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\author{Ondra Kamenik}
\title{Multidimensional Tensor Library \\ for perturbation methods applied to DSGE
models}

\date{2004}
\maketitle

\section{Introduction}

The design of the library was driven by the needs of perturbation
methods for solving Stochastic Dynamic General Equilibrium models. The
aim of the library is not to provide an exhaustive interface to
multidimensional linear algebra. The tensor library's main purposes
include:
\begin{itemize}

\item Define types for tensors, for a multidimensional index of a
tensor, and types for folded and unfolded tensors. The tensors defined
here have only one multidimensional index and one reserved
one-dimensional index. The tensors should allow modelling of higher
order derivatives with respect to a few vectors with different sizes
(for example $\left[g_{y^2u^3}\right]$). The tensors should allow
folded and unfolded storage modes and conversion between them. A
folded tensor stores symmetric elements only once, while an unfolded
stores data as a whole multidimensional cube.

\item Define both sparse and dense tensors. We need only one particular
type of sparse tensor. This in contrast to dense tensors, where we
need much wider family of types.

\item Implement the Faa Di Bruno multidimensional formula. So, the main
purpose of the library is to implement the following step of Faa Di Bruno:
$$\left[B_{s^k}\right]_{\alpha_1\ldots\alpha_k}
=\left[h_{y^l}\right]_{\gamma_1\ldots\gamma_l}
\left(\sum_{c\in M_{l,k}}
\prod_{m=1}^l\left[g_{s^{\vert c_m\vert}}\right]^{\gamma_m}_{c_m(\alpha)}\right)$$
where $s$ can be a compound vector of variables, where $h_{y^l}$ and $g_{s^i}$
are tensors, $M_{l,k}$ is a set of
all equivalences of $k$ element set having $l$ classes, $c_m$ is
$m$-th class of equivalence $c$, $\vert c_m\vert$ is its
   cardinality, and $c_m(\alpha)$ is a tuple of
picked indices from $\alpha$ by class $c_m$.

Note that the sparse tensors play a role of $h$ in the Faa Di Bruno, not
of $B$ nor $g$.

\end{itemize}

\section{Classes}

The following list is a road-map to various classes in the library.

\begin{description}

\item[Tensor]
Virtual base class for all dense tensors, defines \emph{index} as the
multidimensonal iterator
\item[FTensor]
Virtual base class for all folded tensors
\item[UTensor]
Virtual base class for all unfolded tensors
\item[FFSTensor]
Class representing folded full symmetry dense tensor,
for instance $\left[g_{y^3}\right]$
\item[FGSTensor]
Class representing folded general symmetry dense tensor,
for instance $\left[g_{y^2u^3}\right]$
\item[UFSTensor]
Class representing unfolded full symmetry dense tensor,
for instance $\left[g_{y^3}\right]$
\item[UGSTensor]
Class representing unfolded general symmetry dense tensor,
for instance $\left[g_{y^2u^3}\right]$
\item[URTensor]
Class representing unfolded/folded full symmetry, row-oriented,
dense tensor. Row-oriented tensors are used in the Faa Di Bruno
above as some part (few or one column) of a product of $g$'s. Their
fold/unfold conversions are special in such a way, that they must
yield equivalent results if multiplied with folded/unfolded
column-oriented counterparts.
\item[URSingleTensor]
Class representing unfolded/folded full symmetry, row-oriented,
single column, dense tensor. Besides use in the Faa Di Bruno, the
single column row oriented tensor models also higher moments of normal
distribution.
\item[UPSTensor]
Class representing unfolded, column-oriented tensor whose symmetry
is not that of the $\left[B_{y^2u^3}\right]$ but rather of something
as $\left[B_{yuuyu}\right]$. This tensor evolves during the product
operation for unfolded tensors and its basic operation is to add
itself to a tensor with nicer symmetry, here $\left[B_{y^2u^3}\right]$.
\item[FPSTensor]
Class representing partially folded, column-oriented tensor whose
symmetry is not that of the $\left[B_{y^3u^4}\right]$ but rather
something as $\left[B_{yu\vert y^3u\vert u^4}\right]$, where the
portions of symmetries represent folded dimensions which are combined
in unfolded manner. This tensor evolves during the Faa Di Bruno
for folded tensors and its basic operation is to add itself to a
tensor with nicer symmetry, here folded $\left[B_{y^3u^4}\right]$.
\item[USubTensor]
Class representing unfolded full symmetry, row-oriented tensor which
contains a few columns of huge product
$\prod_{m=1}^l\left[g_{s^{\vert c_m\vert}}\right]^{\gamma_m}_{c_m(\alpha)}$. This is
needed during the Faa Di Bruno for folded matrices.
\item[IrregTensor]
Class representing a product of columns of derivatives
$\left[z_{y^ku^l}\right]$, where $z=[y^T,v^T,w^T]^T$. Since the first
part of $z$ is $y$, the derivatives contain many zeros, which are not
stored, hence the tensor's irregularity. The tensor is used when
calculating one step of Faa Di Bruno formula, i.e.
$\left[h_{y^l}\right]_{\gamma_1\ldots\gamma_l}
\left(\sum_{c\in M_{l,k}}
\prod_{m=1}^l\left[g_{s^{\vert c_m\vert}}\right]^{\gamma_m}_{c_m(\alpha)}\right)$.
\item[FSSparseTensor]
Class representing full symmetry, column-oriented, sparse tensor. It
is able to store elements keyed by the multidimensional index, and
multiply itself with one column of row-oriented tensor.
\item[FGSContainer]
Container of \texttt{FGSTensor}s. It implements the Faa Di Bruno with
unfolded or folded tensor $h$ yielding folded $B$. The methods are
\texttt{FGSContainer::multAndAdd}.
\item[UGSContainer]
Container of \texttt{UGSTensor}s. It implements the Faa Di Bruno with
unfolded tensor $h$ yielding unfolded $B$. The method is
\texttt{UGSContainer::multAndAdd}.
\item[StackContainerInterface]
Virtual pure interface describing all logic
of stacked containers for which we will do the Faa Di Bruno operation.
\item[UnfoldedStackContainer]
Implements the Faa Di Bruno operation for stack of
containers of unfolded tensors.
\item[FoldedStackContainer]
Implements the Faa Di Bruno for stack of
containers of folded tensors.
\item[ZContainer]
The class implements the interface \texttt{StackContainerInterface} according
to $z$ appearing in context of DSGE models. By a simple inheritance,
we obtain \texttt{UnfoldedZContainer} and also \texttt{FoldedZContainer}.
\item[GContainer]
The class implements the interface \texttt{StackContainerInterface} according
to $G$ appearing in context of DSGE models. By a simple inheritance,
we obtain \texttt{UnfoldedGContainer} and also \texttt{FoldedGContainer}.
\item[Equivalence]
The class represents an equivalence on $n$-element set. Useful in the
Faa Di Bruno.
\item[EquivalenceSet]
The class representing all equivalences on $n$-element set. Useful in the
Faa Di Bruno.
\item[Symmetry]
The class defines a symmetry of general symmetry tensor. This is it
defines a basic shape of the tensor. For $\left[B_{y^2u^3}\right]$,
the symmetry is $y^2u^3$.
\item[Permutation]
The class represents a permutation of $n$ indices. Useful in the
Faa Di Bruno.
\item[IntSequence]
The class represents a sequence of integers. Useful everywhere.
\item[TwoDMatrix and ConstTwoDMatrix]
These classes provide an interface to a code handling two-dimensional
matrices. The code resides in Sylvester module, in directory {\tt
sylv/cc}. There is
no similar interface to \texttt{Vector} and \texttt{ConstVector} classes from the
Sylvester module and they are used directly.
\item[KronProdAll]
The class represents a Kronecker product of a sequence of arbitrary
matrices and is able to multiply a matrix from the right without
storing the Kronecker product in memory.
\item[KronProdAllOptim]
The same as \texttt{KronProdAll} but it optimizes the order of matrices in
the product to minimize the used memory during the Faa Di Bruno
operation. Note that it is close to optimal flops.
\item[FTensorPolynomial and UTensorPolynomial]
Abstractions representing a polynomial whose coefficients are
folded/unfolded tensors and variable is a column vector. The classes
provide methods for traditional and horner-like polynomial
evaluation. This is useful in simulation code.
\item[FNormalMoments and UNormalMoments]
These are containers for folded/unfolded single column tensors for
higher moments of normal distribution. The code contains an algorithm
for generating the moments for arbitrary covariance matrix.
\item[TLStatic]
The class encapsulates all static information needed for the
library. It includes precalculated equivalence sets.
\item[TLException]
Simple class thrown as an exception.
\end{description}

\section{Multi-threading}

The tensor library is multi-threaded. The basic property of the
thread implementation in the library is that we do not allow running
more concurrent threads than the preset limit. This prevents threads
from competing for memory in such a way that the OS constantly switches
among threads with frequent I/O for swaps. This may occur since one
thread might need much own memory. The threading support allows for
detached threads, the synchronization points during the Faa Di Bruno
operation are relatively short, so the resulting load is close to the
preset maximum number parallel threads.

\section{Tests}

A few words to the library's test suite. The suite resides in
directory {\tt tl/testing}. There is a file {\tt tests.cc} which
contains all tests and {\tt main()} function. Also there are files
{\tt factory.hh} and {\tt factory.cc} implementing random generation
of various objects. The important property of these random objects is
that they are the same for all object's invocations. This is very
important in testing and debugging. Further, one can find files {\tt
monoms.hh} and {\tt monoms.cc}. See below for their explanation.

There are a few types of tests:
\begin{itemize}
\item We test for tensor indices. We go through various tensors with
various symmetries, convert indices from folded to unfolded and
vice-versa. We test whether their coordinates are as expected.
\item We test the Faa Di Bruno by comparison of the results of
\texttt{FGSContainer::multAndAdd} against the results of \texttt{UGSContainer::multAndAdd}. The two
 implementations are pretty different, so this is a good test.
\item We use a code in {\tt monoms.hh} and {\tt monoms.cc} to generate a
 random vector function $f(x(y,u))$ along with derivatives of
 $\left[f_x\right]$, $\left[x_{y^ku^l}\right]$, and
 $\left[f_{y^ku^l}\right]$. Then we calculate the resulting derivatives
 $\left[f_{y^ku^l}\right]$ using \texttt{multAndAdd} method of \texttt{UGSContainer}
 or \texttt{FGSContainer} and compare the derivatives provided by {\tt
 monoms}. The functions generated in {\tt monoms} are monomials with
 integer exponents, so the implementation of {\tt monoms} is quite
 easy.
\item We do a similar thing for sparse tensors. In this case the {\tt monoms}
 generate a function $f(y,v(y,u),w(y,u))$, provide all the derivatives
 and the result $\left[f_{y^ku^l}\right]$. Then we calculate the
 derivatives with \texttt{multAndAdd} of \texttt{ZContainer} and compare.
\item We test the polynomial evaluation by evaluating a folded and
 unfolded polynomial in traditional and Horner-like fashion. This gives
 four methods in total. The four results are compared.
\end{itemize}


\end{document}