abcmb.ABCMBTools module

Script for helper numerical tools

abcmb.ABCMBTools.bilinear_interp(x, y, z, xq, yq)

Bilinear interpolation on 2D regular grid.

Performs bilinear interpolation to evaluate function at query point (xq, yq) given values on a regular 2D grid.

Parameters:

• x array

  1D array of x-coordinates (must be sorted)

• y array

  1D array of y-coordinates (must be sorted)

• z array

  2D array of function values, shape (len(y), len(x))

• xq float

  Query x-coordinate

• yq float

  Query y-coordinate

Returns:

float

Interpolated value at (xq, yq)

Notes:

Uses standard bilinear interpolation formula with four nearest grid points.

abcmb.ABCMBTools.d00(mu, ells)

Compute Wigner d-matrix elements d^ell_{00}.

Parameters:

• mu array

  Cosine of rotation angle

• ells array

  Multipole values starting from ell=2

Returns:

array

d^ell_{00} elements for ells >= 2

abcmb.ABCMBTools.d1n(mu, ells, n)

Compute Wigner d-matrix elements d^ell_{1n}.

Parameters:

• mu array

  Cosine of rotation angle

• ells array

  Multipole values

• n int

  Second index (abs(n) <= 1)

Returns:

array

d^ell_{1n} elements

abcmb.ABCMBTools.d2n(mu, ells, n)

Compute Wigner d-matrix elements d^ell_{2n}.

Parameters:

• mu array

  Cosine of rotation angle

• ells array

  Multipole values

• n int

  Second index (abs(n) <= 2)

Returns:

array

d^ell_{2n} elements

abcmb.ABCMBTools.d3n(mu, ells, n)

Compute Wigner d-matrix elements d^ell_{3n}.

Parameters:

• mu array

  Cosine of rotation angle

• ells array

  Multipole values

• n int

  Second index (abs(n) <= 3)

Returns:

array

d^ell_{3n} elements, zero-padded for ell < 3

abcmb.ABCMBTools.d4n(mu, ells, n)

Compute Wigner d-matrix elements d^ell_{4n}.

Parameters:

• mu array

  Cosine of rotation angle

• ells array

  Multipole values

• n int

  Second index (abs(n) <= 4)

Returns:

array

d^ell_{4n} elements, zero-padded for ell < 4

abcmb.ABCMBTools.fast_interp(x, xp_min, xp_max, fp)

Fast 1D linear interpolation for uniformly-spaced grids.

Optimized interpolation that avoids searchsorted by exploiting uniform grid spacing. Significantly faster than jnp.interp for large arrays.

Parameters:

• x float or array

  Query points for interpolation

• xp_min float

  Minimum value of interpolation grid

• xp_max float

  Maximum value of interpolation grid

• fp array

  Function values on uniform grid

Returns:

float or array

Interpolated values at query points

Notes:

Credit: JAX issue #16182 (https://github.com/jax-ml/jax/issues/16182) Assumes fp is uniformly spaced between xp_min and xp_max.

abcmb.ABCMBTools.gauss_legendre_weights(n, tol=1e-16, max_it=50)

Iteratively finds the roots and weights for Gauss-Legendre quadrature integration between -1 and 1, given the number of roots n requested.

Parameters:

• n int

  Number of roots desired, typically set by lmax of the lensed power spectrum.

• tol jnp.float64

  Accuracy tolerance on the Newton root finder.

• max_it int

  Maximum iteration on the Newton root finder.

Returns:

(mu, w)(jnp.array, jnp.array)

The roots mu and weights w.

abcmb.ABCMBTools.wigner_d_matrix(mu, ells, m, n)

Compute Wigner d-matrix elements for rotation.

Recursively computes reduced Wigner d-matrix elements d^ell_{mn}(beta) for CMB lensing calculations using three-term recurrence relation.

Parameters:

• mu array

  Cosine of rotation angle beta

• ells array

  Multipole values [m, m+1, m+2, …, ellmax]

• m int

  First index (must be positive and >= abs(n))

• n int

  Second index (must satisfy abs(n) <= m)

Returns:

array

Wigner d-matrix elements, shape (len(mu), len(ells))