Locating Roots and Iterative Methods

A root is an input x where f(x) = 0. To evaluate f(a), replace every x in the rule with a, then calculate powers before multiplication and subtraction. A function is continuous on an interval if its graph has no break there. Opposite endpoint signs then guarantee at least one root strictly between the endpoints. If an endpoint value is zero, that endpoint is itself a root.

Opposite signs do not prove there is only one root. Same signs do not rule out roots: (x - 3)² is positive at 2 and 4 but zero at 3. Continuity matters too. The function 1/(x - 2) changes sign between 1 and 3, but is undefined at 2 and never equals zero. When searching a table, compare signs at neighbouring inputs to find a bracket.

To form x = g(x), use equivalent algebra and keep any restrictions. For x³ + 2x = 9, subtract 2x and take cube roots: x = the cube root of (9 - 2x). For x² = 2x + 8, divide every term by x: x = 2 + 8/x. This rule requires a nonzero input. Check that each proposed starting value is allowed.

Write an update as x_next = g(x). The notation x0, also written x_0, names the starting value. Then x1 is the first output and x2 is the second. Always use the latest output as the next input. For x_next = (x + 4)/2, starting at x0 = 0 gives x1 = 2, x2 = 3 and x3 = 3.5. Keep calculator precision between updates and round only the reported answer.

A fixed point r satisfies r = g(r). For g(x) = (x + 4)/2, 2r = r + 4 gives r = 4. A valid rearrangement need not converge. For a linear rule g(x) = mx + b, subtract r = mr + b: g(x) - r = m(x - r). Each absolute error is multiplied by |m|. It shrinks when |m| < 1 and grows when |m| > 1, unless the start is r.

Absolute error is |estimate - root|. If an initial error bound is B and each update multiplies errors by c, where 0 < c < 1, multiply the bound by c after each update. Stop when it is below the requested tolerance. Here 1-decimal accuracy means error below 0.05; 2-decimal accuracy means below 0.005. A sign bracket's midpoint is at most half its width from a root inside it. Correct rounding also needs a bracket within one rounding interval.

Suppose a root is bracketed between 4.314 and 4.316. The interval crosses 4.315, the boundary between rounding to 4.31 and rounding to 4.32. Values inside the bracket can round either way. Refine the bracket before reporting the root to two decimal places.

Matching rounded iterates do not certify the root. For x_next = 0.9999x + 0.0001, the fixed point is 1. Starting at zero gives 0.0001 and then 0.00019999. Both round to 0.000 at three decimals, although both are far from 1. Use a justified error bound or a continuous-function sign bracket that certifies the requested accuracy.

A small function value is not an x-error bound either. For f(x) = 0.001(x - 10), the root is 10. At x = 9, f(9) = -0.001, but the estimate is still 1 away from the root. The function's scale affects its values. Check the distance using a justified bound or bracket.