Functions
template<typename NumberType >
std::array< NumberType, 3 >	givens_rotation (const NumberType &x, const NumberType &y)

template<typename NumberType >
std::array< NumberType, 3 >	hyperbolic_rotation (const NumberType &x, const NumberType &y)

template<typename OperatorType , typename VectorType >
double	lanczos_largest_eigenvalue (const OperatorType &H, const VectorType &v0, const unsigned int k, VectorMemory< VectorType > &vector_memory, std::vector< double > *eigenvalues=nullptr)

template<typename OperatorType , typename VectorType >
void	chebyshev_filter (VectorType &x, const OperatorType &H, const unsigned int n, const std::pair< double, double > unwanted_spectrum, const double tau, VectorMemory< VectorType > &vector_memory)

Detailed Description

A collection of linear-algebra utilities.

Function Documentation

◆ givens_rotation()

template<typename NumberType >

std::array<NumberType, 3> Utilities::LinearAlgebra::givens_rotation	(	const NumberType &	x,
		const NumberType &	y
	)

Return the elements of a continuous Givens rotation matrix and the norm of the input vector.

That is for a given pair x and y, return $c$ , $s$ and $\sqrt{x^2+y^2}$ such that

$\begin{bmatrix} c & s \\ -s & c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \sqrt{x^2+y^2} \\ 0 \end{bmatrix}$

Note: The function is implemented for real valued numbers only.

Author: Denis Davydov, 2017

◆ hyperbolic_rotation()

template<typename NumberType >

std::array<NumberType, 3> Utilities::LinearAlgebra::hyperbolic_rotation	(	const NumberType &	x,
		const NumberType &	y
	)

Return the elements of a hyperbolic rotation matrix.

That is for a given pair x and y, return $c$ , $s$ and $r$ such that

$\begin{bmatrix} c & -s \\ -s & c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} r \\ 0 \end{bmatrix}$

Real valued solution only exists if $|x|>|g|$ , the function will throw an error otherwise.

Note: The function is implemented for real valued numbers only.

Author: Denis Davydov, 2017

◆ lanczos_largest_eigenvalue()

template<typename OperatorType , typename VectorType >

double Utilities::LinearAlgebra::lanczos_largest_eigenvalue	(	const OperatorType &	H,
		const VectorType &	v0,
		const unsigned int	k,
		VectorMemory< VectorType > &	vector_memory,
		std::vector< double > *	eigenvalues = `nullptr`
	)

Estimate an upper bound for the largest eigenvalue of H by a k -step Lanczos process starting from the initial vector v0. Typical values of k are below 10. This estimator computes a k-step Lanczos decomposition $H V_k=V_k T_k+f_k e_k^T$ where $V_k$ contains k Lanczos basis, $V_k^TV_k=I_k$ , $T_k$ is the tridiagonal Lanczos matrix, $f_k$ is a residual vector $f_k^TV_k=0$ , and $e_k$ is the k-th canonical basis of $R^k$ . The returned value is $ ||T_k||_2 + ||f_k||_2$ . If eigenvalues is not nullptr, the eigenvalues of $T_k$ will be written there.

vector_memory is used to allocate memory for temporary vectors. OperatorType has to provide vmult operation with VectorType.

This function implements the algorithm from

@article{Zhou2006,
  Title   = {Self-consistent-field Calculations Using Chebyshev-filtered
             Subspace Iteration},
  Author  = {Zhou, Yunkai and Saad, Yousef and Tiago, Murilo L. and
             Chelikowsky, James R.},
  Journal = {Journal of Computational Physics},
  Year    = {2006},
  Volume  = {219},
  Pages   = {172--184},
}

Note: This function uses Lapack routines to compute the largest eigenvalue of .; This function provides an alternate estimate to that obtained from several steps of SolverCG with SolverCG<VectorType>::connect_eigenvalues_slot().

Author: Denis Davydov, 2017

◆ chebyshev_filter()

template<typename OperatorType , typename VectorType >

void Utilities::LinearAlgebra::chebyshev_filter	(	VectorType &	x,
		const OperatorType &	H,
		const unsigned int	n,
		const std::pair< double, double >	unwanted_spectrum,
		const double	tau,
		VectorMemory< VectorType > &	vector_memory
	)

Apply Chebyshev polynomial of the operator H to x. For a non-defective operator $H$ with a complete set of eigenpairs $H \psi_i = \lambda_i \psi_i$ , the action of a polynomial filter $p$ is given by $p(H)x =\sum_i a_i p(\lambda_i) \psi_i$ , where $x=: \sum_i a_i \psi_i$ . Thus by appropriately choosing the polynomial filter, one can alter the eigenmodes contained in $x$ .

This function uses Chebyshev polynomials of first kind. Below is an example of polynomial $T_n(x)$ of degree $n=8$ normalized to unity at $-1.2$ .

By introducing a linear mapping $L$ from unwanted_spectrum to $[-1,1]$ , we can dump the corresponding modes in x. The higher the polynomial degree $n$ , the more rapid it grows outside of the . In order to avoid numerical overflow, we normalize polynomial filter to unity at tau. Thus, the filtered operator is $p(H) = T_n(L(H))/T_n(L(\tau))$ .

The action of the Chebyshev filter only requires evaluation of vmult() of H and is based on the recursion equation for Chebyshev polynomial of degree $n$ : $T_{n}(x) = 2x T_{n-1}(x) - T_{n-2}(x)$ with $T_0(x)=1$ and $T_1(x)=x$ .

vector_memory is used to allocate memory for temporary objects.

This function implements the algorithm (with a minor fix of sign of $\sigma_1$ ) from

@article{Zhou2014,
  Title   = {Chebyshev-filtered subspace iteration method free of sparse
             diagonalization for solving the Kohn--Sham equation},
  Author  = {Zhou, Yunkai and Chelikowsky, James R and Saad, Yousef},
  Journal = {Journal of Computational Physics},
  Year    = {2014},
  Volume  = {274},
  Pages   = {770--782},
}

Note: If tau is equal to std::numeric_limits<double>::infinity(), no normalization will be performed.

Author: Denis Davydov, 2017