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:

y = f(x) …… (1)

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

Determine the steps of each iterations.

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

$∆y\left(i\right)=y-f\left[x\left(i\right)\right]$ …… (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) .

$J\left(i\right)=\left[\begin{array}{cc}1& 1\\ x_2\left(0\right)& x_1\left(0\right)\end{array}\right]$ …… (2)

Here, the $J\left(i\right)$is Jacobian matrix,$x_1\left(0\right)$is first output vector and $x_2\left(0\right)$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 $J\left(i\right)$is Jacobian matrix,$∆x\left(i\right)$is back substitution equation and $∆y\left(i\right)$is converge first step iteration equation.

Determine the fourth iterations$x(i+1)$

Let, taking fourth iterations equation

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

Here, $∆x\left(i\right)$is back substitution equation,$x(i+1)$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.