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Reference documentation for deal.II version 9.3.3
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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 ,
and
such that
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 ,
and
such that
Real valued solution only exists if , 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 where
contains k Lanczos basis,
,
is the tridiagonal Lanczos matrix,
is a residual vector
, and
is the k-th canonical basis of
. The returned value is
. If
eigenvalues
is not nullptr
, the eigenvalues of 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 with a complete set of eigenpairs
, the action of a polynomial filter
is given by
, where
. Thus by appropriately choosing the polynomial filter, one can alter the eigenmodes contained in
.
This function uses Chebyshev polynomials of first kind. Below is an example of polynomial of degree
normalized to unity at
.
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By introducing a linear mapping from
unwanted_spectrum
to , we can dump the corresponding modes in
x
. The higher the polynomial degree , 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 .
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 :
with
and
.
vector_memory
is used to allocate memory for temporary objects.
This function implements the algorithm (with a minor fix of sign of ) from
tau
is equal to std::numeric_limits<double>::infinity()
, no normalization will be performed.