Question

The number of iterations required for convergence is

dependent/independent of the dimension N for Newton-Raphson

method. Choose one.

Short answer

The correct term is “Independent”

Step-by-step solution

Definition of Newton-Raphson method.

Newton-Raphson method is used for finding a real root polynomial of a given differential function. It is defined as:

$\mathrm{y}=\mathrm{f}\left(\mathrm{x}\right).......\left(1\right)$

Here, x , y and f are vectors for the power flow problem.

Determine the steps of each iterations.

Determine the first step of iterations $△y\left(i\right)$

localid="1655187782674" $△y\left(i\right)=y-f\left[x\left(i\right)\right]$ …… (2)

Here, $△y\left(i\right)$is to converge first step iteration equation, y is input vector and

$f\left[x\left(i\right)\right]$is vector functions.

Determine the second step of iterations J ( i ).

$\mathrm{J}\left(\mathrm{i}\right)=\left[\frac{1}{x_2\left(0\right)}|\frac{1}{{\mathrm{x}}_2\left(0\right)}\right]........\left(2\right)$

Here, the $J\left(i\right)$is Jacobian matrix, localid="1655188292786" $x_2\left(0\right)$is first output vector and is

second output vector.

Determine Gauss elimination and back substitution method

Let, taking third step iterations equation

$J\left(i\right)∆x\left(i\right)=△y\left(i\right)$ …… (3)

Here, the $\mathrm{J}\left(\mathrm{i}\right)$is Jacobian matrix,$△x\left(i\right)$is back substitution equation and $△\mathrm{y}\left(\mathrm{i}\right)$is converge first step iteration equation.

Determine the fourth iterations$x\left(i+1\right)$

Let, taking fourth iterations equation

$△x\left(i\right)=x\left(i+1\right)-x\left(i\right)$ …… (8)

Here, $△x\left(i\right)$is back substitution equation,$x\left(i+1\right)$is new value of linear equations and $x\left(i\right)$is an old value of linear equation.

Therefore, the number of iterations required for convergence is independent

of the dimension N for Newton-Raphson, but increases with N for Jacobi and

Gauss-Seidel.