double_gram_schmidt_step Interface

public interface double_gram_schmidt_step

Description

Given an array $X$ of abstract_vector and an abstract_vector (or array of abstract_vectors) $y$, this subroutine returns a modified vector $y$ orthogonal to all columns of $X$, i.e.

$$y \leftarrow \left( I - XX^H \right) y,$$

using a double Gram-Schmidt process. On exit, $y$ is orthogonal to $X$ albeit does not have unit-norm. Note moreover that $X$ is assumed to be an orthonormal set of vectors. The function can also return the projection coefficients $\beta = X^H y$.

Syntax

    call double_gram_schmidt_step(y, X, info [, if_chk_orthonormal] [, beta])

Arguments

y : abstract_vector (or array of abstract_vector) that needs to be orthogonalize in-place against $X$.

X : Array of abstract_vector against which $y$ needs to be orthogonalized. Note the function assumes that $X$ is an orthonormal set of vectors, i.e. $X^H X = I$. If it this is not the case, the result are meaningless.

info : integer Information flag.

if_chk_orthonormal (optional) : logical flag (default .true.) to check whether $X$ is an orthonormal set of vectors or not. If the orthonormality returns .false., the function throws an error. Note that this check is however computationally expensive and can be disable for the sake of performances.

beta (optional) : real or complex array containing the coefficients $\beta = X^H y$.

Module Procedures