tlpipe.timestream.map_making.eigh¶

tlpipe.timestream.map_making.eigh(a, b=None, lower=True, eigvals_only=False, overwrite_a=False, overwrite_b=False, turbo=True, eigvals=None, type=1, check_finite=True)[source]¶

Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix.

Find eigenvalues w and optionally eigenvectors v of matrix a, where b is positive definite:

              a v[:,i] = w[i] b v[:,i]
v[i,:].conj() a v[:,i] = w[i]
v[i,:].conj() b v[:,i] = 1

Parameters:

a ((M, M) array_like) – A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed.
b ((M, M) array_like, optional) – A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed.
lower (bool, optional) – Whether the pertinent array data is taken from the lower or upper triangle of a. (Default: lower)
eigvals_only (bool, optional) – Whether to calculate only eigenvalues and no eigenvectors. (Default: both are calculated)
turbo (bool, optional) – Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None)
eigvals (tuple (lo, hi), optional) – Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo <= hi <= M-1. If omitted, all eigenvalues and eigenvectors are returned.
type (int, optional) –
Specifies the problem type to be solved:

type = 1: a v[:,i] = w[i] b v[:,i]
type = 2: a b v[:,i] = w[i] v[:,i]

type = 3: b a v[:,i] = w[i] v[:,i]
overwrite_a (bool, optional) – Whether to overwrite data in a (may improve performance)
overwrite_b (bool, optional) – Whether to overwrite data in b (may improve performance)
check_finite (boolean, optional) – Whether to check the input matrixes contain only finite numbers. Disabling may give a performance gain, but may result to problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns:

w ((N,) float ndarray) – The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity.
v ((M, N) complex ndarray) – (if eigvals_only == False)

The normalized selected eigenvector corresponding to the eigenvalue w[i] is the column v[:,i].

Normalization:

type 1 and 3: v.conj() a v = w

type 2: inv(v).conj() a inv(v) = w

type = 1 or 2: v.conj() b v = I

type = 3: v.conj() inv(b) v = I

Raises:

LinAlgError : – If eigenvalue computation does not converge, an error occurred, or b matrix is not definite positive. Note that if input matrices are not symmetric or hermitian, no error is reported but results will be wrong.