Solving equations numerically
Trap a root of an equation with no formula by a change of sign, close in on it by repeating a rule, and go faster with the Newton–Raphson method.
- سوريا: الصف ١١
- ألمانيا: الصف ١١
- فرنسا: الصف ١٢
- بريطانيا: السنة ١٢
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
If f(a) and f(b) have opposite signs and the graph of f is continuous between a and b, the graph must cross the x-axis: there is a root between a and b.
In iteration we start from a number and apply a rule again and again: xₙ₊₁ = g(xₙ). If the numbers settle towards a value, that value solves x = g(x).
Newton–Raphson uses the tangent: the tangent at xₙ meets the x-axis at xₙ₊₁ = xₙ − f(xₙ)f′(xₙ), which is usually nearer the root than xₙ.