The equation has one root between and

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Mathematical Methods for Physics, Ch Root finding, interpolation, numerical linear algebra and quadrature

Problem

The equation x3−2x−2=0x^3-2x-2=0 has one root between x=1x=1 and x=2x=2, at x∗=1.769292354x_* = 1.769292354. Carry at least nine significant figures. ((a)) Take three bisection steps from [1,2][1,2]. List each midpoint, its error and the width of the bracket that remains. ((b)) Take three Newton steps from x0=2x_0 = 2 and list the errors ∣e0∣|e_0| to ∣e3∣|e_3|. ((c)) Take three secant steps from x0=2x_0 = 2, x1=1.8x_1 = 1.8 and list the errors ∣e1∣|e_1| to ∣e4∣|e_4|. ((d)) Estimate each method's order of convergence from its numbers, and explain why bisection's midpoint errors do not fall at every step although its order is one.

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