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SHOGUN
4.1.0
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Abstract base class of the rational approximation of a function of a linear operator (A) times vector (v) using Cauchy's integral formula -
\[f(\text{A})\text{v}=\oint_{\Gamma}f(z)(z\text{I}-\text{A})^{-1} \text{v}dz\]
Computes eigenvalues of linear operator and uses Jacobi elliptic functions and conformal maps [2] for quadrature rule for discretizing the contour integral and computes complex shifts, weights and constant multiplier of the rational approximation of the above expression as
\[f(\text{A})\text{v}\approx \eta\text{A}\Im-\left(\sum_{l=1}^{N}\alpha_{l} (\text{A}-\sigma_{l}\text{I})^{-1}\text{v}\right)\]
where \(\alpha_{l},\sigma_{l}\in\mathbb{C}\) are respectively the shifts and weights of the linear systems generated from the rational approximation, and \(\eta\in\mathbb{R}\) is the constant multiplier, equals to \(\frac{-8K(\lambda_{m}\lambda_{M})^{\frac{1}{4}}}{k\pi N}\).
The number of shifts is automatically computed based on a previously specified accuracy \(\epsilon\) using the error bound
\[ -1.5\left(\log\left( \frac{\lambda_\text{max}}{\lambda_\text{min}}\right)+6.0 \right)\frac{\log(\epsilon)}{2\pi^2}. \]
It can also manually be set.
Reference: [1] Aune, E., D. Simpson, and J. Eidsvik (2012). Parameter estimation in high dimensional gaussian distributions. Technical Report Statistics 5/2012, NTNU.
[2] Nicholas Hale, Nicholas J. Higham and Lloyd N. Trefethen (2008). Computing \(A^{\alpha}\) , \(log(A)\) and related matrix functions by contour integrals. SIAM Journal of Numerical Analysis, 46:2505-2523
Note: The implementation of compute_weights_shifts_const function has been adapted from KRYLSTAT (Copyright 2011 by Erlend Aune erlenda@math.ntnu.no) under GPL2+. See https://github.com/Froskekongen/KRYLSTAT.
在文件 RationalApproximation.h 第 63 行定义.
Public 属性 | |
| SGIO * | io |
| Parallel * | parallel |
| Version * | version |
| Parameter * | m_parameters |
| Parameter * | m_model_selection_parameters |
| Parameter * | m_gradient_parameters |
| uint32_t | m_hash |
Protected 成员函数 | |
| virtual void | load_serializable_pre () throw (ShogunException) |
| virtual void | load_serializable_post () throw (ShogunException) |
| virtual void | save_serializable_pre () throw (ShogunException) |
| virtual void | save_serializable_post () throw (ShogunException) |
default constructor
在文件 RationalApproximation.cpp 第 25 行定义.
| CRationalApproximation | ( | CLinearOperator< float64_t > * | linear_operator, |
| CIndependentComputationEngine * | computation_engine, | ||
| CEigenSolver * | eigen_solver, | ||
| float64_t | desired_accuracy, | ||
| EOperatorFunction | function_type | ||
| ) |
Constructor. Number of shifts will be computed using a specified accuracy.
| linear_operator | real valued linear operator for this operator function |
| computation_engine | engine that computes the independent jobs |
| eigen_solver | eigen solver for computing min and max eigenvalues needed for computing shifts, weights and constant multiplier |
| desired_accuracy | desired error bound on approximation. Computes the number of shifts automatically |
| function_type | operator function type |
在文件 RationalApproximation.cpp 第 33 行定义.
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virtual |
destructor
在文件 RationalApproximation.cpp 第 52 行定义.
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inherited |
Builds a dictionary of all parameters in SGObject as well of those of SGObjects that are parameters of this object. Dictionary maps parameters to the objects that own them.
| dict | dictionary of parameters to be built. |
在文件 SGObject.cpp 第 597 行定义.
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virtualinherited |
Creates a clone of the current object. This is done via recursively traversing all parameters, which corresponds to a deep copy. Calling equals on the cloned object always returns true although none of the memory of both objects overlaps.
在文件 SGObject.cpp 第 714 行定义.
| int32_t compute_num_shifts_from_accuracy | ( | ) |
Computes the number of shifts from the current set accuracy \(\epsilon\) using
\[ -1.5\left(\log\left( \frac{\lambda_\text{max}}{\lambda_\text{min}}\right)+6.0 \right)\frac{\log(\epsilon)}{2\pi^2}, \]
在文件 RationalApproximation.cpp 第 129 行定义.
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virtualinherited |
A deep copy. All the instance variables will also be copied.
在文件 SGObject.cpp 第 198 行定义.
Recursively compares the current SGObject to another one. Compares all registered numerical parameters, recursion upon complex (SGObject) parameters. Does not compare pointers!
May be overwritten but please do with care! Should not be necessary in most cases.
| other | object to compare with |
| accuracy | accuracy to use for comparison (optional) |
| tolerant | allows linient check on float equality (within accuracy) |
在文件 SGObject.cpp 第 618 行定义.
| float64_t get_constant_multiplier | ( | ) | const |
在文件 RationalApproximation.cpp 第 96 行定义.
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inherited |
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inherited |
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inherited |
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inherited |
在文件 SGObject.cpp 第 498 行定义.
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inherited |
Returns description of a given parameter string, if it exists. SG_ERROR otherwise
| param_name | name of the parameter |
在文件 SGObject.cpp 第 522 行定义.
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inherited |
Returns index of model selection parameter with provided index
| param_name | name of model selection parameter |
在文件 SGObject.cpp 第 535 行定义.
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virtual |
重载 COperatorFunction< float64_t > .
被 CLogRationalApproximationIndividual , 以及 CLogRationalApproximationCGM 重载.
在文件 RationalApproximation.h 第 142 行定义.
| index_t get_num_shifts | ( | ) | const |
在文件 RationalApproximation.cpp 第 101 行定义.
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inherited |
在文件 OperatorFunction.h 第 81 行定义.
| SGVector< complex128_t > get_shifts | ( | ) | const |
在文件 RationalApproximation.cpp 第 86 行定义.
| SGVector< complex128_t > get_weights | ( | ) | const |
在文件 RationalApproximation.cpp 第 91 行定义.
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virtualinherited |
If the SGSerializable is a class template then TRUE will be returned and GENERIC is set to the type of the generic.
| generic | set to the type of the generic if returning TRUE |
在文件 SGObject.cpp 第 296 行定义.
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virtualinherited |
Load this object from file. If it will fail (returning FALSE) then this object will contain inconsistent data and should not be used!
| file | where to load from |
| prefix | prefix for members |
在文件 SGObject.cpp 第 369 行定义.
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protectedvirtualinherited | |||||||||||||
Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_POST is called.
| ShogunException | will be thrown if an error occurs. |
被 CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel , 以及 CExponentialKernel 重载.
在文件 SGObject.cpp 第 426 行定义.
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protectedvirtualinherited | |||||||||||||
Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_PRE is called.
| ShogunException | will be thrown if an error occurs. |
被 CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool > , 以及 CDynamicObjectArray 重载.
在文件 SGObject.cpp 第 421 行定义.
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virtualinherited |
在文件 SGObject.cpp 第 262 行定义.
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virtual |
precompute method that computes extremal eigenvalues using the eigensolver and then computes complex shifts, weights and constant multiplier coming from rational approximation of operator function times vector
Automatically computes the number of shifts if they have not been specified or are zero using set_shifts_from_accuracy().
实现了 COperatorFunction< float64_t >.
在文件 RationalApproximation.cpp 第 111 行定义.
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inherited |
prints all parameter registered for model selection and their type
在文件 SGObject.cpp 第 474 行定义.
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virtualinherited |
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virtualinherited |
Save this object to file.
| file | where to save the object; will be closed during returning if PREFIX is an empty string. |
| prefix | prefix for members |
在文件 SGObject.cpp 第 314 行定义.
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protectedvirtualinherited | |||||||||||||
Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_POST is called.
| ShogunException | will be thrown if an error occurs. |
被 CKernel 重载.
在文件 SGObject.cpp 第 436 行定义.
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protectedvirtualinherited | |||||||||||||
Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_PRE is called.
| ShogunException | will be thrown if an error occurs. |
被 CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool > , 以及 CDynamicObjectArray 重载.
在文件 SGObject.cpp 第 431 行定义.
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inherited |
在文件 SGObject.cpp 第 41 行定义.
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inherited |
在文件 SGObject.cpp 第 46 行定义.
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inherited |
在文件 SGObject.cpp 第 51 行定义.
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inherited |
在文件 SGObject.cpp 第 56 行定义.
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inherited |
在文件 SGObject.cpp 第 61 行定义.
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inherited |
在文件 SGObject.cpp 第 66 行定义.
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inherited |
在文件 SGObject.cpp 第 71 行定义.
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inherited |
在文件 SGObject.cpp 第 76 行定义.
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inherited |
在文件 SGObject.cpp 第 81 行定义.
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inherited |
在文件 SGObject.cpp 第 86 行定义.
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inherited |
在文件 SGObject.cpp 第 91 行定义.
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inherited |
在文件 SGObject.cpp 第 96 行定义.
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inherited |
在文件 SGObject.cpp 第 101 行定义.
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inherited |
在文件 SGObject.cpp 第 106 行定义.
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inherited |
在文件 SGObject.cpp 第 111 行定义.
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inherited |
set generic type to T
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inherited |
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inherited |
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inherited |
| void set_num_shifts | ( | index_t | num_shifts | ) |
| num_shifts | number of shifts |
在文件 RationalApproximation.cpp 第 106 行定义.
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virtualinherited |
A shallow copy. All the SGObject instance variables will be simply assigned and SG_REF-ed.
被 CGaussianKernel 重载.
在文件 SGObject.cpp 第 192 行定义.
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pure virtual |
abstract method that creates a job result aggregator, then creates a number of jobs based on its implementation, attaches the aggregator with all those jobs, hands over the responsility of those to the computation engine and then returns the aggregator for collecting the job results
| sample | the vector for which new computation job(s) are to be created |
实现了 COperatorFunction< float64_t >.
在 CLogRationalApproximationIndividual , 以及 CLogRationalApproximationCGM 内被实现.
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inherited |
unset generic type
this has to be called in classes specializing a template class
在文件 SGObject.cpp 第 303 行定义.
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virtualinherited |
Updates the hash of current parameter combination
在文件 SGObject.cpp 第 248 行定义.
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inherited |
io
在文件 SGObject.h 第 369 行定义.
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protectedinherited |
the computation engine
在文件 OperatorFunction.h 第 115 行定义.
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protected |
constant multiplier
在文件 RationalApproximation.h 第 158 行定义.
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protected |
desired accuracy from which number of shifts might be computed
在文件 RationalApproximation.h 第 164 行定义.
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protected |
the eigen solver for computing extremal eigenvalues
在文件 RationalApproximation.h 第 149 行定义.
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protectedinherited |
the linear operator function type
在文件 OperatorFunction.h 第 118 行定义.
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inherited |
parameters wrt which we can compute gradients
在文件 SGObject.h 第 384 行定义.
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inherited |
Hash of parameter values
在文件 SGObject.h 第 387 行定义.
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protectedinherited |
the linear operator
在文件 OperatorFunction.h 第 112 行定义.
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inherited |
model selection parameters
在文件 SGObject.h 第 381 行定义.
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protected |
number of shifts
在文件 RationalApproximation.h 第 161 行定义.
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inherited |
parameters
在文件 SGObject.h 第 378 行定义.
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protected |
complex shifts in the systems coming from rational approximation
在文件 RationalApproximation.h 第 152 行定义.
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protected |
complex weights in the systems coming from rational approximation
在文件 RationalApproximation.h 第 155 行定义.
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inherited |
parallel
在文件 SGObject.h 第 372 行定义.
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inherited |
version
在文件 SGObject.h 第 375 行定义.
