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Reference documentation for deal.II version 9.1.1
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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) |
A collection of linear-algebra utilities.
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
\[ \begin{bmatrix} c & s \\ -s & c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \sqrt{x^2+y^2} \\ 0 \end{bmatrix} \]
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.
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
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 \(-1.2\).
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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 \([-1,1]\). 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
tau
is equal to std::numeric_limits<double>::infinity()
, no normalization will be performed.