# Frequency Grids¶

GYRE searches for sign changes of the discriminant function \(\Dfunc(\omega)\) on a grid \(\{\omega_{1},\omega_{2},\ldots,\omega_{M}\}\) in the dimensionless frequency \(\omega \equiv \sqrt{R^{3}/GM} \sigma\). The computational cost of a calculation scales with the total number of points \(M\) in this grid, while the grid’s resolution — i.e., the spacing between adjacent points — impacts the completeness of the modes found by GYRE (see the Limitations of the Numerical Method section for a discussion of these behaviors in the context of the stretched string BVP).

GYRE constructs a fresh frequency grid for each combination of
harmonic degree \(\ell\) and azimuthal order \(m\) specified
in the `&mode`

namelist groups (see the
Namelist Input Files chapter for more details).

The starting

point for each of these grids is the *scaffold grid*, which comprises
the following:

- an inner point \(x=\xin\);
- an outer point \(x=\xout\);
- the subset of points of the input model grid satisfying \(\xin < x < \xout\)

By default, \(\xin\) and \(\xout\) are obtained from the input
model grid as well, meaning that the scaffold grid is identical to the
model grid. However, either or both can be overridden using the
`x_i`

and `x_o`

parameters, respectively, of the
`&grid`

namelist group.

The grid is established