`sandalwood` Tutorial: Advanced Functionality

This notebook explores advanced topics in sandalwood, including substitution, composition, accessing coefficients, and the use of other elementary functions.

1. Initialization

As always, we begin by initializing the global settings for sandalwood.

[1]:

import numpy as np
import sympy as sp
from IPython.display import display

from sandalwood import ComplexMultivariateTaylorFunction, mtf

if not mtf.get_mtf_initialized_status():
    mtf.initialize_mtf(max_order=8, max_dimension=3)
else:
    print("MTF globals are already initialized.")

# Define some variables for our examples
x = mtf.var(1)
y = mtf.var(2)
z = mtf.var(3)

2. Substitution and Composition

A powerful feature of sandalwood is the ability to substitute variables with constants or even other Taylor functions.

Substitution with a Constant

Replace a variable in a Taylor function with a constant value.

[2]:

f_xyz = x + y + z**4
print(f"Original function f(x, y, z):\n{f_xyz}\n")

# Substitute x with 0.1
g_yz = f_xyz.substitute_variable(1, 0.1)
print(f"After substituting x=0.1, we get g(y, z):\n{g_yz}")

Original function f(x, y, z):
         Coefficient  Order  Exponents
0 1.000000000000e+00      1  (1, 0, 0)
1 1.000000000000e+00      1  (0, 1, 0)
2 1.000000000000e+00      4  (0, 0, 4)


After substituting x=0.1, we get g(y, z):
         Coefficient  Order  Exponents
0 1.000000000000e-01      0  (0, 0, 0)
1 1.000000000000e+00      1  (0, 1, 0)
2 1.000000000000e+00      4  (0, 0, 4)

Composition with other Taylor Functions

Substitute variables with other Taylor functions to compose them.

[3]:

# Outer function: f(x, y) = x^2 + y^3
f_xy = x**2 + y**3

# Substituting functions:
# g(y) = 1 + y
# h(y, z) = 1 + y*z
g_y = 1 + y
h_yz = 1 + y * z

# Perform composition: f(g(y), h(y,z))
composed_f = f_xy.compose({1: g_y, 2: h_yz})
print(f"Composed function f(g(y), h(y,z)):\n{composed_f}")

Composed function f(g(y), h(y,z)):
         Coefficient  Order  Exponents
0 2.000000000000e+00      0  (0, 0, 0)
1 2.000000000000e+00      1  (0, 1, 0)
2 1.000000000000e+00      2  (0, 2, 0)
3 3.000000000000e+00      2  (0, 1, 1)
4 3.000000000000e+00      4  (0, 2, 2)
5 1.000000000000e+00      6  (0, 3, 3)

3. Accessing Coefficients

You can directly access the coefficients and exponents of a Taylor function.

[4]:

sin_x = mtf.sin(x)
coefficients = {tuple(exp): coeff for exp, coeff in zip(sin_x.exponents, sin_x.coeffs)}

print("Coefficients of sin(x):")
for exp, coeff in coefficients.items():
    print(f"  Exponent {exp}: {coeff}")

Coefficients of sin(x):
  Exponent (np.int32(1), np.int32(0), np.int32(0)): (1+0j)
  Exponent (np.int32(3), np.int32(0), np.int32(0)): (-0.16666666666666666+0j)
  Exponent (np.int32(5), np.int32(0), np.int32(0)): (0.008333333333333333+0j)
  Exponent (np.int32(7), np.int32(0), np.int32(0)): (-0.0001984126984126984+0j)

4. Other Elementary Functions

sandalwood supports a variety of other elementary functions.

[5]:

print("--- Example: mtf.exp(x+2*y) ---")
x = mtf.var(1)
y = mtf.var(2)
mtf_sin = mtf.sin(x + 2 * y)
sympy_sin_expr = mtf_sin.symprint()
print("The following is a SymPy object, which will render beautifully in a notebook.")
display(sympy_sin_expr)

# --- Example with ComplexMultivariateTaylorFunction ---
print("--- Example: mtf.exp(i*x) ---")
i = 1j

x_complex = ComplexMultivariateTaylorFunction.from_variable(var_index=1, dimension=1)
mtf_complex_exp = mtf.exp(i * x_complex)
sympy_complex_expr = mtf_complex_exp.symprint(symbols=["x"])
print("The following is a SymPy object for a complex function.")
display(sympy_complex_expr)

# --- Example with custom coefficient formatting ---
print("--- Example: Custom coefficient formatting ---")


def custom_formatter(c, p):
    # A simple rational formatter
    if np.iscomplexobj(c):
        return sp.Rational(c.real).limit_denominator(10**p) + sp.I * sp.Rational(
            c.imag
        ).limit_denominator(10**p)
    else:
        return sp.Rational(c).limit_denominator(10**p)


# Re-initialize for the 2D mtf_sin
x = mtf.var(1)
y = mtf.var(2)
mtf_sin = mtf.sin(x + 2 * y)

sympy_custom_format_expr = mtf_sin.symprint(
    precision=3, coeff_formatter=custom_formatter
)
print("The following is a SymPy object with custom rational formatting.")
display(sympy_custom_format_expr)

--- Example: mtf.exp(x+2*y) ---
The following is a SymPy object, which will render beautifully in a notebook.

$\displaystyle - 0.000198413 x^{7} - 0.00277778 x^{6} y - 0.0166667 x^{5} y^{2} + 0.00833333 x^{5} - 0.0555556 x^{4} y^{3} + 0.0833333 x^{4} y - 0.111111 x^{3} y^{4} + 0.333333 x^{3} y^{2} - 0.166667 x^{3} - 0.133333 x^{2} y^{5} + 0.666667 x^{2} y^{3} - 1.0 x^{2} y - 0.0888889 x y^{6} + 0.666667 x y^{4} - 2.0 x y^{2} + 1.0 x - 0.0253968 y^{7} + 0.266667 y^{5} - 1.33333 y^{3} + 2.0 y$

--- Example: mtf.exp(i*x) ---
The following is a SymPy object for a complex function.

$\displaystyle 2.48016 \cdot 10^{-5} x^{8} - 0.000198413 i x^{7} - 0.00138889 x^{6} + 0.00833333 i x^{5} + 0.0416667 x^{4} - 0.166667 i x^{3} - 0.5 x^{2} + 1.0 i x + 1.0$

--- Example: Custom coefficient formatting ---
The following is a SymPy object with custom rational formatting.

$\displaystyle - \frac{x^{6} y}{360} - \frac{x^{5} y^{2}}{60} + \frac{x^{5}}{120} - \frac{x^{4} y^{3}}{18} + \frac{x^{4} y}{12} - \frac{x^{3} y^{4}}{9} + \frac{x^{3} y^{2}}{3} - \frac{x^{3}}{6} - \frac{2 x^{2} y^{5}}{15} + \frac{2 x^{2} y^{3}}{3} - x^{2} y - \frac{4 x y^{6}}{45} + \frac{2 x y^{4}}{3} - 2 x y^{2} + x - \frac{8 y^{7}}{315} + \frac{4 y^{5}}{15} - \frac{4 y^{3}}{3} + 2 y$

sandalwood Tutorial: Advanced Functionality