Write down the formula used get the approximation x_i given x_i-1 to the squarer

Question

Write down the formula used get the approximation x_i given x_i-1 to the squareroots of f(x) and g(x) (given below) using Newton's squareroot finding algorithm and implement the algorithm. f(x) = 2x^3 - 5x^2 + 3x - 1 g(x) = exp (0.3 x) - exp (-0.6x) - 1 For both functions, start Newton's algorithm at x = 0 and terminate after it converges to a stationary point such that the candidate squareroot points x found at two successive steps do not differ up to the fourth significant fractional digit, i.e. |x_i - x_i - 1| lessthanorequalto 0.0001. Show the results of each step until convergence. For example, for the function f(x) = log(x) - 1 + exp(-x), you would show:

Explanation / Answer

a)

newton mehod - > xi = xi-1 - f(i-1)/f'(i-1)

b)

n x accuracy(x) f(x) 1 0.3333333333333333333333 0.3 -1 2 1.01010101010101010101 1 -2.481481481481481481482 3 1.524189773619015052472 2 -15.23241701784707600441 4 1.890883375890567403754 2 -26.17886609228787517175 5 2.162386276526968610676 2 -30.62742883203710996826 6 2.366259288844343930799 2 -30.52163620885221643721 7 2.520414150038979930345 3 -27.8878469977464002819 8 2.637438157469994742805 3 -24.16374992690669503337 9 2.726494880466854062087 2.7 -20.21983911476303660234 10 2.794379119015691170124 2.8 -16.5196923925119680698 11 2.846182741944123480779 2.8 -13.27023241227149901995 12 2.88574647560915072468 2.9 -10.53049683027852255276 13 2.915979676556449642469 2.9 -8.281892997946588014341 14 2.93909254832890840113 2.9 -6.470354898767695373072 15 2.956767536430985993762 3 -5.030065030456337889049 16 2.97028718646479344601 3 -3.895841256340803439274 17 2.980630214292211309794 3 -3.008899979350527984489 18 2.988544053374262539943 2.99 -2.318939488296236625439 19 2.994599833513129522038 2.99 -1.784304125370111981712 20 2.999234152209249691306 3 -1.371242142912542548247 21 3.00278087074960182356 3 -1.052815934581854996183 22 3.00549535059111772252 3.01 -0.8077562838563779749806 23 3.007572946380619967277 3.01 -0.6194001343138883529606 24 3.009163127490221064528 3.01 -0.4747677980051557194678 25 3.01038026767108910668 3.01 -0.3637917917183972467748

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