Create a 2D complex map from a set of complex spherical harmonic coefficients sampled on the Gauss-Legendre quadrature nodes.

## Usage

gridglq = MakeGridGLQC (cilm, zero, [lmax, norm, csphase, lmax_calc, extend])

## Returns

- gridglq : complex, dimension (nlat, nlong)
- A 2D complex map of the function sampled on the Gauss-Legendre quadrature nodes, dimensioned as (lmax+1, 2*lmax+1) if extend is 0 or (lmax+1, 2*lmax+2) if extend is 1.

## Parameters

- cilm : complex, dimension (2, lmaxin+1, lmaxin+1)
- The complex spherical harmonic coefficients of the function. When evaluating the function, the maximum spherical harmonic degree considered is the minimum of lmax, lmaxin, or lmax_calc (if specified). The first index specifies the coefficient corresponding to the positive and negative order of m, respectively, with Clm=cilm[0,l,m+] and Cl,-m=cilm[1,l,m].
- zero : float, dimension (lmax+1)
- The nodes used in the Gauss-Legendre quadrature over latitude, calculated by a call to SHGLQ.
- lmax : optional, integer, default = lmaxin
- The maximum spherical harmonic bandwidth of the function. This determines the sampling nodes and dimensions of the output grid.
- norm : optional, integer, default = 1
- 1 (default) = Geodesy 4-pi normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
- csphase : optional, integer, default = 1
- 1 (default) = do not apply the Condon-Shortley phase factor to the associated Legendre functions; -1 = append the Condon-Shortley phase factor of (-1)^m to the associated Legendre functions.
- lmax_calc : optional, integer, default = lmax
- The maximum spherical harmonic degree used in evaluating the function. This must be less than or equal to lmax.
- extend : input, optional, bool, default = False
- If True, compute the longitudinal band for 360 E.

## Description

MakeGridGLQC will create a 2-dimensional complex map from a set of input complex spherical harmonic coefficients sampled on the Gauss-Legendre quadrature nodes. This is the inverse of the routine SHExpandGLQC. The latitudinal nodes correspond to the zeros of the Legendre polynomial of degree lmax+1, and the longitudinal nodes are equally spaced with an interval of 360/(2*lmax+1) degrees. When evaluating the function, the maximum spherical harmonic degree that is considered is the minimum of lmax, lmaxin, or lmax_calc (if specified).

The redundant longitudinal band for 360 E is excluded from the grid by default, but this can be computed by specifying the optional argument extend. The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments norm and csphase; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m. The normalized legendre functions are calculated using the scaling algorithm of Holmes and Featherstone (2002), which are accurate to about degree 2800. The unnormalized functions are accurate only to about degree 15.

The reconstruction of the spherical harmonic function may be speeded up by precomputing the Legendre functions on the Gauss-Legendre quadrature nodes in the routine SHGLQ with the optional parameter cnorm set to 1. However, given that this array contains on the order of lmax**3 entries, this is only feasible for moderate values of lmax.

## Reference

Holmes, S. A., and W. E. Featherstone, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions, J. Geodesy, 76, 279-299, 2002.

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