How Can Approximation Search Efficiently Find Values and Parameters for Non-Monotonic Functions?

Approximating Values and Parameters for Function Fitting and Equation Solving

Understanding the inner workings of approximation search is crucial in solving problems related to fitting polynomials, parametric functions, and transcendental equations. This approach allows you to estimate values and parameters effectively, even when dealing with functions that are not strictly monotonic.

How Approximation Search Works

Approximation search mimics the concept of binary search, but it doesn't impose the constraint of the searched function being strictly monotonic. Here's how it works:

1. Probe sample points evenly distributed across a known interval .
2. Calculate the error or distance between the function value f(x) at each sample point and a target value y0.
3. Identify the sample point with the minimum error (denoted by aa).
4. Repeat steps 1-3 within a narrower interval centered around aa to increase accuracy.
5. Continue recursion until the desired accuracy is achieved or a maximum recursion count is reached.

Key Features

• Ability to handle non-monotonic functions or functions that are not strict functions.
• Provides a reliable and efficient algorithm with O(log(n)) complexity.
• Allows for user-defined precision and error calculation.

Implementation

Here's a simplified C implementation of approximation search:

class approx {
public:
    void init(double a0, double a1, double da, int n, double *e) {
        a0 = min(a0, a1); a1 = max(a0, a1); // Ensure a0 < a1
        da = abs(da);
        this->n = n; e = e;
        e0 = -1.0; i = 0; a = a0; aa = a0;
        done = false; stop = false;
    }
    void step() {
        if ((e0 < 0.0) || (e0 > *e)) { e0 = *e; aa = a; }  // Better solution
        if (stop) {
            i++;
            if (i >= n) { done = true; a = aa; return; }  // Final solution
            a0 = aa - fabs(da);
            a1 = aa + fabs(da);
            a = a0; da *= 0.1;
            a0 += da; a1 -= da;
            stop = false;
        } else {
            a += da;
            if (a > a1) { a = a1; stop = true; }  // Next point
        }
    }
};

Usage Example

Suppose you want to find the value of x such that y = f(x) is closest to y0. Here's how to use the approx class:

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

Advantages

• Increased accuracy of solutions for transcendental equations and curve fitting.
• Handles non-monotonic functions and allows for multiple solutions.
• Efficient and recursive approach for complex equations.

Considerations

• Choosing an appropriate search interval is crucial for efficiency.
• Initial step size da affects the trade-off between speed and accuracy.
• Recursion depth can impact performance for highly complex problems.

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