Learn Taylor Expansion for Multivariate Functions

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The Taylor expansion is a powerful tool that allows you to approximate a function near a specific point using its derivatives. For functions of a single variable, you have already seen how the Taylor series provides a polynomial that closely matches the function near a point. When you move to functions of several variables, the idea is similar, but the formula involves gradients and higher-order derivatives like the Hessian matrix.

For a function $f(x, y)$ that is sufficiently differentiable, the second-order Taylor expansion about the point $(a, b)$ is:

\begin{aligned} f(x, y) \approx \underbrace{f(a, b)}_{\text{Base value}} &+ \underbrace{f_x(a, b)(x - a) + f_y(a, b)(y - b)}_{\text{Linear approximation (1st order)}} \\ &+ \underbrace{\frac{1}{2} \Big[ f_{xx}(a, b)(x - a)^2 + 2f_{xy}(a, b)(x - a)(y - b) + f_{yy}(a, b)(y - b)^2 \Big]}_{\text{Quadratic correction (2nd order)}} \end{aligned}

Here's how you build this expansion step by step:

Evaluate the function at the expansion point: $f(a, b)$ ;
Add the first-order terms, which use the gradient (the vector of partial derivatives): $f_x(a, b)*(x - a)$ and $f_y(a, b)*(y - b)$ ;
Add the second-order terms, which use the second partial derivatives (entries of the Hessian matrix): $f_{xx}(a, b)*(x - a)^2$ , $f_{yy}(a, b)*(y - b)^2$ , and the mixed partial $f_{xy}(a, b)*(x - a)*(y - b)$ ;
Each second-order term is multiplied by $0.5$ to account for the Taylor expansion formula.

This approximation gives you a quadratic surface that matches the function's value, slope, and curvature at the point $(a, b)$ . The more derivatives you include, the more accurate your approximation becomes near the expansion point.


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import sympy as sp
import numpy as np

# 1. Define symbols and the function
x, y = sp.symbols('x y')
f = x**2 * y + y**3

# 2. SymPy automatically computes the Gradient and the Hessian matrix
grad_f = sp.Matrix([sp.diff(f, x), sp.diff(f, y)])
hessian_f = sp.hessian(f, (x, y))

# 3. Set the coordinates
a, b = 1.0, 2.0
x_val, y_val = 1.1, 2.05

# 4. Substitute the base point (a, b) and convert to NumPy arrays
subs_dict = {x: a, y: b}

f_ab = float(f.subs(subs_dict))
grad_ab = np.array(grad_f.subs(subs_dict), dtype=float).flatten() # Vector (2,)
hessian_ab = np.array(hessian_f.subs(subs_dict), dtype=float)     # Matrix (2, 2)

# 5. Create the difference vector (Delta)
delta = np.array([x_val - a, y_val - b])

# 6. Vector-matrix Taylor formula (using @ for matrix multiplication)
taylor_approx = f_ab + grad_ab @ delta + 0.5 * (delta @ hessian_ab @ delta)

print(f"Second-order Taylor approximation at ({x_val}, {y_val}): {taylor_approx:.4f}")

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Section 1. Chapter 9

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Section 1. Chapter 9