagator` and the function `propagator`, another solver that also computes time propagators. 
 
-Let's start by importing the necessary packages.
+Here, we import the necessary packages.
 
 ```python
 from qutip.solver.dysolve_propagator import DysolvePropagator, dysolve_propagator
@@ -33,15 +33,15 @@ import numpy as np
 
 ### One qubit example using `DysolvePropagator`
 
-We have to define what $H_0$, $X$ and $\omega$ will be. Let's say $H(t) = \sigma_z + \cos(10t)\sigma_x$.
+We have to define what $H_0$, $X$ and $\omega$ will be. For example, $H(t) = \sigma_z + \cos(10t)\sigma_x$.
 
 ```python
 H_0 = sigmaz()
 X = sigmax()
 omega = 10.0
 ```
 
-Some options can be defined. `max_order` will be the order of approximation used when calculating a propagator. The higher this integer is, the more precise the results will be (at a cost of taking more time to calculate). `a_tol` is simply the absolute tolerance used in the calculations. Finally, a time propagator can be computed using subpropagators of time increment `max_dt`. Let's say `max_dt` is set to 0.25, then the propagator $U(1, 0)$ will come from the multiplication of the supropagators $U(0.25, 0)$, $U(0.5, 0.25)$, $U(0.75, 0.5)$ and $U(1, 0.75)$. This allows for more precise results when the evolution is over a long period of time. In our case, let's keep `a_tol` and `max_dt` to their default value, but let's change `max_order`.
+Some options can be defined. `max_order` will be the order of approximation used when calculating a propagator. The higher this integer is, the more precise the results will be (at a cost of taking more time to calculate). `a_tol` is simply the absolute tolerance used in the calculations. Finally, a time propagator can be computed using subpropagators of time increment `max_dt`. If `max_dt` is set to 0.25, then the propagator $U(1, 0)$ will come from the multiplication of the supropagators $U(0.25, 0)$, $U(0.5, 0.25)$, $U(0.75, 0.5)$ and $U(1, 0.75)$. This allows for more precise results when the evolution is over a long period of time. In our case, we keep `a_tol` and `max_dt` to their default value, but we change `max_order` to 5.
 
 ```python
 options = {'max_order': 5}
@@ -61,7 +61,7 @@ t_f = 1
 U = dy(t_f, t_i)
 ```
 
-This returns a single time propagator $U(t_f = 1, t_i = -1)$. To verify that the $U$ is correct, let's compare it to what `propagator` would return.
+This returns a single time propagator $U(t_f = 1, t_i = -1)$. To verify that the $U$ is correct, we compare it to what `propagator` would return.
 
 ```python
 # Solve using propagator
@@ -99,7 +99,7 @@ times = [-0.1, 0, 0.1]
 Us = dysolve_propagator(H_0, X, omega, times)
 ```
 
-Again, let's compare the results with `propagator`.
+Again, we compare the results with `propagator`.
 
 ```python
 # Solve using propagator