Consider the following statements regarding the convergence of the Newton-Raphson procedure:  1. It does not converge to a root when the second differential coefficient changes sign 2. It is preferred when the graph of (X) is nearly horizontal where it crosses the X-axis 3. It is used to solve algebraic and transcendental equations Which of these statements are correct?

Consider the following statements regarding the convergence of the Newton-Raphson procedure:  1. It does not converge to a root when the second differential coefficient changes sign 2. It is preferred when the graph of (X) is nearly horizontal where it crosses the X-axis 3. It is used to solve algebraic and transcendental equations Which of these statements are correct? Correct Answer 1 and 3 only

Newton- Raphson method:

• The Newton - Raphson method is the type of open method (Extrapolation method).
• It is a powerful technique for solving algebraic and transcendental equations f( x ) = 0, numerically.
• It is an iteration method for solving a set of various nonlinear equations with an equal number of unknowns.

Advantages:

• It possesses quadratic convergence characteristics. Therefore, the convergence is very fast.
• The number of iterations is independent of the size of the system.
• The Newton-Raphson Method convergence is not sensitive to the choice of slack bus.
• Overall, there is a saving in computation time since a fewer number of iterations are required.

Disadvantages:

• It does not converge to a root when the second differential coefficient changes sign

Important Points:

• It is a multiplexing technique that allows the transmission of multiple signals over a common channel but in different time slots.
• Each signal will get transmitted very quickly over the channel but at a time only one signal will be transmitted.
• The formula converges provided the initial approximation x0 is chosen sufficiently close to the root.
• It is generally used to improve the result obtained by other methods.
• It has quadratic convergence i.e. order of convergence is 2. The subsequent error at each step is proportional to the square of the error at the previous step.

• It is sensitive to the starting value. Convergence fails if the starting point is not near the root.