Hi, Today we'll learn about Newton-Raphson Method "Newton's Method". this method is for finding successively better approximate solutions to the roots of a function (real valued). so are you ready for this............you will.
Newton's Method:
The Newton's method is one of the most powerful and well known method used for finding a root of f(X) = 0 there are many ways to derive Newton-Raphson method. the simplest way to derive this formula is by using the first two terms in the Taylor series expansion of the form.
Instead of working with the chord joining points on the graph of y= f(X) as in bisection method and false position methods. Newton's method uses the tangent at one point on the graph of y= f(X) therefor it requires only one instead of two initial guesses.
Suppose X is an initial guesses draw the tangent line at X= X0 and procedure it to meet the X - axis at X = X1
which shows in picture above. now from right angle triangle
tanO f(Xo) / ( Xo - X1 )
As we knew that tanO is a slop it means tanO = f ' (Xo)
The root a is given by the point of intersection of the curve y = f(X) and the x-axis if an iterative procedure is being designed to approximate real root a, one simple approach can be to replace the curve by a straight line the intersection of which with the x-axis can easily be found. starting with an arbitrary initial approximation Xo , we then calculate a sequence of iterates, X1, X2, X3,........
Now the question is how to select the direction of the straight line, in Newton Raphson method the direction of the straight line is that of the tangent to the curve at the given point that is why this method is also known as Newton's Method of tangents.
Examples: here
for more please visit: Newton's method
Newton's Method:
The Newton's method is one of the most powerful and well known method used for finding a root of f(X) = 0 there are many ways to derive Newton-Raphson method. the simplest way to derive this formula is by using the first two terms in the Taylor series expansion of the form.
Instead of working with the chord joining points on the graph of y= f(X) as in bisection method and false position methods. Newton's method uses the tangent at one point on the graph of y= f(X) therefor it requires only one instead of two initial guesses.
Suppose X is an initial guesses draw the tangent line at X= X0 and procedure it to meet the X - axis at X = X1
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| Newton's Method |
tan
As we knew that tan
The root a is given by the point of intersection of the curve y = f(X) and the x-axis if an iterative procedure is being designed to approximate real root a, one simple approach can be to replace the curve by a straight line the intersection of which with the x-axis can easily be found. starting with an arbitrary initial approximation Xo , we then calculate a sequence of iterates, X1, X2, X3,........
Now the question is how to select the direction of the straight line, in Newton Raphson method the direction of the straight line is that of the tangent to the curve at the given point that is why this method is also known as Newton's Method of tangents.
Examples: here
for more please visit: Newton's method
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