• symmetric, positive-definite (SPD) matrices of specific sizes
• associated constructs such as column matrices ("vectors") and Nx2 matrices ("biVectors") which are useful in Kalman filter calculations and related tracking computations.

• The idea is to sacrifice flexibility, where necessary, to achieve optimum performance.
Thus for example a given matrix or vector will be of fixed size, thus greatly simplifying storage strategy.
• The operations provided are those which make sense in the realm of SPD matrices, and those which are useful in Kalman filter computation.
• Although the case for general N is provided, the most important classes expected to be used are specializations for 4x4, 5x5, and 6x6 SPD matrices.
• The algorithms provided are optimized, tailored to matrix size and taking full advantage of the SPD properties. Thus for example, inversion can proceed significantly faster than would be possible for general symmetric matrices.
• The design philosophy is to keep the package down at the lowest suitable level of C++ feature sophistication. Thus it should be suitable for use outside the ZOOM environment.
• Problems are reported via returned status values.
• No use of exception mechanism.
• No spontaneous outputs to cerr or cout.
• Classes are not templated.

Classes

* std::vector < double > is used to represent a 1-column matrix of arbitrary length.

Constructors

It is the user's responsibility to know that these form a positive definite matrix.

A Vector may be constructed from from its elements, or from an array conaining them.

A BiVector may be constructed from two vectors, or from its elements in the order of first one vector than the other.

Operations

  M = M1 + M2;
  bool OK = M.invert();
  double d = M.determinant();
  Vector5 eigenvals; Vector[5] eigenvecs[5]; // here we assume M is 5x5
  M.diagonalize(eigenvals, eigenvecs);
  M.trace();
  cout << M; // nicely formatted output
  M = M1 - M2; // See note
  M*v;
  M*w;
  v*M;
  w*M;
  v+v; v-v;
  w+w; w-w;

Element access

All elements of the upper part of the matrix, in row order, may be dumped into an array of doubles.

  CovMatrix5 M;
  double elements[15];
  M.dump(elements);

Speed

The bottom line is that for 5x5 and 6x6 matrices, inversion proceeds almost twice as fast using the CovMatrices package as it does using CLHEP Matrices.

Performing matrix addition timings table  (more condensed form of the same table: only the minimum values are reported)
Performing matrix subtraction timings table  (more condensed form of the same table: only the minimum values are reported)
Performing matrix to matrix multiplication timings table  (more condensed form of the same table: only the minimum values are reported)
  (Note: matrix*matrix can't be performed in CovMatrices)
Performing matrix to vector multiplication timings table  (more condensed form of the same table: only the minimum values are reported)
Performing matrix to bivector multiplication timings table (more condensed form of the same table: only the minimum values are reported)
  (Note: since matrix*bivector can't be performed in LinearAlgebra and CLHEP, matrix*matrix with a 2-column matrix as the 2nd matrix was done)
Performing vector to vector multiplication timings table  (more condensed form of the same table: only the minimum values are reported)
  (Note: in order to perform vector*vector in CLHEP, we have to transpose one of the two vectors. However, when we transpose a HepVector we get back a HepMatrix. Thus, we end up performing vector*matrix multiplication and not vector*vector.)
Performing matrix inversion timings table  (more condensed form of the same table: only the minimum values are reported)
Performing random matrix inversion timings table  (more condensed form of the same table: only the minimum values are reported)
Calculating matrix determinant timings table  (more condensed form of the same table: only the minimum values are reported)
Calculating matrix trace timings table  (more condensed form of the same table: only the minimum values are reported)
 

Correctness Validation

The following pages compare the results of various matrix operations in CovMatrices with the same operations in LinearAlgebra and CLHEP.
The link on each matrix operation leads to a comparison summary for the matrix operation, whereas the link on each package leads to the actual test.

Matrix addition:  LinearAlgebra, CLHEP, CLHEP (Symmetric matrices), CovMatrices
Matrix subtraction:  LinearAlgebra, CLHEP, CLHEP (Symmetric matrices), CovMatrices
Matrix * Matrix multiplication: LinearAlgebra, CLHEP, CLHEP (Symmetric matrices), CovMatrices (dot product sums) (*)
Matrix * Vector multiplication: LinearAlgebra, CLHEP, CLHEP (Symmetric matrices), CovMatrices
Vector * Matrix multiplication: LinearAlgebra, CLHEP (**), CLHEP (Symmetric matrices) (**), CovMatrices
(Column)Vector * (Row)Vector multiplication: LinearAlgebra, CLHEP (***), CovMatrices (dot product) (*)
(Row)Vector * (Column)Vector multiplication: LinearAlgebra, CLHEP (***), CovMatrices (dot product) (*)
Matrix trace:  LinearAlgebra, CLHEP, CLHEP (Symmetric matrices), CovMatrices
Matrix determinant:  LinearAlgebra, CLHEP, CLHEP (Symmetric matrices), CovMatrices
Matrix inversion:  LinearAlgebra, CLHEP, CLHEP (Symmetric matrices), CovMatrices

(*) Dot products do not test the ability of the package itself to perform the clculations, they don't use/test a method native to the package; they are provided for symmetry's sake.
(**) In order to perform vector*matrix in CLHEP we have to transpose the vector. However, when we transpose a HepVector we get back a HepMatrix. Thus, we end up performing matrix*matrix multiplication and not vector*matrix.
(***) In order to perform vector*vector in CLHEP we have to transpose one of the two vectors. However, when we transpose a HepVector we get back a HepMatrix. Thus, we end up performing vector*matrix multiplication and not vector*vector.