How Can Approximation Search Efficiently Find Approximate Solutions Without Monotonicity?

Understanding Approximation Search

Approximation search mimics the efficiency of binary search without the strict monotonicity constraint. It enables approximation of values or parameters within a specified domain, such as real numbers (double precision).

Algorithm Explanation:

Given a function y=f(x) and a desired y-value (y0), the algorithm seeks to find x0 within a range [a0, a1] such that f(x0) approaches y0. It iteratively evaluates points x(i) within this range with a defined step size (da) and selects the point aa that minimizes the error |f(x(i)) - y0|.

Recursive Accuracy Enhancement:

To increase accuracy, the algorithm recursively refines the search range around aa, reducing da by a factor of 0.1. This process continues until a desired accuracy or a maximum number of recursions is reached.

Implementation:

A C class called "approx" implements this algorithm. It allows initialization with search parameters (a0, a1, da, n, e), where n specifies the number of recursions and e is a pointer to the error variable. The "step()" method iterates through points x(i), updates the best solution aa, and adjusts search parameters for recursive refinement.

Sample Usage:

approx aa;
double ee, x, y, x0, y0; // Input parameters and solution
for (aa.init(0.0, 10.0, 0.1, 6, &ee); !aa.done; aa.step())
{
    x = aa.a;
    y = f(x); // Evaluate the function
    ee = fabs(y - y0); // Calculate the error
}

Notes:

This approximation can be nested for multidimensional applications. However, it is crucial to carefully define search intervals and select the appropriate step size to optimize efficiency.

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