@q $Id: tensor.hweb 741 2006-05-09 11:12:46Z kamenik $ @>
@q Copyright 2004, Ondra Kamenik @>

@*2 Tensor concept. Start of {\tt tensor.h} file.

Here we define a tensor class. Tensor is a mathematical object
corresponding to a $(n+1)$-dimensional array. An element of such array
is denoted $[B]_{\alpha_1\ldots\alpha_n}^\beta$, where $\beta$ is a
special index and $\alpha_1\ldots\alpha_n$ are other indices. The
class |Tensor| and its subclasses view such array as a 2D matrix,
where $\beta$ corresponds to one dimension, and
$\alpha_1\ldots\alpha_2$ unfold to the other dimension. Whether
$\beta$ correspond to rows or columns is decided by tensor subclasses,
however, most of our tensors will have rows indexed by $\beta$, and
$\alpha_1\ldots\alpha_n$ will unfold column-wise.

There might be some symmetries in the tensor data. For instance, if
$\alpha_1$ is interchanged with $\alpha_3$ and the both elements equal
for all possible $\alpha_i$, and $\beta$, then there is a symmetry
of $\alpha_1$ and $\alpha_3$.

For any symmetry, there are basically two possible storages of the
data. The first is unfolded storage, which stores all elements
regardless the symmetry. The other storage type is folded, which
stores only elements which do not repeat. We declare abstract classes
for unfolded tensor, and folded tensor.

Also, here we also define a concept of tensor index which is the
$n$-tuple $\alpha_1\ldots\alpha_n$. It is an iterator, which iterates
in dependence of symmetry and storage of the underlying tensor.

Although we do not decide about possible symmetries at this point, it
is worth noting that we implement two kinds of symmetries. The first
one is a full symmetry where all indices are interchangeable. The
second one is a generalization of the first. We define tensor of a
symmetry, where there are a few groups of indices interchangeable
within a group and not across. Moreover, the groups are required to be
consequent partitions of the index $n$-tuple. This is, we do not allow
$\alpha_1$ be interchangeable with $\alpha_3$ and not with $\alpha_2$
at the same time.

However, some intermediate results are, in fact, tensors of a symmetry
not fitting to our concept. We develop the tensor abstraction for it,
but these objects are not used very often. They have limited usage
due to their specialized constructor.

@c

#ifndef TENSOR_H
#define TENSOR_H

#include "int_sequence.h"
#include "twod_matrix.h"

@<index class definition@>;
@<|Tensor| class declaration@>;
@<|UTensor| class declaration@>;
@<|FTensor| class declaration@>;

#endif

@ The index represents $n$-tuple $\alpha_1\ldots\alpha_n$. Since its
movement is dependent on the underlying tensor (with storage and
symmetry), we maintain a pointer to that tensor, we maintain the
$n$-tuple (or coordinates) as |IntSequence| and also we maintain the
offset number (column, or row) of the index in the tensor. The pointer
is const, since we do not need to change data through the index.

Here we require the |tensor| to implement |increment| and |decrement|
methods, which calculate following and preceding $n$-tuple. Also, we
need to calculate offset number from the given coordinates, so the
tensor must implement method |getOffset|. This method is used only in
construction of the index from the given coordinates. As the index is
created, the offset is automatically incremented, and decremented
together with index. The|getOffset| method can be relatively
computationally complex. This must be kept in mind.  Also we generally
suppose that n-tuple of all zeros is the first offset (first columns
or row).

What follows is a definition of index class, the only
interesting point is |operator==| which decides only according to
offset, not according to the coordinates. This is useful since there
can be more than one of coordinate representations of past-the-end
index.

@s _Tptr int
@s _Self int

@<index class definition@>=
template<class _Tptr> class _index {
	typedef _index<_Tptr> _Self;
	_Tptr tensor;
	int offset;
	IntSequence coor;
public:@;
	_index(_Tptr t, int n) 
		: tensor(t), offset(0), coor(n, 0)@+ {}
	_index(_Tptr t, const IntSequence& cr, int c)
		: tensor(t), offset(c), coor(cr)@+ {}
	_index(_Tptr t, const IntSequence& cr)
		: tensor(t), offset(tensor->getOffset(cr)), coor(cr)@+ {}
	_index(const _index& ind)
		: tensor(ind.tensor), offset(ind.offset), coor(ind.coor)@+ {}
	const _Self& operator=(const _Self& in)
		{@+ tensor = in.tensor;@+ offset = in.offset;@+ coor = in.coor;
		return *this;@+}
	_Self& operator++()
		{@+ tensor->increment(coor);@+ offset++;@+ return *this;@+}
	_Self& operator--()
		{@+ tensor->decrement(coor);@+ offset--;@+ return *this;@+}
	int operator*() const
		{@+ return offset;@+}
	bool operator==(const _index& n) const
		{@+ return offset == n.offset;@+}
	bool operator!=(const _index& n) const
		{@+ return offset != n.offset;@+}
	const IntSequence& getCoor() const
		{@+ return coor;@+}
	void print() const
		{@+ printf("%4d: ", offset);@+  coor.print();@+}
};

@ Here is the |Tensor| class, which is nothing else than a simple subclass
of |TwoDMatrix|. The unique semantically new member is |dim| which is tensor
dimension (length of $\alpha_1\ldots\alpha_n$). We also declare
|increment|, |decrement| and |getOffset| methods as pure virtual.

We also add members for index begin and index end. This is useful,
since |begin| and |end| methods do not return instance but only
references, which prevent making additional copy of index (for example
in for cycles as |in != end()| which would do a copy of index for each
cycle). The index begin |in_beg| is constructed as a sequence of all
zeros, and |in_end| is constructed from the sequence |last| passed to
the constructor, since it depends on subclasses. Also we have to say,
along what coordinate is the multidimensional index. This is used only
for initialization of |in_end|.

Also, we declare static auxiliary functions for $\pmatrix{n\cr k}$
which is |noverk| and $a^b$, which is |power|.

@s indor int

@<|Tensor| class declaration@>=
class Tensor : public TwoDMatrix {
public:@;
	enum indor {along_row, along_col};
	typedef _index<const Tensor*> index;
protected:@;
	const index in_beg;
	const index in_end;
	int dim;
public:@;
	Tensor(indor io, const IntSequence& last, int r, int c, int d)
		: TwoDMatrix(r, c),
		  in_beg(this, d),
		  in_end(this, last, (io == along_row)? r:c),
		  dim(d)@+ {}
	Tensor(indor io, const IntSequence& first, const IntSequence& last,
		   int r, int c, int d)
		: TwoDMatrix(r, c),
		  in_beg(this, first, 0),
		  in_end(this, last, (io == along_row)? r:c),
		  dim(d)@+ {}
	Tensor(int first_row, int num, Tensor& t)
		: TwoDMatrix(first_row, num, t),
		  in_beg(t.in_beg),
		  in_end(t.in_end),
		  dim(t.dim)@+ {}
	Tensor(const Tensor& t)
		: TwoDMatrix(t),
		  in_beg(this, t.in_beg.getCoor(), *(t.in_beg)),
		  in_end(this, t.in_end.getCoor(), *(t.in_end)),
		  dim(t.dim)@+ {}
	virtual ~Tensor()@+ {}
	virtual void increment(IntSequence& v) const =0;
	virtual void decrement(IntSequence& v) const =0;
	virtual int getOffset(const IntSequence& v) const =0;
	int dimen() const
		{@+ return dim;@+}

	const index& begin() const
		{@+ return in_beg;@+}
	const index& end() const
		{@+ return in_end;@+}

	static int noverk(int n, int k);
	static int power(int a, int b);
	static int noverseq(const IntSequence& s)
		{
			IntSequence seq(s);
			return noverseq_ip((IntSequence&)s);
		}
private:@;
	static int noverseq_ip(IntSequence& s);
};

@ Here is an abstraction for unfolded tensor. We provide a pure
virtual method |fold| which returns a new instance of folded tensor of
the same symmetry. Also we provide static methods for incrementing and
decrementing an index with full symmetry and general symmetry as
defined above.

@<|UTensor| class declaration@>=
class FTensor;
class UTensor : public Tensor {
public:@;
	UTensor(indor io, const IntSequence& last, int r, int c, int d)
		: Tensor(io, last, r, c, d)@+ {}
	UTensor(const UTensor& ut)
		: Tensor(ut)@+ {}
	UTensor(int first_row, int num, UTensor& t)
		: Tensor(first_row, num, t)@+ {}
	virtual ~UTensor()@+ {}
	virtual FTensor& fold() const =0;

	static void increment(IntSequence& v, int nv);
	static void decrement(IntSequence& v, int nv);
	static void increment(IntSequence& v, const IntSequence& nvmx);
	static void decrement(IntSequence& v, const IntSequence& nvmx);
	static int getOffset(const IntSequence& v, int nv);
	static int getOffset(const IntSequence& v, const IntSequence& nvmx);
};

@ This is an abstraction for folded tensor. It only provides a method
|unfold|, which returns the unfolded version of the same symmetry, and
static methods for decrementing indices.

We also provide static methods for decrementing the |IntSequence| in
folded fashion and also calculating an offset for a given
|IntSequence|. However, this is relatively complex calculation, so
this should be avoided if possible.

@<|FTensor| class declaration@>=
class FTensor : public Tensor {
public:@;
	FTensor(indor io, const IntSequence& last, int r, int c, int d)
		: Tensor(io, last, r, c, d)@+ {}
	FTensor(const FTensor& ft)
		: Tensor(ft)@+ {}
	FTensor(int first_row, int num, FTensor& t)
		: Tensor(first_row, num, t)@+ {}
	virtual ~FTensor()@+ {}
	virtual UTensor& unfold() const =0;

	static void decrement(IntSequence& v, int nv);
	static int getOffset(const IntSequence& v, int nv)
		{@+IntSequence vtmp(v);@+ return getOffsetRecurse(vtmp, nv);@+}
private:@;
	static int getOffsetRecurse(IntSequence& v, int nv);
};

@ End of {\tt tensor.h} file.