matrix

© 2005,2008 John Abbott
GNU Free Documentation License, Version 1.2



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User documentation for the classes matrix, MatrixView and ConstMatrixView

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 MatrixViews).

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.

Constructors and Pseudo-constructors:

The following create matrix views (see MatrixViews):

A B
C D
A
C
A B
A 0
0 B
0 A
B 0

The possible operations on a ConstMatrixView are:

    BaseRing(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)
    IsZeroRow(M,i) -- true iff row i of M is zero
    IsZeroCol(M,j) -- true iff column j of M is zero
    M(i,j)         -- the (i,j) entry of M  (NB both indices start from 0)
    out << M       -- print the value of the matrix on ostream out

The following come from MatrixArith, see there for more details.

    det(M)         -- determinant of M (M must be square)
    rank(M)        -- rank of M (the base ring must be an integral domain)
    inverse(M)     -- inverse of M as a dense matrix
    adjoint(M)     -- adjoint of M as a dense matrix

The extra operations on a MatrixView are:

    M->myIsWritable(i,j)-- true iff posn (i,j) can be written to
    SetEntry(M,i,j,r)   -- assign the RingElem r to entry (i,j) of matrix M
    SetEntry(M,i,j,N)   -- assign the integer n to entry (i,j) of matrix M
    SetEntry(M,i,j,n)   -- assign the integer n to entry (i,j) of matrix M
    AssignZero(M)       -- set all entries of M to zero
  ???  mul(MV, CMV1, CMV2) -- MV = CMV1*CMV2;
  ??? M->myRefEntry(i,j)   -- gives a reference to the writable entry (i,j)

NOTE: You cannot set a matrix entry the obvious way, i.e. M(i,j) = NewValue; You must use SetEntry. Calling SetEntry on a position which is not writable will throw CoCoA::ERR::BadMatrixSetEntry

The extra operations on a true matrix are:

    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 MatrixViews because of variou 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 zero. 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).

Documentation for library contributors

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:

    myBaseRing()       -- 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)
    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,r)  -- set entry (i,j) to r where r is RingElem
    mySetEntry(i,j,N)  -- set entry (i,j) to r where N is big integer (of type [``ZZ`` ZZ.html])
    mySetEntry(i,j,n)  -- set entry (i,j) to r where n is long
    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: - myMulByRow, myMulByCol, myIsZeroRow, myIsZeroCol, myOutput all have default "dense" definitions - myDet and myRank have default definitions which use gaussian elimination

Maintainer documentation for the matrix classes

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!

Bugs, Shortcomings and other ideas

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?

I am uncertain whether myRefEntry should be documented (i.e. made available for public use). It could be handy if you want to call a function which has a RefRingElem arg -- without myRefEntry, you'd have to use a (named) temporary.

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?