CoCoALib offers two distinct concepts for dealing with matrices: one
is an explicit implementation of a matrix, the other is a way to "view"
an 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).
In contrast, a true matrix offers further operations for changing rows,
columns and 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 matrix views: for instance, changing an entry in
RowMat(v) will also change the vector v,
see MatrixView PseudoConstructors for more details.
ZeroMat(R, r, c)
IdentityMat(R, n)
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::ConstMatEntry if the entry is not writable
AssignZero(M) -- set all entries of M to zero
MV->myRawEntry(i,j) -- raw pointer to (i,j) entry
(may be called only if the (i,j) posn is writable)
MV->myAssignZero() -- sets all entries to zero.
Throws ERR::ConstMatEntry if not all entries can be made zero
NOTE: You cannot set a matrix entry the obvious way,
i.e. M(i,j) = value; You must use SetEntry(M,i,j,value).
Calling SetEntry on a position which is not writable
will throw CoCoA::ERR::BadMatrixSetEntry
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 and matrix are
just reference counting smart-pointers to objects of type derived from
the abstract base classes ConstMatrixViewBase, MatrixViewBase
and MatrixBase respectively;
this is analogous to the way rings 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)
myAssignZero() -- set all entries to zero
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:
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 three 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 true matrix, and someone creates the
view MtM defined to be ConcatHor(M, transpose(M)) then there is non-trivial
aliasing between the entries of MtM. What should happen if you try to
multiply the second row of MtM by 2? What should happen if you try to
add a new column to MtM? In general, resizing MtM 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 three 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?
2012
2011
MatrixSpecial files