Misleading error message in Invariant low Q extrapolation

[jellybean2004]

Describe the bug

As noted by @DrPaulSharp while reviewing PR #3833, when setting the first slider near the minimum value and running the calculation, warnings are thrown:

11:26:12 - WARNING: Low-q calculation failed: Rg expected positive value, but got -0.0
11:26:12 - WARNING: Calculation failed: Rg expected positive value, but got -0.0

The Guinier function: $I = I_0 \exp(-q^2 R_g^2 / 3)$, and $R_g \propto \sqrt{-m}$, requiring $m \leq 0$, which is the slope of $I$ vs $q^2$ plot.

Mathematically, $R_g = 0$ yields a low-q extrapolation with $e^0 = 1$. So, this should proceed, although it is not a useful result.

Lines 140-142 if invariant.py:

        if slope > 0:
            slope = 0.0
        self.radius = math.sqrt(-3 * slope)

So, the error message is misleading and prints the wrong value.

To Reproduce

Steps to reproduce the behavior:

1. Load data onto Invariant perspective and input contrast value (here, used 10000A_sphere_dsm with $contrast = 2e-6$`)
2. Turn on low q extrapolation and set the low q end value close to data minimum ($,4.2e-5$)
3. Run calculation
4. See error

Expected behavior

• Run extrapolation for $R_g = 0$.
• Show actual $R_g$ value in error message

SasView version (please complete the following information):

• Version: 6.1.2 dev

Operating system (please complete the following information):

• OS: Windows 11

Installation method (please complete the following information):

• Installed: pip

[DrPaulSharp]

Of course, the actual $R_g$ value in this case is imaginary, so possibly the correct solution is to move the raising of an error to the code that checks the value of slope. We should also consider the errors on lines 143-149 of invariant.py

# Errors
self.dscale = math.exp(constant) * dconstant
if slope == 0.0:
    n_zero = -1.0e-24
    self.dradius = -3.0 / 2.0 / math.sqrt(-3 * n_zero) * dslope
else:
    self.dradius = -3.0 / 2.0 / math.sqrt(-3 * slope) * dslope

Here, we instead choose to use a small value for slope to ensure a valid math.sqrt can be evaluated. We should choose a single convention here.

[yunliu01]

If my understanding of the discussion is right, I feel that the invariant calculation does not need to have true Rg value. (If I misunderstand something, I am sorry for this. The rest of the discussions may not be relevant then. )

What the invariant calculation is trying to do is to find a function that can be used to extrapolate the value to low Q. What it needs is a function, exp(-1/3 * phi * Q^2), where phi = Rg^2. As long as phi has a value (no matter it is positive or negative), the invariant for low-Q can be calculated. There is no need to require phi > 0. And probably, there is no need to calculate Rg for this case.

For the true radius of gyration calculation (such as Guinier plot), it should be noted that in some unusual cases, Rg^2 can be legitimately a negative value.

[DrPaulSharp]

Thanks @yunliu01, this is my understanding of the situation as well. Given this, I'll put a PR together that enables low-q Guinier extrapolation for negative values of phi.