Provide support for conjugate transpose (adjoint) matrices

[Klaus Iglberger]

Description

Blaze provides support for complex numbers and transpose matrices, but is missing the feature to compute the conjugate transpose of a complex matrix (also called adjoint matrix, Hermetian conjugate matrix, or transjugate). Conceptually, the ctrans operation should work like this:

using blaze::StaticMatrix;

typedef std::complex<double>  cplx;

// Creating the matrix
//    ( (1,0)  (-2,-1) )
//    ( (1,1)  ( 0, 1) )
StaticMatrix<cplx,2UL,2UL> A( cplx(1.0,0.0), cplx(-2.0,-1.0),
                              cplx(1.0,1.0), cplx( 0.0, 1.0) );

// Computing the conjugate transpose matrix
//    ( ( 1,0)  (1,-1) )
//    ( (-2,1)  (0,-1) )
StaticMatrix<cplx,2UL,2UL> B;
B = ctrans( A );

Tasks

• provide the ctrans operation for scalars, including complex numbers
• provide the ctrans operation for dense and sparse vectors
• provide the ctrans operation for dense and sparse matrices
• provide a full documentation of the operation
• ensure compatibility to all existing vector and matrix classes
• ensure compatibility to all existing vector and matrix expressions
• guarantee maximum performance for the operation
• extend existing test cases to also cover the ctrans operation

[Klaus Iglberger]

Instead of calling the function to compute the conjugate transpose of a matrix adjoint, the function name ctrans should be used. There are two reasons for this:

• From a terminology point of view, there is only an adjoint matrix, but no adjoint vector, although it is also possible to compute the conjugate transpose of a vector. Thus adjoint would (from a terminology point of view) limit the operation to matrices.
• In mathematics the term adjoint matrix is also used for adjugate matrices. Therefore an adjoint function could cause misunderstandings.

The choice of ctrans is inspired by MATLAB. MATLAB provides a function called transpose to compute the non-conjugate transpose of a matrix and a ctranspose function to compute the conjugate transpose of a matrix. Since Blaze provides the function trans for computing the non-conjugate transpose of a matrix, the function to compute the conjugate transpose of a matrix should be called ctrans.

[Klaus Iglberger]

The feature has been implemented, tested, optimized (including vectorization and parallelization) and documented as required. The following examples give an impression of the final feature:

using blaze::StaticVector;

typedef std::complex<double>  cplx;

// Creating the vector
//    ( (-2,-1) ( 1, 1) )
StaticVector<cplx,2UL,rowVector> a( cplx(-2.0,-1.0), cplx(1.0,1.0) );

// Computing the conjugate transpose of the vector
//    ( (-2, 1) )
//    ( ( 1,-1) )
StaticVector<cplx,2UL,columnVector> b;
b = ctrans( a );

using blaze::StaticMatrix;

typedef std::complex<double>  cplx;

// Creating the matrix
//    ( (1,0)  (-2,-1) )
//    ( (1,1)  ( 0, 1) )
StaticMatrix<cplx,2UL,2UL> A( cplx(1.0,0.0), cplx(-2.0,-1.0),
                              cplx(1.0,1.0), cplx( 0.0, 1.0) );

// Computing the conjugate transpose of the matrix
//    ( ( 1,0)  (1,-1) )
//    ( (-2,1)  (0,-1) )
StaticMatrix<cplx,2UL,2UL> B;
B = ctrans( A );

The feature is immediately available via cloning the Blaze repository. It will be officially released in Blaze 2.5.