关于quad_mpi.c中double f(double x)函数数学公式含义的技术咨询

Let's break this down clearly — I recognize this function from numerical quadrature (aka numerical integration) tasks, which fits perfectly with the quad_mpi.c filename (since "quad" almost always refers to integration here).

First, let's restate the code's mathematical formula in plain terms:

f(x) = \frac{50}{\pi \cdot (2500x^2 + 1)}

What this function actually is #

This is a scaled Cauchy distribution (Lorentz distribution) probability density function (PDF). The standard Cauchy PDF has the form:

f(x; x_0, \gamma) = \frac{1}{\pi\gamma \left(1 + \left(\frac{x - x_0}{\gamma}\right)^2\right)}

Where:

• x_0 is the location parameter (the center of the distribution, here it's 0)
• γ is the scale parameter (controls the width of the curve)

If you plug in x_0 = 0 and γ = 1/50 into the standard formula, you'll get exactly the function in your code:

• (x - x_0)/γ = 50x
• Squaring that gives 2500x²
• Substitute back: 1/(π*(1/50)*(1 + 2500x²)) = 50/(π*(2500x² + 1))

Why this is in quad_mpi.c #

This function is almost certainly being used as a test function for parallel numerical integration:

• MPI-based quadrature programs like this split an integration interval across multiple processes, compute partial integrals, then combine results.
• The Cauchy distribution is a good test because its integral has a known analytic solution:
  • The integral from -∞ to +∞ of f(x) is 1 (since it's a valid PDF).
  • For finite intervals (e.g., -a to a), you can calculate the exact integral using the arctangent function, then compare it to the numerical result from the MPI code to verify correctness.

Why π is pre-defined instead of calculated #

You were right to notice that π isn't being solved for here — the function inherently depends on π as part of its definition. The code uses a precise value of π to compute the PDF accurately, since the goal is to integrate this function, not compute π itself.

内容的提问来源于stack exchange，提问作者bubaebi