mtflib Tutorial: Advanced Functionality
This notebook explores advanced topics in mtflib, including substitution, composition, accessing coefficients, and the use of other elementary functions.
1. Initialization
As always, we begin by initializing the global settings for mtflib.
[1]:
import numpy as np
import sympy as sp
from IPython.display import display
from mtflib 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)
Initializing MTF globals with: _MAX_ORDER=8, _MAX_DIMENSION=3
Loading/Precomputing Taylor coefficients up to order 8
Global precomputed coefficients loading/generation complete.
Size of precomputed_coefficients dictionary in memory: 464 bytes, 0.45 KB, 0.00 MB
MTF globals initialized: _MAX_ORDER=8, _MAX_DIMENSION=3, _INITIALIZED=True
Max coefficient count (order=8, nvars=3): 165
Precomputed coefficients loaded and ready for use.
/tmp/ipykernel_932/2062807213.py:5: DeprecationWarning: mtflib is deprecated and has been superseded by sandalwood. Please migrate to the sandalwood package. For details, visit: https://github.com/shashi-manikonda/sandalwood
from mtflib import ComplexMultivariateTaylorFunction, mtf
2. Substitution and Composition
A powerful feature of mtflib 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+0.000000000000e+00j 0 (0, 0, 0)
1 1.000000000000e+00+0.000000000000e+00j 1 (0, 1, 0)
2 1.000000000000e+00+0.000000000000e+00j 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)): 1.0
Exponent (np.int32(3), np.int32(0), np.int32(0)): -0.16666666666666666
Exponent (np.int32(5), np.int32(0), np.int32(0)): 0.008333333333333333
Exponent (np.int32(7), np.int32(0), np.int32(0)): -0.0001984126984126984
4. Other Elementary Functions
mtflib 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.
$\displaystyle - \frac{x^{6} y}{360} - \frac{x^{5} y^{2}}{60} + \frac{x^{5}}{120} - \frac{x^{4} y^{3}}{18} + \frac{x^{4} y}{12} - \frac{x^{3} y^{4}}{9} + \frac{x^{3} y^{2}}{3} - \frac{x^{3}}{6} - \frac{2 x^{2} y^{5}}{15} + \frac{2 x^{2} y^{3}}{3} - x^{2} y - \frac{4 x y^{6}}{45} + \frac{2 x y^{4}}{3} - 2 x y^{2} + x - \frac{8 y^{7}}{315} + \frac{4 y^{5}}{15} - \frac{4 y^{3}}{3} + 2 y$