GNU Free Documentation License, Version 1.2

- User documentation for the classes matrix, MatrixView and ConstMatrixView
- Library contributor documentation
- Maintainer documentation for the matrix classes
- Bugs, Shortcomings and other ideas
- Main changes

CoCoALib offers two distinct concepts for dealing with matrices: one
is an explicit implementation of a matrix, the other is a way to "view"
another existing object as though it were a matrix (possibly of a special form).
An example of a `MatrixView`

is seeing a `std::vector<RingElem>`

as a
row matrix (see `MatrixView`

).

There are two categories of matrix view, namely `ConstMatrixView`

and
`MatrixView`

. The only difference between them is that the former
does not allow you to change the entries while the latter allows you
to change them (or at least some of them).

There are also two categories of explicit matrix. A `ConstMatrix`

is
a matrix whose entries and dimensions are fixed. In contrast, a (non-const)
`matrix`

offers operations for assigning to entries, exchanging rows and
columns, and even varying the dimensions -- see the maintainer documentation
if you're curious about why these operations are not allowed on a `MatrixView`

.

Here are some guidelines for writing a function or procedure which takes
matrices as arguments. If the function/procedure does not change the
structure of the matrix, then use `ConstMatrixView`

or `MatrixView`

.
If the structure of the matrix parameter may be modified then you must use
`matrix&`

as the parameter type.

The following create a `matrix`

:

`NewDenseMat(R, r, c)`

-- (see`DenseMatrix`

)`NewSparseMat(R, r, c)`

-- NOT YET IMPLEMENTED!!

The following create a `ConstMatrix`

:

`ZeroMat(R, r, c)`

-- constant matrix:`r`

-by-`c`

zero matrix over`R`

`IdentityMat(R, n)`

-- constant matrix:`n`

-by-`n`

identity matrix over`R`

The following create *matrix views*: for instance, changing an entry in
`RowMat(v)`

will also change the vector `v`

,
see `MatrixView`

PseudoConstructors for more details.

`transpose(M)`

`submat(M, rows, cols)`

`ColMat(v)`

`RowMat(v)`

`DiagMat(v)`

`BlockMat(A, B, C, D)`

`ConcatVer(A, B)`

`ConcatHor(A, B)`

`ConcatDiag(A, B)`

`ConcatAntiDiag(A, B)`

The following create a `matrix`

and come from `MatrixSpecial`

.
See there for more details.

`jacobian(f, indets)`

`TensorMat(M1, M2)`

`RingOf(M)`

-- the ring to which the matrix entries belong`NumRows(M)`

-- the number of rows in`M`

(may be zero)`NumCols(M)`

-- the number of columns in`M`

(may be zero)`out << M`

-- print the value of the matrix on ostream out (with a*dense*representation)

`M1 == M2`

-- true iff`M1(i,j) == M2(i,j)`

for all i,j`IsSymmetric(M)`

-- true iff`M(i,j) == M(j,i)`

for all i,j`IsAntiSymmetric(M)`

-- true iff`M(i,j) == -M(j,i)`

for all i,j`IsDiagonal(M)`

-- true iff`M(i,j) == 0`

for all i!=j`IsMat0x0(M)`

-- true iff`NumRows(M) == 0 && NumCols(M)==0`

**NB indices start from 0**

`M(i,j)`

-- the (`i`

,`j`

) entry of`M`

`IsZeroRow(M,i)`

-- true iff row`i`

of`M`

is zero`IsZeroCol(M,j)`

-- true iff column`j`

of`M`

is zero

The following come from `MatrixArith`

, see there for more details.

`*`

`+`

`-`

`/`

`det(M)`

`rank(M)`

`inverse(M)`

`adjoint(M)`

`void mul(matrix& lhs, M1, M2)`

`LinSolve(M,rhs)`

`LinKer(M)`

`M->myIsWritable(i,j)`

-- true iff posn`(i,j)`

can be written to`SetEntry(M,i,j,val)`

-- set entry`(i,j)`

of matrix`M`

to`val`

(integer, rational, RingElem). Throws`ERR::BadMatrixSetEntry`

if`(i,j)`

is not writable`MV->myRawEntry(i,j)`

-- raw pointer to`(i,j)`

entry (may be called only if the`(i,j)`

posn is writable)

**NOTE:** You cannot set a matrix entry the obvious way,
*i.e.* `M(i,j) = value;`

You must use `SetEntry(M,i,j,value)`

.

With sanity checks

`SwapRows(M,i1,i2)`

-- swap rows`i1`

and`i2`

`SwapCols(M,j1,j2)`

-- swap columns`j1`

and`j2`

`DeleteRow(M,i)`

-- delete row`i`

and moves up the following rows`DeleteCol(M,j)`

-- delete column`j`

and moves up the following cols

Without sanity checks

`M->myResize(r,c)`

-- change size of`M`

to`r`

-by-`c`

(new entries are zero)`M->myRowMul(i,r)`

-- multiply row`i`

by`r`

`M->myColMul(j,r)`

-- multiply column`j`

by`r`

`M->myAddRowMul(i1,i2,r)`

-- add`r`

times row`i2`

to row`i1`

`M->myAddColMul(j1,j2,r)`

-- add`r`

times column`j2`

to column`j1`

`M->mySwapRows(i1,i2)`

-- swap rows`i1`

and`i2`

`M->mySwapCols(j1,j2)`

-- swap columns`j1`

and`j2`

**NOTE:** these are not permitted on `MatrixView`

because of various problems which
could arise *e.g.* with aliasing in block matrices (see maintainer documentation).
`myResize`

simply truncates rows/columns if they are too long, and any new
entries are filled with zeroes. So, if you resize to a smaller matrix, you get
just the "top left hand" part of the original.

At the moment assignment of matrices is not allowed. The only way to make
a copy of a matrix (view) is by calling a genuine constructor (so far only
`NewDenseMat`

comes into this category).

`IsRectangular(VV)`

-- says whether a`vector`

of`vector`

is rectangular

The classes `ConstMatrixView`

, `MatrixView`

, `ConstMatrix`

and `matrix`

are
just reference counting smart-pointers to objects of type derived from
the abstract base classes `ConstMatrixViewBase`

, `MatrixViewBase`

,
`ConstMatrixBase`

and `MatrixBase`

respectively;
this is analogous to the way `ring`

s are implemented. Consequently every
concrete matrix class or matrix view class must be derived from these abstract
classes. At the moment, it is better to derive from `MatrixViewBase`

rather
than `ConstMatrixViewBase`

because of the way `BlockMat`

is implemented.

The base class `ConstMatrixViewBase`

declares the following pure virtual
member fns:

`myRing()`

-- returns the ring to which the matrix entries belong`myNumRows()`

-- returns the number of rows in the matrix`myNumCols()`

-- returns the number of columns in the matrix`myEntry(i,j)`

-- returns ConstRefRingElem aliasing the value of entry`(i,j)`

`IamEqual(M)`

-- true iff *this==M`IamSymmetric()`

-- true iff entry (i,j) == entry (j,i)`IamAntiSymmetric()`

-- true iff entry (i,j) == -entry (j,i)`IamDiagonal()`

-- true iff entry (i,j) == 0 for i!=j`myMulByRow(v,w)`

-- v = w.M, vector-by-matrix product`myMulByCol(v,w)`

-- v = M.w, matrix-by-vector product`myIsZeroRow(i)`

-- true iff row`i`

is zero`myIsZeroCol(j)`

-- true iff column`j`

is zero`myDet(d)`

-- computes determinant into d`myRank()`

-- computes rank (matrix must be over an integral domain)`myOutput(out)`

-- print out the matrix on ostream`out`

`myCheckRowIndex(i)`

-- throws an exception ERR::BadRowIndex if`i`

is too large`myCheckColIndex(j)`

-- throws an exception ERR::BadColIndex if`j`

is too large

These are the additional virtual functions present in `MatrixViewBase`

:

`myIsWritable(i,j)`

-- true iff entry`(i,j)`

can be modified;`i`

&`j`

are unchecked`mySetEntry(i,j,value)`

-- set entry`(i,j)` to ``value`

(integer, rational, RingElem)

The class `ConstMatrixBase`

is almost identical to `ConstMatrixViewBase`

; the
only real difference is that an instance of a concrete class derived from
`ConstMatrixBase`

should be self-contained (*i.e.* not refer to any external
data structure) whereas a `ConstMatrixView`

may refer to an external object
(and typically *should* do so, unless it is derived from `ConstMatrixBase`

).

These are the additional virtual functions present in `MatrixBase`

:

`myRowMul(i,r)`

-- multiply row i by r`myColMul(j,r)`

-- multiply column j by r`myAddRowMul(i1,i2,r)`

--add r times row i2 to row i1`myAddColMul(j1,j2,r)`

--add r times column j2 to column j1`mySwapRows(i1,i2)`

-- swap rows i1 and i2`mySwapCols(j1,j2)`

-- swap columns j1 and j2

**Default definitions:**

- IamEqual, IamSymmetric, IamAntiSymmetric, IamDiagonal,
myMulByRow, myMulByCol, myIsZeroRow, myIsZeroCol, myOutput all have
default
*dense*definitions - myDet and myRank have default definitions which use gaussian elimination

I shall assume that you have already read the *User Documentation* and
*Library Contributor Documentation*.

The implementation underwent a big structural change in April 2008. I believe
most of the design is sensible now, but further important changes could still
occur. The implementation of the four matrix classes is wholly analogous to
that of ring: they are simply reference counting smart-pointer classes (which
may have derived classes). If assignment of matrices becomes permitted then
some extra complication will be needed -- *e.g.* `MakeUnique`

, and the pointed
object must be able to clone itself.

The only delicate part of the implementation is in `myMulByRow`

and
`myMulByCol`

where a buffer is used for the answer so that the fns can be
exception clean and not suffer from aliasing problems between the args.

Recall that by convention member functions of the base class do not
perform sanity checks on their arguments; though it is wise to include
such checks inside `CoCoA_ASSERT`

calls to help during debugging. The
sanity check should be conducted in the functions which present a nice
user interface.

Q: Why did I create both `MatrixView`

and `ConstMatrixView`

?

A: Because the usual C++ *const mechanism* doesn't work the way I want it to.
Consider a function which takes an argument of type `const MatrixView&`

.
One would not expect that function to be able to modify the entries of the
matrix view supplied as argument. However, you can create a new non
const `MatrixView`

using the default copy ctor, and since `MatrixView`

is
a smart pointer the copy refers to the same underlying object. Currently,
a `MatrixView`

object does not perform *copy on write* if the reference
count of the underlying object is greater than 1 -- it is not at all clear
what *copy on write* would mean for a matrix view (Should the underlying
object be duplicated??? I don't like that idea!).

Q: Why are row, column and resizing operations which are allowed on `matrix`

objects not allowed on `MatrixView`

objects?

A: I disallowed them because there are cases where it is unclear what should
happen. For example, suppose `M`

is a square matrix, and someone creates the
view `MtrM`

defined to be `ConcatHor(M, transpose(M))`

then there is non-trivial
aliasing between the entries of `MtrM`

. What should happen if you try to
multiply the second row of `MtrM`

by 2? What should happen if you try to
add a new column to `MtrM`

? In general, resizing `MtrM`

would be problematic.
Here's another case: it is not clear how a resize operation should work on a
matrix view based on a `vector<RingElem>`

; would the underlying vector be
resized too?

I chose to offer member fns for checking indices so that error messages could
be uniform in appearance. I chose to have two index checking member fns
`myCheckRowIndex`

and `myCheckColIndex`

rather than a single unified fn, as a
single fn would have to have the *ugly* possibility of throwing either of two
different exceptions.

I declared (and defined) explicitly the default ctor and dtor of the four base classes, to prohibit/discourage improper use of pointers to these classes.

The default *dense* definition of `MatrixBase::myOutput`

seems a reasonable
starting point -- but see the bugs section below!

The use of `std::vector<RingElem>`

should be replaced by `ModuleElem`

which
automatically guarantees that all its components are in the same ring.

Should the default *dense* definitions of the output functions be removed?
They could be quite inappropriate for a large sparse matrix.

Should the OpenMath output function send the ring with every value sent (given that the ring is also specified in the header)?

Should the index checking fns `myCheckRowIndex`

and `myCheckColIndex`

really
throw? Perhaps there should be an alternative which merely returns a boolean
value? When would the boolean version be genuinely beneficial?

Why can you not simply write `M(i,j) = NewValue;`

? It is non-trivial
because if `M`

is a sparse matrix then use of `M(i,j)`

in that context
will require a structural modification to `M`

if `NewValue`

is non-zero
and currently `M`

has no `[i,j]`

element. This natural syntax could be made
possible by using a proxy class for `M(i,j)`

; in a RHS context it simply
produces a `ConstRefRingElem`

for the value of the entry; in a LHS context
the appropriate action depends on the implementation of the matrix.

I'm quite unsure about the signatures of several functions. I am not happy
about requiring the user to use member functions for self-modifying operations
(*e.g.* swap rows, etc) since elsewhere member functions by convention do not
check the validity of their arguments.

Virtual member fn `myIsWritable`

is not really intended for public use, but an
arcane C++ rule prevents me from declaring it to be `protected`

. Apparently a
`protected`

name in the base class is accessible only through a ptr/ref to the
derived class (and not through one to the base class) -- no idea why!

Should assignment of matrices be allowed? Ref counting should make this
relatively cheap, but must beware of the consequences for iterators
(*e.g.* if it is possible to have a *reference to a row/column of a matrix*).

Would it be useful/helpful/interesting to have row-iterators and col-iterators for matrices?

**2016**

- Sept: added doc for
`ConstMatrix`

**2012**

- April: added fns
`SwapRows`

and`SwapCols`

- March: changed printing style

**2011**

- February: IsSymmetric, IsAntiSymmetric, IsDiagonal, operator== default and some optimized implementations.
- February (v0.9942): first release of
`MatrixSpecial`

files