How Does Approximation Search Efficiently Find Approximate Solutions in Non-Monotonic Domains?

How Approximation Search Works

Approximation search, akin to binary search, enables the efficient approximation of values or parameters within a specified real domain. Unlike binary search, it operates independently of monotonic function restrictions.

Algorithm:

1. Probe evenly dispersed points: Calculate the distance/error for each point within the search interval.
2. Identify minimal error point: Determine the point with the lowest error.
3. Recursively increase accuracy: Adjust the search interval around the minimal error point and refine the search step size.
4. Final solution: Repeat until the desired accuracy is achieved.

Applicability:

Approximation search finds applications in various scenarios, including:

• Approximating solutions to transcendental equations
• Fitting polynomials or parametric functions
• Solving difficult equations when inverse functions are unavailable
• Non-monotonic or non-functional value approximations

Implementation:

The provided C code implements the approximation search algorithm:

class approx { ... };
...
for (aa.init(0.0,10.0,0.1,6,&ee); !aa.done; aa.step()) { ... }

Usage:

• Define an approx object (aa).
• Initialize it with parameters a0, a1, da, n, and a pointer to the error variable ee.
• Iterate through the loop to perform the approximation search. The final solution is stored in aa.a.

Key Points:

• Careful interval and step size selection is crucial.
• The algorithm explores the possibility of multiple solutions for non-functional fits through recursive subdivision.
• Nested multidimensional fits require careful consideration for performance.

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