function [x,state] = struct_invsqrtm(z,task)
%STRUCT_INVSQRTM Matrix inverse square root.
%   [x,state] = struct_invsqrtm(z,[]) computes x as the matrix inverse
%   square root such that inv(z) equals x*x. The structure state stores
%   information which is reused in computing the right and left
%   Jacobian-vector products.
%
%   struct_invsqrtm(z,task) computes the right or left Jacobian-vector
%   product of this transformation, depending on the structure task. Use
%   the structure state and add the field 'r' of the same shape as z or the
%   field 'l' of the same shape as x to obtain the structure task for
%   computing the right and left Jacobian-vector products
%   
%      (dF(:)/dz(:).')*task.r(:) and
%      (dF(:)/dz(:).')'*task.l(:) + conj((dF(:)/dconj(z(:)).')'*task.l(:)),
%   
%   respectively. Here, F(z) represents this transormation, (:) signifies
%   vectorization and the derivative w.r.t. z (conj(z)) is a partial
%   derivative which treats conj(z) (z) as constant. The output has the
%   same shape as x or z for the right and left Jacobian-vector products,
%   respectively.

%   Authors: Laurent Sorber (Laurent.Sorber@cs.kuleuven.be)
%            Marc Van Barel (Marc.VanBarel@cs.kuleuven.be)
%            Lieven De Lathauwer (Lieven.DeLathauwer@kuleuven-kulak.be)
%
%   References:
%   [1] L. Sorber, M. Van Barel, L. De Lathauwer, "Structured data fusion,"
%       J. Sel. Topics Signal Process., IEEE, Vol. 9, No. 4, pp. 586-600,
%       2015.

if nargin < 2, task = []; end
state = [];

if isempty(task) || (isempty(task.l) && isempty(task.r))
    x = z^(-1/2);
    state.invz = inv(z);
    state.invsqrtz = x;
elseif ~isempty(task.r)
    x = lyap(task.invsqrtz,task.invsqrtz,task.invz*task.r*task.invz);
elseif ~isempty(task.l)
    x = lyap(task.invsqrtz',task.invsqrtz',task.invz'*task.l*task.invz');
end

end
