{"id":1692,"date":"2025-05-25T07:47:55","date_gmt":"2025-05-25T07:47:55","guid":{"rendered":"https:\/\/www.mathros.net.ua\/en\/?p=1692"},"modified":"2025-11-06T11:41:51","modified_gmt":"2025-11-06T11:41:51","slug":"adams-method","status":"publish","type":"post","link":"https:\/\/www.mathros.net.ua\/en\/adams-method.html","title":{"rendered":"Adams Method Explained Simply: What It Is and How It Works"},"content":{"rendered":"

The Adams method is one of the most efficient numerical methods for solving ordinary differential equations. It is widely used by students, engineers, and scientists. What makes it so popular<\/strong>? The answer is simple: the Adams method allows you to quickly obtain highly accurate results without requiring extensive calculations. Let’s take a closer look at how exactly the Adams method works and why you should use it for practical problems.<\/p>\n

Single-Step and Multi-Step Methods: What\u2019s the Difference?<\/h2>\n

First, let’s briefly recall Euler\u2019s method<\/a>, which most students are familiar with. It uses only one previous point to find the next value of a function. Such an approach is simple, but is it always accurate<\/strong>? Unfortunately, no. Results often have significant errors.<\/p>\n

\"How<\/p>\n

This is where multi-step methods, including the Adams method, come in handy. They use several previous points\u2014not just one\u2014to determine the next value of a function. Thanks to this, multi-step methods provide much greater accuracy<\/strong>. You get fewer errors without doing additional calculations. Convenient, isn’t it?<\/p>\n

How the Adams Method Works: A Simple Explanation of the Mathematical Basis<\/h2>\n

To understand the essence of the Adams method, let’s assume we have a Cauchy problem of the form:<\/p>\n

\"Cauchy<\/p>\n

Suppose we already have several approximate solution values: yn<\/sub>, yn-1<\/sub>, yn-2<\/sub>,…<\/em>,\u00a0and the corresponding function values fn<\/sub>=f(xn<\/sub>, yn<\/sub>), fn-1<\/sub>=f(xn-1<\/sub>, yn-1<\/sub>), fn-2<\/sub>=f(xn-2<\/sub>, yn-2<\/sub>),…<\/em>.<\/p>\n

Instead of using information from only one point, we construct an interpolation polynomial passing through several previous points. We then integrate this polynomial from xn<\/sub><\/em> to xn+1<\/sub><\/em>. This gives us the next value yn+1<\/sub><\/em><\/strong>:<\/p>\n

\"Adams<\/p>\n

Thus, the Adams method allows us to account for significantly more information, resulting in more accurate results.<\/p>\n

Newton\u2019s Interpolation Polynomial: The Core Idea of the Adams Method<\/h2>\n

Where does this polynomial come from? It is based on Newton\u2019s polynomial<\/a>, which precisely approximates a function at several known points. The general form of Newton\u2019s polynomial is:<\/p>\n

\"Newton\u2019s<\/p>\n

This polynomial is easily constructed from the known values at previous points<\/strong>, and integrating it gives straightforward formulas for calculating the subsequent values of the function. That’s how different variants of the Adams method arise, depending on the number of points we use.<\/p>\n

Adams Method Formulas for Different Numbers of Points: A Brief Overview<\/h2>\n

To see how this approach works in practice, let’s consider a few simple examples:<\/p>\n