% -*- texinfo -*-
% @deftypefn {Function File} {[@var{u}, @var{h}, @var{nu}] =} krylov (@var{a}, @var{v}, @var{k}, @var{eps1}, @var{pflg})
% Construct an orthogonal basis @var{u} of block Krylov subspace
%
% @example
% [v a*v a^2*v @dots{} a^(k+1)*v]
% @end example
%
% @noindent
% Using Householder reflections to guard against loss of orthogonality.
%
% If @var{v} is a vector, then @var{h} contains the Hessenberg matrix
% such that @code{a*u == u*h+rk*ek'}, in which @code{rk =
% a*u(:,k)-u*h(:,k)}, and @code{ek'} is the vector
% @code{[0, 0, @dots{}, 1]} of length @code{k}. Otherwise, @var{h} is
% meaningless.
%
% If @var{v} is a vector and @var{k} is greater than
% @code{length(A)-1}, then @var{h} contains the Hessenberg matrix such
% that @code{a*u == u*h}.
%
% The value of @var{nu} is the dimension of the span of the krylov
% subspace (based on @var{eps1}).
%
% If @var{b} is a vector and @var{k} is greater than @var{m-1}, then
% @var{h} contains the Hessenberg decomposition of @var{a}.
%
% The optional parameter @var{eps1} is the threshold for zero. The
% default value is 1e-12.
%
% If the optional parameter @var{pflg} is nonzero, row pivoting is used
% to improve numerical behavior. The default value is 0.
%
% Reference: Hodel and Misra, 'Partial Pivoting in the Computation of
% Krylov Subspaces', to be submitted to Linear Algebra and its
% Applications
% @end deftypefn