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The lecture notes for the first week of the Numerical Computing course at Indus University. It covers mathematical preliminaries, error analysis, and types of error. The document also introduces the Secant Method, which is an alternative to Newton's approach for approximating roots. The method involves using a linear function based on interpolation, known as a secant line. The formula for the secant line is provided in the document.
Typology: Exercises
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Course Name: Numerical Computing 2
Secant Method
Presented by: Muhammad Owais Qadri(519-2020) Muhammad Muddasir (174-2020)
(^) The tangent line to the curve of y = f(x) with the point of tangency (x 0 , f(x 0 ) was used in Newton’s approach. The graph of the tangent line about x = α is essentially the same as the graph of y = f(x) when x 0 ≈ α. The root of the tangent line was used to approximate α. (^) Consider employing an approximating line based on ‘ interpolation’. Let’s pretend we have two root estimations of root α, say, x 0 and x 1. Then, we have a linear function (^) q(x) = a 0 + a 1 x (^) using q(x 0 ) = f (x 0 ), q(x 1 ) = f (x 1 ). (^) This line is also known as a secant line. Its formula is as follows:
(^) The secant method procedures are given below using equation (1). (^) Step 1: Initialization (^) x 0 and x 1 of α are taken as initial guesses. (^) Step 2: Iteration (^) In the case of n = 1, 2, 3, …, (^) until a specific criterion for termination has been met (i.e., The desired accuracy of the answer or the maximum number of iterations has been attained).
(^) Therefore, f(x 2 ) = – 0. (^) Performing the second approximation, , (^) x 3 = x 2 – [( x 1 – x 2 ) / (f(x 1 ) – f(x 2 ))]f(x 2 ) (^) =(- 0.234375) – [(1 – 0.25)/(-3 – (- 0.234375))](- 0.234375) (^) = 0. (^) Hence, f(x 3 ) = 0. (^) Stay tuned to BYJU’S – The Learning App for more Maths-related articles and videos that help you grasp the concepts quickly.
