Find the third approximation of the differential equation
Answers
Answered by
2
The Newton-Raphson method, also known as Newton’s method, is one of the most powerful numerical methods for solving algebraic and transcendental equations of the form f(x) = 0 . This method is commonly used because of its simplicity and rapid convergence. In this review article, we’ll learn how to use the Newton-Raphson method in different applications, including problems with differential equations.
What is the Newton-Raphson Method?
Consider a real function, f(x) , that crosses thex -axis at some point, x = p . To determine the value of p , we have to solve the equation f(x) = 0 . The function f(x) can be a polynomial, and in this case, the equation f(x) = 0 is an algebraic equation. If the function f(x) involves transcendental functions (for example, \sin xor \ln x ), the equation f(x) = 0 is a transcendental equation. Algebraic and transcendental equations do not always have exact solutions. To solve f(x) = 0 when there is no formula for the exact solution, we can useapproximation techniques.
The Newton-Raphson method (or Newton’s method) is one of the most efficient and simple numerical methods that can be used to find the solution of the equation f(x) = 0 . In this method, we start with an initial guess, x = p_0 (which may be a rough estimate),then calculate progressively better approximations,x = p_1, p_2, ... The method is commonly used because of its simplicity and rapid convergence (which means high accuracy with only a few iterations).
Consider the graph y = f(x) of some function that crosses the x -axis, as illustrated in the image below:
From the graph, we make a first, imprecise approximation, x = p_0 , for the zero of the function f(x) . To get a better approximation, we take the tangent to the curve of y = f(x) at x = p_0 , and compute the point x = p_1 where the tangent line intersects the x -axis:
f^{\prime} (p_0) = \dfrac{f(p_0)}{p_0 - p_1}\,, \quad \text{hence} \quad p_1 = p_0 - \dfrac{f(p_0)}{f^{\prime}(p_0)}
Here, the derivative f^{\prime} (p_0) is the slope of the tangent curve. Next, we repeat the procedure, starting from x = p_1 to get a better approximation:
p_2 = p_1 - \dfrac{f(p_1)}{f^{\prime}(p_1)}
Then, we compute p_3 from p_2 by the same formula, and so on. Since the Newton-Raphson method involves dividing byf^{\prime}(p_n) , we should avoid starting at a point where the derivative is not continuous or close to zero. When the derivative is near zero, the tangent line is almost horizontal, and we may not reach the desired accuracy in a reasonable number of iterations.
What is the Newton-Raphson Method?
Consider a real function, f(x) , that crosses thex -axis at some point, x = p . To determine the value of p , we have to solve the equation f(x) = 0 . The function f(x) can be a polynomial, and in this case, the equation f(x) = 0 is an algebraic equation. If the function f(x) involves transcendental functions (for example, \sin xor \ln x ), the equation f(x) = 0 is a transcendental equation. Algebraic and transcendental equations do not always have exact solutions. To solve f(x) = 0 when there is no formula for the exact solution, we can useapproximation techniques.
The Newton-Raphson method (or Newton’s method) is one of the most efficient and simple numerical methods that can be used to find the solution of the equation f(x) = 0 . In this method, we start with an initial guess, x = p_0 (which may be a rough estimate),then calculate progressively better approximations,x = p_1, p_2, ... The method is commonly used because of its simplicity and rapid convergence (which means high accuracy with only a few iterations).
Consider the graph y = f(x) of some function that crosses the x -axis, as illustrated in the image below:
From the graph, we make a first, imprecise approximation, x = p_0 , for the zero of the function f(x) . To get a better approximation, we take the tangent to the curve of y = f(x) at x = p_0 , and compute the point x = p_1 where the tangent line intersects the x -axis:
f^{\prime} (p_0) = \dfrac{f(p_0)}{p_0 - p_1}\,, \quad \text{hence} \quad p_1 = p_0 - \dfrac{f(p_0)}{f^{\prime}(p_0)}
Here, the derivative f^{\prime} (p_0) is the slope of the tangent curve. Next, we repeat the procedure, starting from x = p_1 to get a better approximation:
p_2 = p_1 - \dfrac{f(p_1)}{f^{\prime}(p_1)}
Then, we compute p_3 from p_2 by the same formula, and so on. Since the Newton-Raphson method involves dividing byf^{\prime}(p_n) , we should avoid starting at a point where the derivative is not continuous or close to zero. When the derivative is near zero, the tangent line is almost horizontal, and we may not reach the desired accuracy in a reasonable number of iterations.
Similar questions