class TMatrixTBase<double>: public TObject

Linear Algebra Package


The present package implements all the basic algorithms dealing
with vectors, matrices, matrix columns, rows, diagonals, etc.
In addition eigen-Vector analysis and several matrix decomposition
have been added (LU,QRH,Cholesky,Bunch-Kaufman and SVD) .
The decompositions are used in matrix inversion, equation solving.

For a dense matrix, elements are arranged in memory in a ROW-wise
fashion . For (n x m) matrices where n*m <=kSizeMax (=25 currently)
storage space is available on the stack, thus avoiding expensive
allocation/deallocation of heap space . However, this introduces of
course kSizeMax overhead for each matrix object . If this is an
issue recompile with a new appropriate value (>=0) for kSizeMax

Sparse matrices are also stored in row-wise fashion but additional
row/column information is stored, see TMatrixTSparse source for
additional details .

Another way to assign and store matrix data is through Use
see for instance stressLinear.cxx file .

Unless otherwise specified, matrix and vector indices always start
with 0, spanning up to the specified limit-1. However, there are
constructors to which one can specify aribtrary lower and upper
bounds, e.g. TMatrixD m(1,10,1,5) defines a matrix that ranges
from 1..10, 1..5 (a(1,1)..a(10,5)).

The present package provides all facilities to completely AVOID
returning matrices. Use "TMatrixD A(TMatrixD::kTransposed,B);"
 and other fancy constructors as much as possible. If one really needs
to return a matrix, return a TMatrixTLazy object instead. The
conversion is completely transparent to the end user, e.g.
"TMatrixT m = THaarMatrixT(5);" and _is_ efficient.

Since TMatrixT et al. are fully integrated in ROOT, they of course
can be stored in a ROOT database.

For usage examples see $ROOTSYS/test/stressLinear.cxx

Acknowledgements

1. Oleg E. Kiselyov
First implementations were based on the his code . We have diverged
quite a bit since then but the ideas/code for lazy matrix and
"nested function" are 100% his .
You can see him and his code in action at http://okmij.org/ftp
2. Chris R. Birchenhall,
We adapted his idea of the implementation for the decomposition
classes instead of our messy installation of matrix inversion
His installation of matrix condition number, using an iterative
scheme using the Hage algorithm is worth looking at !
Chris has a nice writeup (matdoc.ps) on his matrix classes at
ftp://ftp.mcc.ac.uk/pub/matclass/
3. Mark Fischler and Steven Haywood of CLHEP
They did the slave labor of spelling out all sub-determinants
for Cramer inversion  of (4x4),(5x5) and (6x6) matrices
The stack storage for small matrices was also taken from them
4. Roldan Pozo of TNT (http://math.nist.gov/tnt/)
He converted the EISPACK routines for the eigen-vector analysis to
C++ . We started with his implementation
5. Siegmund Brandt (http://siux00.physik.uni-siegen.de/~brandt/datan
We adapted his (very-well) documented SVD routines

How to efficiently use this package


1. Never return complex objects (matrices or vectors)
Danger: For example, when the following snippet:
TMatrixD foo(int n)
{
TMatrixD foom(n,n); fill_in(foom); return foom;
}
TMatrixD m = foo(5);
runs, it constructs matrix foo:foom, copies it onto stack as a
return value and destroys foo:foom. Return value (a matrix)
from foo() is then copied over to m (via a copy constructor),
and the return value is destroyed. So, the matrix constructor is
called 3 times and the destructor 2 times. For big matrices,
the cost of multiple constructing/copying/destroying of objects
may be very large. *Some* optimized compilers can cut down on 1
copying/destroying, but still it leaves at least two calls to
the constructor. Note, TMatrixDLazy (see below) can construct
TMatrixD m "inplace", with only a _single_ call to the
constructor.

2. Use "two-address instructions"
"void TMatrixD::operator += (const TMatrixD &B);"
as much as possible.
That is, to add two matrices, it's much more efficient to write
A += B;
than
TMatrixD C = A + B;
(if both operand should be preserved,
TMatrixD C = A; C += B;
is still better).

3. Use glorified constructors when returning of an object seems
inevitable:
"TMatrixD A(TMatrixD::kTransposed,B);"
"TMatrixD C(A,TMatrixD::kTransposeMult,B);"

like in the following snippet (from $ROOTSYS/test/vmatrix.cxx)
that verifies that for an orthogonal matrix T, T'T = TT' = E.

TMatrixD haar = THaarMatrixD(5);
TMatrixD unit(TMatrixD::kUnit,haar);
TMatrixD haar_t(TMatrixD::kTransposed,haar);
TMatrixD hth(haar,TMatrixD::kTransposeMult,haar);
TMatrixD hht(haar,TMatrixD::kMult,haar_t);
TMatrixD hht1 = haar; hht1 *= haar_t;
VerifyMatrixIdentity(unit,hth);
VerifyMatrixIdentity(unit,hht);
VerifyMatrixIdentity(unit,hht1);

4. Accessing row/col/diagonal of a matrix without much fuss
(and without moving a lot of stuff around):

TMatrixD m(n,n); TVectorD v(n); TMatrixDDiag(m) += 4;
v = TMatrixDRow(m,0);
TMatrixDColumn m1(m,1); m1(2) = 3; // the same as m(2,1)=3;
Note, constructing of, say, TMatrixDDiag does *not* involve any
copying of any elements of the source matrix.

5. It's possible (and encouraged) to use "nested" functions
For example, creating of a Hilbert matrix can be done as follows:

void foo(const TMatrixD &m)
{
TMatrixD m1(TMatrixD::kZero,m);
struct MakeHilbert : public TElementPosActionD {
void Operation(Double_t &element)
{ element = 1./(fI+fJ-1); }
};
m1.Apply(MakeHilbert());
}

of course, using a special method THilbertMatrixD() is
still more optimal, but not by a whole lot. And that's right,
class MakeHilbert is declared *within* a function and local to
that function. It means one can define another MakeHilbert class
(within another function or outside of any function, that is, in
the global scope), and it still will be OK. Note, this currently
is not yet supported by the interpreter CINT.

Another example is applying of a simple function to each matrix
element:

void foo(TMatrixD &m,TMatrixD &m1)
{
typedef  double (*dfunc_t)(double);
class ApplyFunction : public TElementActionD {
dfunc_t fFunc;
void Operation(Double_t &element)
{ element=fFunc(element); }
public:
ApplyFunction(dfunc_t func):fFunc(func) {}
};
ApplyFunction x(TMath::Sin);
m.Apply(x);
}

Validation code $ROOTSYS/test/vmatrix.cxx and vvector.cxx contain
a few more examples of that kind.

6. Lazy matrices: instead of returning an object return a "recipe"
how to make it. The full matrix would be rolled out only when
and where it's needed:
TMatrixD haar = THaarMatrixD(5);
THaarMatrixD() is a *class*, not a simple function. However
similar this looks to a returning of an object (see note #1
above), it's dramatically different. THaarMatrixD() constructs a
TMatrixDLazy, an object of just a few bytes long. A special
"TMatrixD(const TMatrixDLazy &recipe)" constructor follows the
recipe and makes the matrix haar() right in place. No matrix
element is moved whatsoever!

This class is also known as (typedefs to this class)

TMatrixTBase<Double_t>, TMatrixDBase