ROOT Mathematical Libraries
5.34/00

The template ROOT::Math::SMatrix class has 4 template parameters which define, at compile time, its properties.
These are:
The following constructors are available to create a matrix:
Here are some examples on how to create a matrix. We use typedef's in the following examples to avoid the full C++ names for the matrix classes. Notice that for a general matrix the representation has the default value, ROOT::Math::MatRepStd, and it is not needed to be specified. Furtheremore, for a general square matrix, the number of column may be as well omitted.
// typedef definitions used in the following declarations typedef ROOT::Math::SMatrix<double,3> SMatrix33; typedef ROOT::Math::SMatrix<double,2> SMatrix22; typedef ROOT::Math::SMatrix<double,3,3,ROOT::Math::MatRepSym<double,3> > SMatrixSym3; typedef ROOT::Math::SVector>double,2> SVector2; typedef ROOT::Math::SVector>double,3> SVector3; typedef ROOT::Math::SVector>double,6> SVector6;
SMatrix33 m0; // create a zero 3x3 matrix // create an 3x3 identity matrix SMatrix33 i = ROOT::Math::SMatrixIdentity(); double a[9] = {1,2,3,4,5,6,7,8,9}; // input matrix data SMatrix33 m(a,9); // create a matrix using the a[] data // this will produce the 3x3 matrix // ( 1 2 3 // 4 5 6 // 7 8 9 )
Example to create a symmetric matrix from an std::vector:
std::vector<double> v(6); for (int i = 0; i<6; ++i) v[i] = double(i+1); SMatrixSym3 s(v.begin(),v.end()) // this will produce the symmetric matrix // ( 1 2 4 // 2 3 5 // 4 5 6 )
// create a a general matrix from a symmetric matrix. The opposite will not compile SMatrix33 m2 = s;
Example to create a symmetric matrix from a ROOT::Math::SVector contining the lower/upper data block:
ROOT::Math::SVectorr<double, 6> v(1,2,3,4,5,6); SMatrixSym3 s1(v); // lower block (default) // this will produce the symmetric matrix // ( 1 2 4 // 2 3 5 // 4 5 6 )
SMatrixSym3 s2(v,false); // upper block // this will produce the symmetric matrix // ( 1 2 3 // 2 4 5 // 3 5 6 )
The matrix elements can be set using the operator()(irow,icol), where irow and icol are the row and column indexes or by using the iterator interface. Notice that the indexes start from zero and not from one as in FORTRAN. All the matrix elements can be set also by using the ROOT::Math::SetElements function passing a generic iterator.
The elements can be accessed by these same methods and also by using the ROOT::Math::SMatrix::apply function. The apply(i) function has exactly the same behavior for general and symmetric matrices, in contrast to the iterator access methods which behave differently (it follows the data order).
SMatrix33 m; m(0,0) = 1; // set the element in first row and first column *(m.begin()+1) = 2; // set the second element (0,1) double d[9]={1,2,3,4,5,6,7,8,9}; m.SetElements(d,d+9); // set the d[] values in m
double x = m(2,1); // return the element in third row and first column x = m.apply(7); // return the 8th element (row=2,col=1) x = *(m.begin()+7); // return the 8th element (row=2,col=1) // symmetric matrices (note the difference in behavior between apply and the iterators) x = *(m.begin()+4) // return the element (row=2,col=1). x = m.apply(7); // returns again the (row=2,col=1) element
There are methods to place and/or retrieve ROOT::Math::SVector objects as rows or columns in (from) a matrix. In addition one can put (get) a submatrix as another ROOT::Math::SMatrix object in a matrix. If the size of the the subvector or submatrix are larger than the matrix size a static assert ( a compilation error) is produced. The nonconst methods are:
SMatrix33 m; SVector2 v2(1,2); // place a vector of size 2 in the first row starting from element (0,1) : m(0,1) = v2[0] m.Place_in_row(v2,0,1); // place the vector in the second column from (0,1) : m(0,1) = v2[0] m.Place in_col(v2,0,1); SMatrix22 m2; // place the submatrix m2 in m starting from the element (1,1) : m(1,1) = m2(0,0) m.Place_at(m2,1,1); SVector3 v3(1,2,3); // set v3 as the diagonal elements of m : m(i,i) = v3[i] for i=0,1,2 m.SetDiagonal(v3)
The const methods retrieving contents (getting slices of a matrix) are:
a = {1,2,3,4,5,6,7,8,9}; SMatrix33 m(a,a+9); SVector3 irow = m.Row(0); // return as vector the first matrix row SVector3 jcol = m.Col(1); // return as vector the second matrix column // return a slice of the first row from element (0,1) : r2[0] = m(0,1); r2[1] = m(0,2) SVector2 r2 = m.SubRow<SVector2> (0,1); // return a slice of the second column from element (0,1) : c2[0] = m(0,1); c2[1] = m(1,1); SVector2 c2 = m.SubCol<SVector2> (1,0); // return a submatrix 2x2 with the upper left corner at the values (1,1) SMatrix22 subM = m.Sub<SMatrix22> (1,1); // return the diagonal element in a SVector SVector3 diag = m.Diagonal(); // return the upper(lower) block of the matrix m SVector6 vub = m.UpperBlock(); // vub = [ 1, 2, 3, 5, 6, 9 ] SVector6 vlb = m.LowerBlock(); // vlb = [ 1, 4, 5, 7, 8, 9 ]
Only limited linear algebra functionality is available for SMatrix. It is possible for squared matrices NxN, to find the inverse or to calculate the determinant. Different inversion algorithms are used if the matrix is smaller than 6x6 or if it is symmetric. In the case of a small matrix, a faster direct inversion is used. For a large (N > 6) symmetric matrix the BunchKaufman diagonal pivoting method is used while for a large (N > 6) general matrix an LU factorization is performed using the same algorithm as in the CERNLIB routine dinv.
// Invert a NxN matrix. The inverted matrix replace the existing one and returns if the result is successful bool ret = m.Invert() // return the inverse matrix of m. If the inversion fails ifail is different than zero int ifail = 0; mInv = m.Inverse(ifail);
The determinant of a square matrix can be obtained as follows:
double det; // calculate the determinant modyfing the matrix content. Returns if the calculation was successful bool ret = m.Det(det); // calculate the determinant using a temporary matrix but preserving the matrix content bool ret = n.Det2(det);
For additional Matrix functionality see the Matrix and Vector Operators and Functions page