User:Jgirata

$$ A = \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}\mathbf{b} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} $$

$$ F(\mathbf{x}) = \frac{1}{2}(5x^2 + 4xy + y^2) - x - y $$

Taylor's Theorem
$$ f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots + \frac{f^{(n)}(0)}{n!}x^n + R_n(x) $$

$$ \begin{align} R_n(x) & = \frac{1}{n!} \int_0^x f^{(n + 1)}(t)(x - t)^ndt \\ & = \frac{f^{(n+1)}(c)}{(n + 1)!}x^{n+1} \end{align} $$

Methods of numerical approximation
$$ \begin{align} T_n & = \frac{b - a}{n} \left[ \frac{f(x_0) + f(x_1)}{2} + \frac{f(x_1) + f(x_2)}{2} + \dots + \frac{f(x_{n-1}) + f(x_n)}{2} \right] \\ & = \frac{b - a}{2n} \left[ f(x_0) + 2f(x_1) + \dots + 2f(x_{n-1}) + f(x_n) \right] \end{align} $$

$$ S_n = \frac{b - a}{6n} \left \{ f(x_0) + f(x_n) + 2 \left[ f(x_1) + \dots + f(x_{n-1}) \right] + 4 \left[ f \left( \frac{x_0 + x_1}{2} \right) + \dots + f \left( \frac{x_{n-1} + x_n}{2} \right) \right] \right \} $$

Error estimates of numerical approximation
$$ E^T_n = -\frac{(b - a)^3}{12n^2}f''(c) $$

$$ E^S_n = -\frac{(b - a)^5}{2880n^4}f^{(4)}(c) $$

Parallel and perpendicular components of a vector
$$ \begin{array}{lcl} \mathbf{x} \text{, } \mathbf{y} \isin \mathbb{R}^n & & \\ \mathbf{x} = \mathbf{x}_{\parallel} + \mathbf{x}_{\perp} & & \\ \mathbf{x}_{\parallel} = (\mathbf{u} \cdot \mathbf{x})\mathbf{u} & & \mathbf{x}_{\parallel} \text{ is parallel to } \mathbf{y} \\ \mathbf{x}_{\perp} = \mathbf{x} - (\mathbf{u} \cdot \mathbf{x})\mathbf{u} & & \mathbf{x}_{\perp} \text{ is perpendicular to } \mathbf{y} \\ \mathbf{u} = \left( \frac{1}{|\mathbf{y}|} \right)\mathbf{y} & & \mathbf{u} \text{ is the unit vector in the direction of } \mathbf{y} \end{array} $$

Projection of a Matrix
$$ \mathbf{P}_s = \mathbf{V}(\mathbf{V}^t\mathbf{V})^{-1}\mathbf{V}^t $$