Sympy variables are created using unique string identifiers.


```python
import sympy as sp
import numpy as np
x = sp.Symbol("x")
y = sp.Symbol("y")
z = sp.Symbol("z")
```

One can form expression from symbols.
Sympy expressions are made up of numbers, symbols, and sympy functions.


```python
expression = x**2. + y**2. + z ** 2.
expression
```




    x**2.0 + y**2.0 + z**2.0



Two expressions may be added together to form a new one.


```python
other_expression = x**2.
expression += other_expression
expression
```




    2*x**2.0 + y**2.0 + z**2.0



One can form sympy `Matrix` objects.


```python
sp.Matrix([[1,2],[3,4]])
```




    Matrix([
    [1, 2],
    [3, 4]])



An important `Matrix` function is `eye(n)`, which forms a $n \times n$ identity matrix.


```python
sp.eye(3)
```




    Matrix([
    [1, 0, 0],
    [0, 1, 0],
    [0, 0, 1]])



One can stuff expressions into matrices, too.


```python
# One can stuff expressions into matrices
f1 = x**2.+y**2-z**2.
f2 = 2*x + y + z
function_matrix = sp.Matrix([f1,f2])
function_matrix
```




    Matrix([
    [x**2.0 + y**2 - z**2.0],
    [           2*x + y + z]])



One may compute the Jacobian of vector valued functions, too.


```python
function_matrix.jacobian([x,y,z]) # pass in a list of Sympy Symbols to take the Jacobian
```




    Matrix([
    [2.0*x**1.0, 2*y, -2.0*z**1.0],
    [         2,   1,           1]])



Sympy expressions can be evaluated by passing in a Python dictionary mapping Symbol `Symbol`s to specific values.


```python
x_val = 1.0
y_val = 2.0
z_val = 3.0
values={"x":x_val,"y":y_val,"z":z_val}
f1.subs(values)
```




    -4.00000000000000



One can even valuate the Jacobian of functions.


```python
J_mat = function_matrix.jacobian([x,y,z]).subs(values)
J_mat
```




    Matrix([
    [2.0, 4.0, -6.0],
    [  2,   1,    1]])



To convert a Sympy `Matrix` into a Numpy array, one may use the following:


```python
np.array(J_mat)
```




    array([[2.00000000000000, 4.00000000000000, -6.00000000000000],
           [2, 1, 1]], dtype=object)



After evaluating an expression in Sympy, the return type is a `sympy.Float`.
However, this is not readily usable by Numpy. Therefore, consider casting `sympy.Float` to a `numpy.float64`.


```python
J=np.array(J_mat).astype(np.float64)
J
```




    array([[ 2.,  4., -6.],
           [ 2.,  1.,  1.]])



At this point, one cna do all the usual stuff one would in Numpy.


```python
J.T@J
```




    array([[  8.,  10., -10.],
           [ 10.,  17., -23.],
           [-10., -23.,  37.]])



Symp's `Lambdify` can help increase the speed of Sympy's numerical computations.


```python
function_matrix.subs(values)
```




    Matrix([
    [-4.0],
    [ 7.0]])




```python
from sympy.utilities.lambdify import lambdify
array2mat = [{'ImmutableDenseMatrix': np.array}, 'numpy']
lam_f_mat = lambdify((x,y,z), function_matrix, modules=array2mat)
lam_f_mat(1,2,3)
```




    array([[-4.],
           [ 7.]])



Or if it is more convenient to have the function evaluation occur from a list of some sort, the Python `*` operator on lists can help.


```python
lam_f_mat(*[1,2,3])
```




    array([[-4.],
           [ 7.]])



82c6580b4cede372156d7cbe83e275e4187ed594
ipynb
Jupyter Notebook
misc/An Introduction to Sympy.ipynb
JiaheXu/MATH
9cb2b412ba019794702cacf213471742745d17a6
MIT
misc/An Introduction to Sympy.ipynb
JiaheXu/MATH
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MIT
misc/An Introduction to Sympy.ipynb
JiaheXu/MATH
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MIT
Qwen/Qwen-72B
1. YES
2. YES