{"id":1969,"date":"2025-11-09T10:50:33","date_gmt":"2025-11-09T10:50:33","guid":{"rendered":"https:\/\/www.mathros.net.ua\/en\/?p=1969"},"modified":"2025-11-11T14:54:27","modified_gmt":"2025-11-11T14:54:27","slug":"combined-secant-newton-method","status":"publish","type":"post","link":"https:\/\/www.mathros.net.ua\/en\/combined-secant-newton-method.html","title":{"rendered":"Combined Secant-Newton Method: Tangent Speed and Chord Reliability"},"content":{"rendered":"

The combined secant-Newton method blends two strengths of numerical root-finding: it keeps the solution inside a trusted interval and it converges quickly to that solution. We start with an interval that definitely contains a root and, at each step, move its endpoints toward each other: one endpoint is updated by the secant (chord) rule<\/a>, and the other by the Newton (tangent) rule<\/a>. This approach works well in both learning examples and real applications because it keeps the process controlled while still moving toward an accurate result at a solid pace.<\/p>\n

Combined Secant-Newton Method: Update Choice via the Sign f'(x)\u22c5f”(x)<\/em><\/h2>\n

Consider the equation<\/a> f(x)=0<\/em> on the initial interval [a0<\/sub>,b0<\/sub>]=[a,b]<\/em> with f(a0<\/sub>)\u22c5f(b0<\/sub>)<0<\/em>. Assume f<\/em> is sufficiently smooth on this interval. On each iteration, we update one endpoint by a secant step and the other by a Newton step; which side uses which step is determined by the sign of the product f'(x)\u22c5f”(x)<\/em> on the current interval.<\/p>\n

If on [an,bn]<\/em> we have f'(x)\u22c5f”(x)>0<\/em> (cases 1<\/em> and 2<\/em> in the figure), move the left endpoint with the secant and the right endpoint with Newton:<\/p>\n

\"Combined<\/p>\n

In this configuration, the Newton step from the right lands inside the interval, while the left secant step steadily trims the interval.<\/p>\n

\"Geometric<\/p>\n

If f'(x)\u22c5f”(x)<0<\/em> (cases 3<\/em> and 4<\/em>), swap the roles: apply the Newton step on the left and the secant on the right:<\/p>\n

\"Combined<\/p>\n

This choice keeps both new endpoints inside [an<\/sub>,bn<\/sub>]<\/em> and makes the interval length decrease monotonically. The sign of f'(x)\u22c5f”(x)<\/em> tells us from which side Newton\u2019s step remains inside the interval, while the secant step ensures stable shrinking from the opposite side.<\/p>\n

Convergence and Accuracy Control: Stopping Rule and Practical Safeguards<\/h2>\n

It\u2019s convenient to stop the iterations when the current interval becomes smaller than a preset tolerance \u03b5<\/em>, that is, when (bn<\/sub>-an<\/sub>)<\u03b5<\/em>. As an approximation to the root, we often take the midpoint of the interval:<\/p>\n

\"Combined<\/p>\n

in which case the absolute error does not exceed (bn<\/sub>-an<\/sub>)\/2<\/em>. This criterion is easy to implement and immediately gives a clear estimate of result quality without extra checks.<\/p>\n

A few practical tips improve efficiency:<\/p>\n