{"id":1957,"date":"2025-11-02T08:36:49","date_gmt":"2025-11-02T08:36:49","guid":{"rendered":"https:\/\/www.mathros.net.ua\/en\/?p=1957"},"modified":"2025-11-06T11:23:26","modified_gmt":"2025-11-06T11:23:26","slug":"method-of-successive-approximations","status":"publish","type":"post","link":"https:\/\/www.mathros.net.ua\/en\/method-of-successive-approximations.html","title":{"rendered":"Method of Successive Approximations: A Simple Way to Solve Hard Equations"},"content":{"rendered":"

Method of successive approximations is a convenient tool for finding roots of nonlinear equations when an exact analytical solution is difficult or impossible to obtain. The idea is simple: we build a sequence of approximations, each one refining the previous value, and step by step we move toward the desired answer. This approach works well in learning because it shows the logic of calculations and creates a foundation for mastering more powerful techniques. In this article, we focus on the fixed-point idea, the conditions for convergence, and practical iteration choices so that we can confidently move on to examples.<\/p>\n

Method of Successive Approximations: Fixed Point and the Iterative Process<\/h2>\n

Let\u2019s start by reformulating the problem. The equation f(x)=0<\/em> is conveniently written in the form x=\u03c6(x)<\/em>. A solution in this case is a number \u03be<\/em> for which \u03be=\u03c6(\u03be)<\/em> holds; the point \u03be<\/em> is called a fixed point of the transformation \u03c6(x)<\/em>.<\/p>\n

Next, we run iterations from an initial guess x0<\/sub><\/em>:<\/p>\n

\"Method<\/p>\n

To keep computations stable, we work on an interval [a,b]<\/em> where we expect the root and make sure all approximations stay inside this interval. The decisive factor is how “contractive”<\/em> \u03c6(x)<\/em>: is: if on [a,b]<\/em><\/p>\n

\"Method<\/p>\n

then the fixed point is unique, and the sequence {xk<\/sub>}<\/em> converges to \u03be<\/em>. A smaller q<\/em> usually means the error shrinks faster from step to step. In practice, q<\/em> is often estimated approximately: we evaluate the derivative numerically at several points in the interval and take the largest absolute value among those results.<\/p>\n

Building the Formula and Choosing the Parameter: How to Achieve Convergence<\/h2>\n

A convenient way to construct the transformation is the relaxation formula<\/p>\n

\"Method<\/p>\n

where the parameter \u03bb>0<\/em> sets the “step length”<\/em>. If \u03bb<\/em> is too large, the progress becomes unstable and may break convergence; if it\u2019s too small, the march toward the solution slows down. To keep steps under control, we link the choice of \u03bb<\/em> to the behavior of f'(x)<\/em> on the working interval. If<\/p>\n

\"Method<\/p>\n

then \u03c6(x)<\/em> is a contracting transformation and the iterations converge. In practice, it helps to estimate M=maxx\u2208[a,b]<\/sub>|f'(x)|<\/em> and take \u03bb<\/em> within (0,2\/M)<\/em>. A common practical choice is \u03bb\u22481\/M<\/em>: the step is moderate yet sufficient to maintain stable convergence.<\/p>\n

We usually stop the process when |xk+1<\/sub>-xk<\/sub>|<\u03b5<\/em>, where \u03b5<\/em> is the desired accuracy. If, in addition, |\u03c6'(x)|\u2264q<1<\/em> is known, the following a posteriori estimate is useful:<\/p>\n

\"Method<\/p>\n

which allows you to estimate the distance to the true root x\u2217<\/sup><\/em> using the last two iterates and a known (or approximately estimated) constant q<\/em>. If convergence slows down or oscillations appear, it\u2019s worth adjusting \u03bb<\/em> or changing the form of \u03c6(x)<\/em> to reduce the effective q<\/em> and bring the iterations back to a stable regime. This is a natural tuning cycle: estimate the derivative, choose \u03bb<\/em>, check convergence, tweak the parameters, and continue computing until the required accuracy is reached.<\/p>\n

Method of Successive Approximations: Practice on an Example \u2014 Full Calculation<\/h2>\n

Now that we understand the basics, let\u2019s see how the method of successive approximations works on a concrete example. While theory matters, a hands-on calculation gives the best sense of how effective and accurate the approach is. We\u2019ll go through the task from start to finish to see how to obtain an approximate solution step by step.<\/p>\n

Example 1. Find, with accuracy \u03b5=0.01, the solution of the nonlinear equation f(x)=x3<\/sup>+x-5=0 on the interval [-2,2]<\/h6>\n

\"Visualization<\/p>\n

On the segment [-2,2]<\/em> we have f(-2)=-15<0<\/em> and f(2)=5>0<\/em>, so there is at least one root. We will build iterations using the relaxation form \u03c6(x)=x-\u03bb\u22c5f(x)<\/em>. The derivative f'(x)=3\u22c5x2<\/sup>+1<\/em> on [-2,2]<\/em> satisfies 1\u2264f'(x)\u226413<\/em>. Therefore, we take M=max\u2208[-2,2]<\/sub>|f'(x)|=13<\/em> and choose \u03bb=1\/M=1\/13\u22480.077<\/em>. Then the iterative process is<\/p>\n

\"Method<\/p>\n

Start the computations from the initial guess x0<\/sub>=-2<\/em>. We get:<\/p>\n

\"Method<\/p>\n

The stopping criterion is satisfied on the transition from x11<\/sub><\/em> to x12<\/sub><\/em>, because |x12<\/sub>\u200b-x11<\/sub>|=0.005<0.01<\/em>. Therefore, we can finish the calculations, and the approximate solution with the stated accuracy is x=1.513<\/em>. This result is consistent with the behavior of the function on the given interval and shows how a well-chosen parameter \u03bb<\/em> ensures stable convergence of the iterations to the root.<\/p>\n

Next Level: Three Tools for Confident Decisions<\/h2>\n

You\u2019ve already felt the logic of the method in practice. Now let\u2019s expand the toolkit and see what to master next to work with nonlinear equations<\/a> faster, more reliably, and more flexibly.<\/p>\n

    \n
  1. Bisection Method: Guaranteed Convergence without Unnecessary Complications<\/a> \u2014 It narrows the interval step by step, ensuring a predictable move toward the root and transparent accuracy control at every stage.<\/li>\n
  2. Newton\u2019s Method: Maximum Speed with a Good Starting Point<\/a> \u2014 It uses tangent-based approximation to rapidly reduce the error when the initial guess is chosen well and the function behaves stably.<\/li>\n
  3. Secant (Chord) Method: Reliable Approximation without Derivatives<\/a> \u2014 Instead of the tangent, it uses the line segment between two points on the graph, allowing stable steps even when computing derivatives is impractical or impossible.<\/li>\n<\/ol>\n

    Final Stage: The Flowchart Becomes a Program<\/h2>\n

    If you enjoy combining mathematics with programming, try turning the flowchart below into code. See how successive updates gradually bring you closer to the root. You\u2019ll feel the logic of the process, be able to experiment with initial values, and compare the speed of convergence with other approaches. In practice, you\u2019ll see why a clear algorithm structure is important for a stable result. Isn\u2019t it interesting to test this on your own examples and watch your program confidently produce the answer?<\/p>\n

    \"Flowchart<\/p>\n","protected":false},"excerpt":{"rendered":"

    Method of successive approximations is a convenient tool for finding roots of nonlinear equations when an exact analytical solution is<\/p>\n","protected":false},"author":1,"featured_media":1958,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"template-centered.php","format":"standard","meta":{"footnotes":""},"categories":[390],"tags":[129,405,399,134,398],"class_list":["post-1957","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-numerical-methods-for-solving-single-variable-equations","tag-iterative-methods","tag-method-of-successive-approximations","tag-nonlinear-equations","tag-numerical-methods","tag-root-finding"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/1957","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/comments?post=1957"}],"version-history":[{"count":1,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/1957\/revisions"}],"predecessor-version":[{"id":1959,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/1957\/revisions\/1959"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/media\/1958"}],"wp:attachment":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/media?parent=1957"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/categories?post=1957"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/tags?post=1957"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}