class TDecompBase: public TObject

Decomposition Base class

This class forms the base for all the decompositions methods in the
linear algebra package .
It or its derived classes have installed the methods to solve
equations,invert matrices and calculate determinants while monitoring
the accuracy.

Each derived class has always the following methods available:

Condition() :
In an iterative scheme the condition number for matrix inversion is
calculated . This number is of interest for estimating the accuracy
of x in the equation Ax=b
For example:
A is a (10x10) Hilbert matrix which looks deceivingly innocent
and simple, A(i,j) = 1/(i+j+1)
b(i) = Sum_j A(i,j), so a sum of a row in A

the solution is x(i) = 1. i=0,.,9

However,
TMatrixD m....; TVectorD b.....
TDecompLU lu(m); lu.SetTol(1.0e-12); lu.Solve(b); b.Print()
gives,

{1.000,1.000,1.000,1.000,0.998,1.000,0.993,1.001,0.996,1.000}

Looking at the condition number, this is in line with expected the
accuracy . The condition number is 3.957e+12 . As a simple rule of
thumb, a condition number of 1.0e+n means that you lose up to n
digits of accuracy in a solution . Since doubles are stored with 15
digits, we can expect the accuracy to be as small as 3 digits .

Det(Double_t &d1,Double_t &d2)
The determinant is d1*TMath::Power(2.,d2)
Expressing the determinant this way makes under/over-flow very
unlikely .

Decompose()
Here the actually decomposition is performed . One can change the
matrix A after the decomposition constructor has been called
without effecting the decomposition result

Solve(TVectorD &b)
Solve A x = b . x is supplied through the argument and replaced with
the solution .

TransSolve(TVectorD &b)
Solve A^T x = b . x is supplied through the argument and replaced
with the solution .

MultiSolve(TMatrixD    &B)
Solve A X = B . where X and are now matrices . X is supplied through
the argument and replaced with the solution .

Invert(TMatrixD &inv)
This is of course just a call to MultiSolve with as input argument
the unit matrix . Note that for a matrix a(m,n) with m > n  a
pseudo-inverse is calculated .

Tolerances and Scaling

The tolerance parameter (which is a member of this base class) plays
a crucial role in all operations of the decomposition classes . It
gives the user a powerful tool to monitor and steer the operations
Its default value is sqrt(epsilon) where 1+epsilon = 1

If you do not want to be bothered by the following considerations,
like in most other linear algebra packages, just set the tolerance
with SetTol to an arbitrary small number .

The tolerance number is used by each decomposition method to decide
whether the matrix is near singular, except of course SVD which can
handle singular matrices .
For each decomposition this will be checked in a different way; in LU
the matrix is considered singular when, at some point in the
 decomposition, a diagonal element < fTol . Therefore, we had to set in
the example above of the (10x10) Hilbert, which is near singular, the
tolerance on 10e-12 . (The fact that we have to set the tolerance <
sqrt(epsilon) is a clear indication that we are losing precision .)

If the matrix is flagged as being singular, operations with the
decomposition will fail and will return matrices/vectors that are
invalid .

The observant reader will notice that by scaling the complete matrix
 by some small number the decomposition will detect a singular matrix .
 In this case the user will have to reduce the tolerance number by this
 factor . (For CPU time saving we decided not to make this an automatic
procedure) .

Code for this could look as follows:
const Double_t max_abs = Abs(a).Max();
const Double_t scale = TMath::Min(max_abs,1.);
a.SetTol(a.GetTol()*scale);

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