1. **Stating the problem:** We want to find the result of the second iteration using Newton's method for the function $$f(x) = (x_1^2 + x_2 - 11)^2 + (x_1 + x_2^2 - 7)^2$$ starting from the initial point $x^{(0)} = [6,6]$. 2. **Newton's method formula for multivariable functions:** $$x^{(k+1)} = x^{(k)} - H_f(x^{(k)})^{-1} \nabla f(x^{(k)})$$ where $\nabla f(x)$ is the gradient vector and $H_f(x)$ is the Hessian matrix of $f$ at $x$. 3. **Calculate the gradient $\nabla f(x)$:** Let $$g_1 = x_1^2 + x_2 - 11, \quad g_2 = x_1 + x_2^2 - 7$$ Then $$f = g_1^2 + g_2^2$$ Gradient components: $$\frac{\partial f}{\partial x_1} = 2g_1 \cdot 2x_1 + 2g_2 \cdot 1 = 4x_1 g_1 + 2 g_2$$ $$\frac{\partial f}{\partial x_2} = 2g_1 \cdot 1 + 2g_2 \cdot 2x_2 = 2 g_1 + 4 x_2 g_2$$ 4. **Calculate the Hessian matrix $H_f(x)$:** Second derivatives: $$\frac{\partial^2 f}{\partial x_1^2} = 4 g_1 + 8 x_1^2 + 0 = 8 x_1^2 + 4 g_1$$ $$\frac{\partial^2 f}{\partial x_2^2} = 0 + 4 g_2 + 8 x_2^2 = 4 g_2 + 8 x_2^2$$ $$\frac{\partial^2 f}{\partial x_1 \partial x_2} = 4 x_1 + 4 x_2$$ Symmetric Hessian: $$H_f = \begin{bmatrix} 8 x_1^2 + 4 g_1 & 4 x_1 + 4 x_2 \\ 4 x_1 + 4 x_2 & 4 g_2 + 8 x_2^2 \end{bmatrix}$$ 5. **Evaluate at initial point $x^{(0)} = [6,6]$:** Calculate $g_1$ and $g_2$: $$g_1 = 6^2 + 6 - 11 = 36 + 6 - 11 = 31$$ $$g_2 = 6 + 6^2 - 7 = 6 + 36 - 7 = 35$$ Gradient: $$\nabla f = \begin{bmatrix}4 \cdot 6 \cdot 31 + 2 \cdot 35 \\ 2 \cdot 31 + 4 \cdot 6 \cdot 35 \end{bmatrix} = \begin{bmatrix}744 + 70 \\ 62 + 840 \end{bmatrix} = \begin{bmatrix}814 \\ 902 \end{bmatrix}$$ Hessian: $$H_f = \begin{bmatrix}8 \cdot 36 + 4 \cdot 31 & 4 \cdot 6 + 4 \cdot 6 \\ 4 \cdot 6 + 4 \cdot 6 & 4 \cdot 35 + 8 \cdot 36 \end{bmatrix} = \begin{bmatrix}288 + 124 & 24 + 24 \\ 24 + 24 & 140 + 288 \end{bmatrix} = \begin{bmatrix}412 & 48 \\ 48 & 428 \end{bmatrix}$$ 6. **Compute the inverse of Hessian $H_f^{-1}$:** Determinant: $$\det(H_f) = 412 \times 428 - 48 \times 48 = 176336 - 2304 = 174032$$ Inverse: $$H_f^{-1} = \frac{1}{174032} \begin{bmatrix}428 & -48 \\ -48 & 412 \end{bmatrix}$$ 7. **Calculate the Newton step:** $$\Delta x = H_f^{-1} \nabla f = \frac{1}{174032} \begin{bmatrix}428 & -48 \\ -48 & 412 \end{bmatrix} \begin{bmatrix}814 \\ 902 \end{bmatrix}$$ Multiply: $$= \frac{1}{174032} \begin{bmatrix}428 \times 814 - 48 \times 902 \\ -48 \times 814 + 412 \times 902 \end{bmatrix} = \frac{1}{174032} \begin{bmatrix}348392 - 43296 \\ -39072 + 371224 \end{bmatrix} = \frac{1}{174032} \begin{bmatrix}305096 \\ 332152 \end{bmatrix}$$ Divide: $$\Delta x \approx \begin{bmatrix}1.752 \\ 1.909 \end{bmatrix}$$ 8. **Update to get $x^{(1)}$:** $$x^{(1)} = x^{(0)} - \Delta x = \begin{bmatrix}6 \\ 6 \end{bmatrix} - \begin{bmatrix}1.752 \\ 1.909 \end{bmatrix} = \begin{bmatrix}4.248 \\ 4.091 \end{bmatrix}$$ 9. **Second iteration: Evaluate gradient and Hessian at $x^{(1)}$:** Calculate $g_1$ and $g_2$: $$g_1 = (4.248)^2 + 4.091 - 11 = 18.05 + 4.091 - 11 = 11.141$$ $$g_2 = 4.248 + (4.091)^2 - 7 = 4.248 + 16.73 - 7 = 13.978$$ Gradient: $$\nabla f = \begin{bmatrix}4 \cdot 4.248 \cdot 11.141 + 2 \cdot 13.978 \\ 2 \cdot 11.141 + 4 \cdot 4.091 \cdot 13.978 \end{bmatrix} = \begin{bmatrix}189.3 + 27.956 \\ 22.282 + 228.7 \end{bmatrix} = \begin{bmatrix}217.256 \\ 250.982 \end{bmatrix}$$ Hessian: $$H_f = \begin{bmatrix}8 \cdot (4.248)^2 + 4 \cdot 11.141 & 4 \cdot 4.248 + 4 \cdot 4.091 \\ 4 \cdot 4.248 + 4 \cdot 4.091 & 4 \cdot 13.978 + 8 \cdot (4.091)^2 \end{bmatrix}$$ Calculate terms: $$8 \times 18.05 + 44.564 = 144.4 + 44.564 = 188.964$$ $$4 \times 4.248 + 4 \times 4.091 = 16.992 + 16.364 = 33.356$$ $$4 \times 13.978 + 8 \times 16.73 = 55.912 + 133.84 = 189.752$$ So $$H_f = \begin{bmatrix}188.964 & 33.356 \\ 33.356 & 189.752 \end{bmatrix}$$ 10. **Compute inverse of $H_f$ at $x^{(1)}$:** Determinant: $$188.964 \times 189.752 - 33.356^2 = 35868.5 - 1113.9 = 34754.6$$ Inverse: $$H_f^{-1} = \frac{1}{34754.6} \begin{bmatrix}189.752 & -33.356 \\ -33.356 & 188.964 \end{bmatrix}$$ 11. **Calculate Newton step at $x^{(1)}$:** $$\Delta x = H_f^{-1} \nabla f = \frac{1}{34754.6} \begin{bmatrix}189.752 & -33.356 \\ -33.356 & 188.964 \end{bmatrix} \begin{bmatrix}217.256 \\ 250.982 \end{bmatrix}$$ Multiply: $$= \frac{1}{34754.6} \begin{bmatrix}189.752 \times 217.256 - 33.356 \times 250.982 \\ -33.356 \times 217.256 + 188.964 \times 250.982 \end{bmatrix}$$ Calculate: $$= \frac{1}{34754.6} \begin{bmatrix}41222.5 - 8370.3 \\ -7243.3 + 47418.3 \end{bmatrix} = \frac{1}{34754.6} \begin{bmatrix}32852.2 \\ 40175.0 \end{bmatrix}$$ Divide: $$\Delta x \approx \begin{bmatrix}0.945 \\ 1.156 \end{bmatrix}$$ 12. **Update to get $x^{(2)}$:** $$x^{(2)} = x^{(1)} - \Delta x = \begin{bmatrix}4.248 \\ 4.091 \end{bmatrix} - \begin{bmatrix}0.945 \\ 1.156 \end{bmatrix} = \begin{bmatrix}3.303 \\ 2.935 \end{bmatrix}$$ **Final answer:** The result of the second iteration using Newton's method is approximately $$x^{(2)} = [3.303, 2.935]$$