Basic linear algebra operations

The same choice has been made as in MTL to provide basic operations as functions not as operators. The advantages are that it is clearer to see where are the linear algebra operations in the program and the programming of optimized basic linear operations is greatly simplified.

scale and scaled

gmm::scale is used to multiply a vector or a matrix with a scalar factor:

gmm::scale(V, 10.0);  // V * 10.0 ---> V

If one not needs to multiply the vector but wants to use the multiplied vector in an expression gmm::scaled gives a reference to a multiplied vector. This is only a reference, no operation is made until this reference is used somewhere. For instance:

std::cout << gmm::scaled(V, 10.0) << std::endl;

print to the standard output the vector V multiplied by 10.0 without changing V.

transposition

gmm::transposed(M) gives a possibily modifiable reference on the transposed matrix of M.

imaginary and real part

For a complex matrix M or a complex vector V, gmm::real_part(M), gmm::real_part(V), gmm::imag_part(M) or gmm::imag_part(V) give a possibily modifiable reference on the real or imaginary part of the matrix or vector (for instance gmm::clear(gmm::imag_part(M)) will set to zero the imaginary part of a matrix M). These functions cannot be applied to real matrices or vectors.

conjugate

For a matrix M or a vector V, gmm::conjugated(M) and gmm::conjugated(V) give a constant reference on the conjugated vector or matrix. Of course, for a real vectors this has no effect (and no cost at all). Note : gmm::conjugated(M) transposes the matrix M so that this is the hermitian conjugate of M. If you need only the conjugate of each component you have to use both transposition and conjugate with gmm::conjugated(gmm::transposed(M)) or equivalently gmm::transposed(gmm::conjugated(M)).

add

addition of vectors or matrices. It is alway possible to mix different type of vector or matrices in the operations. The following operations are valid:

std::vector<double> V1(10);
gmm::wsvector<double> V2(10);
gmm::clear(V1);
...
gmm::add(V1, V2); // V1 + V2 --> V2
cout << V2;

gmm::add(V1, gmm::scaled(V2, -2.0), V2); // V1 - 2.0 * V2 --> V2
cout << V2;

gmm::row_matrix< std::vector<double> > M1(10, 10);
gmm::col_matrix< gmm::wsvector<double> > M2(1000, 1000);

// M1 + (sub matrix of M2) ---> (sub matrix of M2)
gmm::add(M1, gmm::sub_matrix(M2, gmm::sub_interval(4,10)));

IMPORTANT : all the vectors have to have the same size, no resize will be automatically done. If a vector has not the good size, an error will be thrown.

mult

Matrix-vector or matrix-matrix multiplication. Again, all the matrices and vectors have to have the good size. The following operations are valid:

std::vector<double> V1(10);
gmm::wsvector<double> V2(10);
...
gmm::row_matrix< std::vector<double> > M1(10, 10);
...

gmm::mult(M1, V2, V1);  // M1 * V2 --> V1

gmm::mult(M1, V2, V2, V1);  // M1 * V2 + V2 --> V1

gmm::mult_add(M1, V2, V1);  // M1 * V2 + V1 --> V1

gmm::mult(M1, gmm::scaled(V2, -1.0), V2, V1);  // M1 * (-V2) + V2 --> V1

gmm::col_matrix< gmm::wsvector<double> > M2(10, 10);
gmm::col_matrix< gmm::vsvector<double> > M3(10, 10);
...

gmm::mult(M1, M2, M3); // M1 * M2 ---> M3

gmm::mult(gmm::sub_matrix(M1, sub_interval(0, 3)),
          gmm::sub_matrix(M2, sub_interval(4, 3)),
          gmm::sub_matrix(M3, sub_interval(2, 3)));

norms

gmm::vect_norm1(V)  // sum of the modulus of the components of vector V.
gmm::vect_norm2(V)  // Euclidean norm of vector V.
gmm::vect_dist2(V1, V2)  // Euclidean distance between V1 and V2.
gmm::vect_norminf(V)    // infinity norm of vector V.
gmm::mat_euclidean_norm(M) // Euclidean norm of matrix ``M``
                           // (called also Frobenius norm).
gmm::mat_maxnorm(M) // Max norm (defined as max(|m_ij|; i,j ))
gmm::mat_norm1(M)   // max(sum(|m_ij|, i), j)
gmm::mat_norminf(M) // max(sum(|m_ij|, j), i)

trace

gmm::mat_trace(M) gives the trace of matrix M.

scalar product

for vectors only, gmm::vect_sp(V1, V2) gives the scalar product between V1 and V2. For complex vectors, this do not conjugate V1, you can use gmm::vect_sp(V1, gmm::conjugated(V2)) or gmm::vect_hp(V1, V2) which is equivalent.