Question

For the iterative solutions to nonlinear algebraic equations with the Newton-Raphson method, the Jacobian matrix $\mathbf{J}\mathbf{\left(}\mathbf{i}\mathbf{\right)}$ consists of the partial derivatives. Write down the elements of first row of $\mathbf{J}\left(i\right)$.

Short answer

Therefore, the elements of first row of $J\left(i\right)$is$\left[\frac{a{f}_1}{a{x}_1}\frac{a{f}_1}{a{x}_2}\dots \frac{a{f}_1}{a{x}_N}\right].$

Step-by-step solution

Determine the formula of Jacobian matrix.

for the iterative solutions to nonlinear algebraic equations with the Newton-Raphson method, the Jacobian matrix $J\left(i\right)$consists of the partial derivatives.

Write the Jacobian matrix for Newton Raphson method.

$$\begin{gathered} \mathbf{J}\left(i\right)\mathbf{=}{\overline{)\frac{\mathbf{df}}{\mathbf{dx}}}}_{\mathbf{x}\mathbf{=}\mathbf{x}\left(i\right)} \\ \mathbf{=}{\left[\begin{array}{cccc}\frac{{\mathrm{af}}_1}{{\mathrm{ax}}_1}& \frac{{\mathrm{af}}_1}{{\mathrm{ax}}_2}& \dots & \frac{{\mathrm{af}}_1}{{\mathrm{ax}}_N}\\ \frac{{\mathrm{af}}_2}{{\mathrm{ax}}_1}& \frac{{\mathrm{af}}_2}{{\mathrm{ax}}_2}& \dots & \frac{\mathrm{af}2}{{\mathrm{ax}}_N}\\ .& .& .& .\\ \frac{\mathrm{afN}}{{\mathrm{ax}}_1}& \frac{{\mathrm{af}}_N}{\mathrm{ax}2}& \dots & \frac{\mathrm{afN}}{{\mathrm{ax}}_N}\end{array}\right]}_{\mathbf{x}\mathbf{=}\mathbf{x}\left(i\right)} \end{gathered}$$ ……. (1)

Here, $J\left(i\right)$is Jacobian matrix.

Determine the answer.

From equation (1), the first row elements are,

$\left[\frac{a{f}_1}{a{x}_1}\frac{a{f}_1}{a{x}_2}\dots \frac{a{f}_1}{a{x}_N}\right]$

Therefore, the elements of first row of $\mathrm{J}\left(\mathrm{i}\right)$is$\left[\frac{a{f}_1}{a{x}_1}\frac{a{f}_1}{a{x}_2}\dots \frac{a{f}_1}{a{x}_N}\right]$.