Question
Download Solution PDFThe real root of the equation x3 − x − 5 = 0 lying between 1 and 2 after first iteration by Newton-Raphson method is ________, if initial approximation is taken as x0 = 2 ∈ [1, 2] :
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe correct answer is 1.909.
Key Points
- The Newton-Raphson method is an iterative numerical method used to find the roots of a real-valued function.
- Given a function
and its derivative
, the Newton-Raphson iteration formula is:
- For the given equation
, we first need to compute its derivative:
- Starting with the initial approximation
, we apply the Newton-Raphson iteration formula:
- Substituting
:
Additional Information
- The Newton-Raphson method is highly efficient and converges quickly if the initial approximation is close to the actual root.
- It is important to ensure that the derivative is not zero at the initial approximation to avoid division by zero.
- The method may fail to converge or converge to a different root if the initial approximation is not chosen carefully.
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