Blas functions

Those functions are available both for dense and sparse vectors. In the case of dense vectors, Blas subroutines are called if SELDON_WITH_CBLAS is defined.

Mlt	multiplies the elements of the vector/matrix by a scalar, or performs a matrix-vector product
MltAdd	performs a matrix-vector product or a matrix-matrix product
Add	adds two vectors or two matrices
Copy	copies a vector into another one, or a matrix into another matrix
Swap	exchanges two vectors
GenRot	computes givens rotation
GenModifRot	computes givens roation
ApplyRot	applies givens rotation
ApplyModifRot	applies givens rotation
DotProd	scalar product between two vectors
DotProdConj	scalar product between two vectors, first vector being conjugated
Conjugate	conjugates all elements of a vector
GetMaxAbsIndex	returns index where highest absolute value is reached
Norm1	returns 1-norm of a vector
Norm2	returns 2-norm of a vector
Rank1Update	Adds a contribution X.Y' to a matrix
Rank2Update	Adds a contribution X.Y' + Y.X' to a symmetric matrix
Solve	solves a triangular system

Mlt

Syntax :

  void Mlt(T0, Vector&);
  void Mlt(T0, Matrix&);
  void Mlt(Matrix&, const Vector&, Vector&);

This function multiplies all elements of vector/matrix by a constant. It can also perform a matrix-vector product. This function is implemented both for dense and sparse vectors/matrices.

Example :

Vector<double> X(3);
X.Fill();
// we multiply all elements by 2
Mlt(2.0, X);

// similar method for matrix
Matrix<double> A(3,3);
A.Fill();
// we multiply all elements by -3.4
Mlt(-3.4, A);

// and sparse matrix as well
Matrix<double, General, ArrayRowSparse> Asp(3,3);
Asp(0, 1) = 1.4;
Asp(1, 1) = -0.3; 
Asp(2, 0) = 2.1;
Mlt(2.1, Asp);

// with Mlt, you can compute Y = Asp*X or Y = A*X
Vector<double> Y(3);
Mlt(A, X, Y);
Mlt(Asp, X, Y);

Location :

Functions_Vector.cxx
Functions_MatVect.cxx
Functions_MatrixArray.cxx
Blas_1.cxx
Blas_2.cxx

MltAdd

Syntax :

  void MltAdd(T0, const Matrix&, const Vector&, T0, Vector&);
  void MltAdd(T0, SeldonTrans, const Matrix&, const Vector&, T0, Vector&);
  void MltAdd(T0, const Matrix&, const Matrix&, T0, Matrix&);
  void MltAdd(T0, SeldonTrans, const Matrix&, SeldonTrans, const Matrix&,
              T0, Matrix&);

This function performs a matrix-vector product or a matrix-matrix product. You can specify a product with the transpose of the matrix, or the conjugate of the transpose. The size of matrices need to be compatible in order to complete the matrix-matrix product. The matrix-matrix product is not available for sparse matrices.

Example :

// matrix-vector product
Matrix<double> A(3,3);
Vector<double> X(3), B(3);
A.Fill();
X.Fill();
B.Fill(1.5);
double beta = 2, alpha = 3;
// computation of B = beta*B + alpha*A*X
MltAdd(alpha, A, X, beta, B);

// same function for sparse matrices
Matrix<double, General, RowSparse> Asp(3, 3, nnz, values, ptr, ind);
// computation of B = beta*B + alpha*Asp*X
MltAdd(alpha, Asp, X, beta, B);

// you can compute B = beta*B + alpha*transpose(Asp)*X
MltAdd(alpha, SeldonTrans, Asp, X, beta, B);

// similar method for matrix-matrix product
Matrix<double> M(3,4), N(4,3);
// computation of A = beta*A + alpha*M*N
MltAdd(alpha, M, N, beta, A);

// and you can use SeldonTrans, SeldonNoTrans, SeldonConjTrans

// computation of A = beta*A + alpha*transpose(M)*N
MltAdd(alpha, SeldonTrans, M, SeldonNoTrans, N, beta, A);

// SeldonConjTrans is meaningfull for complex matrices
Matrix<complex<double> > Ac(3, 3), Bc(3, 3), Cc(3, 3);

// computation of Ac = beta * Ac
//                     + alpha * conjugate(transpose(Bc)) * transpose(Cc)
MltAdd(alpha, SeldonConjTrans, Bc, SeldonTrans, Cc, beta, Ac);

Location :

Functions_MatVect.cxx
Functions_Matrix.cxx
Functions_MatrixArray.cxx
Blas_2.cxx
Blas_3.cxx

Add

Syntax :

  void Add(T0, const Vector&, Vector&);
  void Add(T0, const Matrix&, Matrix&);

This function adds two vectors or two matrices. This function is available both for sparse or dense vectors/matrices. The size of the vectors or the matrices to add should be the same.

Example :

Vector<double> X(3), Y(3);
X.Fill(); Y.Fill(1);
double alpha = 2;
// computation of Y = Y + alpha*X
Add(alpha, X, Y);

// similar method for matrix
Matrix<double> A(3,3), B(3,3);
A.Fill();
B.SetIdentity();
Add(alpha, A, B);

// and sparse matrix as well
Matrix<double, General, ArrayRowSparse> Asp(3,3), Bsp(3,3);
Asp(0, 1) = 1.4;
Asp(1, 1) = -0.3; 
Asp(2, 0) = 2.1;
Bsp.SetIdentity();
Add(alpha, Asp, Bsp);

Location :

Functions_Vector.cxx
Functions_Matrix.cxx
Functions_MatrixArray.cxx
Blas_1.cxx

Copy

Syntax :

  void Copy(const Vector&, Vector&);
  void Copy(const Matrix& Matrix&);

A vector is copied into another one. If BLAS is enabled, the two dense vectors have to be of the same size. The operator = is more flexible. For sparse matrices, the function Copy is overloaded to enable the conversion between the different formats.

Example :

// vectors need to have the same size
Vector<float> V(3), W(3);
V.Fill();
Copy(V, W);
// it is equivalent to write W = V;
W = V;

// for sparse matrices, you can use Copy as a convert tool
Matrix<double, General, ArrayRowSparse> A(3, 3);
A.AddInteraction(0, 2, 2.5);
A.AddInteraction(1, 1, -1.0);

Matrix<double, General, RowSparse> B;
Copy(A, B);

Location :

Functions_Vector.cxx
Blas_1.cxx
Matrix_Conversions.cxx

Swap

Syntax :

  void Swap(Vector& Vector&);

The two vectors are exchanged. If BLAS is enabled, the two dense vectors have to be of the same size.

Example :

// vectors need to have the same size
Vector<float> V(3), W(3);
V.Fill(); 
W.Fill(1.0);
Swap(V, W);

Location :

Functions_Vector.cxx
Blas_1.cxx

GenRot, ApplyRot, GenModifRot, ApplyModifRot

Syntax :

  void GenRot(T&, T&, T&, T&);
  void ApplyRot(Vector&, Vector&, T, T); 
  void GenModifRot(T&, T&, T&, T&, T*);
  void ApplyModifRot(Vector&, Vector&, T*);

This function constructs the Givens rotation G = [cos_, sin_; -sin_, cos_] that zeros the second entry of the two vector (a,b).

Example :

double a, b, cos_, sin_;
a = 2.0;
b = 1.0;
// we compute cos_ and sin_ so that
// G = [cos_, sin_; -sin_, cos_] zeros second entry of vector [a;b]
// i.e. G*[a;b] = [r;0]
GenRot(a, b, cos_, sin_);

// then we can apply rotation G to vectors X, Y
// G*[x_i, y_i] for all i
Vector<double> X(10), Y(10);
X.FillRand(); 
Y.FillRand();
ApplyRot(X, Y, cos_, sin_);

Location :

Functions_Vector.cxx
Blas_1.cxx

DotProd, DotProdConj

Syntax :

  T DotProd(const Vector&, const Vector&);
  T DotProdConj(const Vector&, const Vector&);

This function returns the scalar product between two vectors. For DotProdConj, the first vector is conjugated. In the case of dense vectors, they have to be of the same size.

Example :

// we construct two vectors of same size
Vector<double> X(10), Y(10);
X.FillRand(); 
Y.FillRand();

// computation of X*Y
double scal = DotProd(X, Y);

// for complex numbers
Vector<complex<double> > Xc(10), Yc(10);
Xc.Fill(); Yc.Fill();
// computation of conj(X)*Y
complex<double> scalc = DotProdConj(X, Y);

Location :

Functions_Vector.cxx
Blas_1.cxx

Conjugate

Syntax :

  void Conjugate(Vector&);

Each component of the vector is conjugated. In the case of real numbers, the vector is not modified.

Example :

// complex vector
Vector<complex<double> > X(10);
X.FillRand(); 

// we take the conjugate
Conjugate(X);

Location :

Functions_Vector.cxx
Blas_1.cxx

GetMaxAbsIndex

Syntax :

   int GetMaxAbsIndex(const Vector&);

This function returns the index for which the vector reached its highest absolute value.

Example :

// complex vector
Vector<complex<double> > X(10);
X.FillRand(); 

int imax = GetMaxAbsIndex(X);
// maximum ?
double maximum = abs(X(imax));

Location :

Functions_Vector.cxx
Blas_1.cxx

Norm1

Syntax :

   T Norm1(const Vector&);

This function returns the 1-norm of the vector.

Example :

// complex vector
Vector<complex<double> > X(10);
X.Fill(); 

// 1-norm
double norme = Norm1(X);

Location :

Functions_Vector.cxx
Blas_1.cxx

Norm2

Syntax :

   T Norm2(const Vector&);

This function returns the 2-norm of the vector.

Example :

// complex vector
Vector<complex<double> > X(10);
X.Fill(); 

// 2-norm
double norme = Norm2(X);

Location :

Functions_Vector.cxx
Blas_1.cxx

Rank1Update

Syntax :

  void Rank1Update(T, const Vector&, const Vector&, Matrix&);
  void Rank1Update(T, const Vector&, SeldonConj, const Vector&, Matrix&);
  void Rank1Update(T, const Vector&, Matrix&);

This function adds a contribution of rank 1 to matrix.

Example :

// two complex vectors
Vector<complex<double> > X(10), Y(10);
X.FillRand(); 
Y.FillRand(); 

// complex matrix
Matrix<complex<double> > A(10, 10);
A.SetIdentity();
complex<double> alpha(1, 1);

// we compute A = A + alpha*X*transpose(Y)
Rank1Update(alpha, X, Y, A);
// we can also compute A = A + alpha*X*conjugate(transpose(Y))
Rank1Update(alpha, X, SeldonConj, Y, A);

// for hermitian matrices, use of X only
Matrix<complex<double>, General, RowHermPacked> B(10, 10);
// we compute B = B + alpha*X*conjugate(transpose(X))
Rank1Update(alpha, X, B);

// for real numbers, use of symmetric matrices
Vector<double> Xr(10);
Xr.FillRand(); 
Matrix<double, Symmetric, RowSymPacked> Br(10, 10);
Br.Zero();
double beta = 2;
// computation of Br = Br + beta*Xr*tranpose(Xr)
Rank1Update(beta, Xr, Br);

Location :

Blas_2.cxx

Rank2Update

Syntax :

  void Rank2Update(T, const Vector&, const Vector&, Matrix&);

This function adds a contribution of rank 2 to a symmetric or hermitian matrix.

Example :

// two complex vectors
Vector<complex<double> > X(10), Y(10);
X.FillRand(); 
Y.FillRand(); 

// hermitian matrix
Matrix<complex<double>, General, RowHermPacked> A(10, 10);
A.SetIdentity();
complex<double> alpha(1, 1);

// we compute A = A + alpha * (X * conjugate(transpose(Y))
//                + Y * conjugate(transpose(X)))
Rank2Update(alpha, X, Y, A);

// for real numbers, use of symmetric matrices
Vector<double> Xr(10), Yr(10);
Xr.FillRand(); 
Yr.FillRand(); 
Matrix<double, Symmetric, RowSymPacked> Br(10, 10);
Br.Zero();
double beta = 2;
// computation of Br = Br + beta*(Xr*transpose(Yr) + Yr*transpose(Xr))
Rank2Update(beta, Xr, Yr, Br);

Location :

Blas_2.cxx

Solve

Syntax :

  void Solve(const Matrix&, Vector&);
  void Solve(SeldonTrans, SeldonNonUnit, const Matrix&, Vector&);

This function solves a triangular linear system.

Example :

// right hand side
Vector<double> B(3);
B.Fill(); 

// triangular matrix
Matrix<double, General, RowUpTriangPacked> T(3, 3);
T.Fill();

// now you can solve T*X = B, the result overwrites B
Solve(T, B);

// if you want to solve transpose(T)*X = B
Solve(SeldonTrans, SeldonNonUnit, T, B);

// SeldonUnit can be used if you are assuming that T has 
// a unitary diagonal (with 1)

// we force unitary diagonal
for (int i = 0; i < T.GetM(); i++)
  T(i, i) = 1.0;

// then we can call Solve with Seldonunit to solve T*X = B
Solve(SeldonNoTrans, SeldonUnit, T, B);

Location :

Blas_2.cxx

Blas functions

Mlt

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MltAdd

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Add

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Copy

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Swap

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GenRot, ApplyRot, GenModifRot, ApplyModifRot

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DotProd, DotProdConj

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Conjugate

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GetMaxAbsIndex

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Norm1

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Norm2

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Rank1Update

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Rank2Update

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Solve

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