How Does Approximation Search Find Solutions Without Strict Monotonicity?

How Approximation Search Works

To understand how approximation search functions, let's consider the analogy of a classic binary search. In binary search, we search for a specific value within a sorted list by repeatedly dividing the search interval in half. However, approximation search differs from binary search in that it does not require the function we are searching to be strictly monotonic, meaning it can handle both increasing and decreasing values.

Algorithm Overview:

1. Define Search Interval: Specify the initial interval [a0, a1] within which we expect the solution to lie.
2. Probe Points: Evenly distribute points x(i) within the interval [a0, a1] using a step size da.
3. Calculate Errors (ee): For each point x(i), compute the error or distance ee between the function output y=f(x(i)) and the target value y0.
4. Identify Solution Point: Keep track of the point aa with the minimum error ee.
5. Repeat Recursively: Stop the search when all points x(i) have been probed or a certain accuracy has been reached. If not, recursively increase the accuracy by narrowing the search interval and refining the step size da.

Example Implementation:

In C , we can use the following class to implement approximation search:

class approx {
public:
    double a, aa, a0, a1, da, *e, e0;
    int i, n;
    bool done, stop;
};

To use this class:

approx aa;
double ee, x, y, x0, y0;
aa.init(0.0, 10.0, 0.1, 6, &ee);
while (!aa.done) {
    x = aa.a;
    y = f(x);
    ee = fabs(y - y0);
    aa.step();
}

Key Considerations:

• The search interval [a0, a1] must be chosen carefully to encompass the solution while minimizing its width.
• The initial step size da should be selected appropriately to avoid missing local minima or maxima while balancing speed.
• Approximation search can be used to solve a variety of problems, including fitting polynomials, finding unknown parameters, and solving transcendental equations.

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