CAPD::DynSys Library  6.0.0
Classes | Functions
capd::pdes Namespace Reference

Classes

class  C0DoubletonSetGeometricTail
 
class  C0HODoubletonSetGeometricTail
 This class uses representation of subset of R^n inherited from template parameter. More...
 
class  C1DoubletonSetGeometricTail
 
class  DissipativeVectorField
 The class provides a common interface for a dissipative vector required by the class PdeSolver. More...
 
class  GeometricBound
 The class is class represents a subset of a countable infinite dimensional space. More...
 
class  OneDimKSSineVectorField
 The class implements vector field of the one-dimensional real Kuramoto-Shivashinsky PDE under the following assumptions 1 .The solutions are represented in the Fourier basis. More...
 
class  PdeAbstractSection
 
class  PdeAffineSection
 TimeMap class provides class that serves as an affine Poincare section. More...
 
class  PdeCoordinateSection
 TimeMap class provides class that serves as Poincare section of the form x_i = c. More...
 
class  PdeCurve
 This class is a data structure for storing of a parametric curve together with first order derivatives with respect to initial point. More...
 
struct  ComputeOneStepSectionEnclosure
 
struct  ComputeOneStepSectionEnclosure< false >
 
struct  PdeSectionDerivativesEnclosure
 
class  PdeSolver
 
class  PolyLogBound
 The class is class represents a subset of a countable infinite dimensional space. More...
 
class  PolynomialBound
 

Functions

template<class ScalarT >
GeometricBound< ScalarT > operator+ (const GeometricBound< ScalarT > &x, const GeometricBound< ScalarT > &y)
 operator+ More...
 
template<class ScalarT >
GeometricBound< ScalarT > operator- (const GeometricBound< ScalarT > &x, const GeometricBound< ScalarT > &y)
 operator- More...
 
template<class ScalarT >
GeometricBound< ScalarT > operator* (const ScalarT &s, const GeometricBound< ScalarT > &x)
 operator* More...
 
template<class ScalarT >
GeometricBound< ScalarT > operator* (const GeometricBound< ScalarT > &x, const ScalarT &s)
 operator* More...
 
template<class ScalarT >
GeometricBound< ScalarT > operator* (const capd::vectalg::Matrix< ScalarT, 0, 0 > &A, const GeometricBound< ScalarT > &x)
 operator* More...
 
template<class ScalarT >
std::ostream & operator<< (std::ostream &s, const GeometricBound< ScalarT > &x)
 operator<< More...
 
template<class ScalarT >
GeometricBound< ScalarT > intersection (const GeometricBound< ScalarT > &x, const GeometricBound< ScalarT > &y)
 
template<class ScalarT >
void split (const GeometricBound< ScalarT > &X, GeometricBound< ScalarT > &x, GeometricBound< ScalarT > &dx)
 
template<class ScalarT >
GeometricBound< ScalarT > midVector (const GeometricBound< ScalarT > &x)
 
template<class ScalarT >
void swap (GeometricBound< ScalarT > &a, GeometricBound< ScalarT > &b)
 
PolyLogBound operator+ (const PolyLogBound &x, const PolyLogBound &y)
 operator+ More...
 
PolyLogBound operator- (const PolyLogBound &x, const PolyLogBound &y)
 operator- More...
 
PolyLogBound operator* (const interval &s, const PolyLogBound &x)
 operator* More...
 
PolyLogBound operator* (const PolyLogBound &x, const interval &s)
 operator* More...
 
PolyLogBound operator* (const IMatrix &A, const PolyLogBound &x)
 operator* More...
 
std::ostream & operator<< (std::ostream &s, const PolyLogBound &x)
 operator<< More...
 
PolyLogBound intersection (const PolyLogBound &x, const PolyLogBound &y)
 
void split (const PolyLogBound &X, PolyLogBound &x, PolyLogBound &dx)
 
PolyLogBound midVector (const PolyLogBound &x)
 
template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > operator+ (const PolynomialBound< Scalar, Exponent, M > &x, const PolynomialBound< Scalar, Exponent, M > &y)
 this operator realizes addition of two PolynomialBounds More...
 
template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > operator- (const PolynomialBound< Scalar, Exponent, M > &v1, const PolynomialBound< Scalar, Exponent, M > &v2)
 this operator realizes subtraction of two PolynomialBounds More...
 
template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > operator* (const Scalar &s, const PolynomialBound< Scalar, Exponent, M > &v)
 this operator realizes multiplication of any coefficient in PolynomialBound by some scalar More...
 
template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > operator* (const PolynomialBound< Scalar, Exponent, M > &p, const Scalar &s)
 this operator realizes multiplication of any coefficient in PolynomialBound by some scalar More...
 
template<typename Scalar , typename Exponent , int M, int dim>
PolynomialBound< Scalar, Exponent, M > operator* (const capd::vectalg::Matrix< Scalar, dim, dim > &A, const PolynomialBound< Scalar, Exponent, M > &p)
 this operator realizes change of coordiante on some PolynomialBound More...
 
template<typename Scalar , typename Exponent , int M>
std::ostream & operator<< (std::ostream &s, const PolynomialBound< Scalar, Exponent, M > &p)
 this operator writes a PolynomialBound object to a given stream in the following form {{p_1,p_2,...,p_M},C,exponent} where M is a number of main coefficients in an object p More...
 
template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > computeQF (const PolynomialBound< Scalar, Exponent, M > &y, const PolynomialBound< Scalar, Exponent, M > &w)
 the function computes value of operator QF on two PolynomialBounds y and w given by QF_i = $ \sum_{k=1}^{i-1} y_k w_{i-k} $ More...
 
template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > computeQI (const PolynomialBound< Scalar, Exponent, M > &y, const PolynomialBound< Scalar, Exponent, M > &w)
 the function computes value of operator QI on two PolynomialBounds y and w given by QI_i = $ \sum_{k=1}^\infty y_k w_{k+i} $ More...
 
template GeometricBound< interval > operator+<interval > (const GeometricBound< interval > &x, const GeometricBound< interval > &y)
 
template GeometricBound< interval > operator-<interval > (const GeometricBound< interval > &x, const GeometricBound< interval > &y)
 
template GeometricBound< interval > operator*<interval > (const interval &s, const GeometricBound< interval > &x)
 
template GeometricBound< interval > operator*<interval > (const GeometricBound< interval > &x, const interval &s)
 
template GeometricBound< interval > operator*<interval > (const capd::vectalg::Matrix< interval, 0, 0 > &A, const GeometricBound< interval > &v)
 
template std::ostream & operator<<< interval > (std::ostream &out, const GeometricBound< interval > &x)
 
template void split< interval > (const GeometricBound< interval > &X, GeometricBound< interval > &x, GeometricBound< interval > &dx)
 
template GeometricBound< interval > intersection< interval > (const GeometricBound< interval > &x, const GeometricBound< interval > &y)
 

Function Documentation

◆ computeQF()

template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > capd::pdes::computeQF ( const PolynomialBound< Scalar, Exponent, M > &  y,
const PolynomialBound< Scalar, Exponent, M > &  w 
)

the function computes value of operator QF on two PolynomialBounds y and w given by QF_i = $ \sum_{k=1}^{i-1} y_k w_{i-k} $

Parameters
[in]yobject of class PolynomialBound
[in]wobject of class PolynomialBound
Returns
value of operator QF on polynomial bounds y and w

◆ computeQI()

template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > capd::pdes::computeQI ( const PolynomialBound< Scalar, Exponent, M > &  y,
const PolynomialBound< Scalar, Exponent, M > &  w 
)

the function computes value of operator QI on two PolynomialBounds y and w given by QI_i = $ \sum_{k=1}^\infty y_k w_{k+i} $

Parameters
[in]yobject of class PolynomialBound
[in]wobject of class PolynomialBound
Returns
value of operator QI on polynomial bounds y and w

◆ intersection() [1/2]

template<class ScalarT >
GeometricBound< ScalarT > capd::pdes::intersection ( const GeometricBound< ScalarT > &  x,
const GeometricBound< ScalarT > &  y 
)

◆ intersection() [2/2]

PolyLogBound capd::pdes::intersection ( const PolyLogBound &  x,
const PolyLogBound &  y 
)

◆ intersection< interval >()

template GeometricBound<interval> capd::pdes::intersection< interval > ( const GeometricBound< interval > &  x,
const GeometricBound< interval > &  y 
)

◆ midVector() [1/2]

template<class ScalarT >
GeometricBound<ScalarT> capd::pdes::midVector ( const GeometricBound< ScalarT > &  x)
inline

◆ midVector() [2/2]

PolyLogBound capd::pdes::midVector ( const PolyLogBound &  x)
inline

◆ operator*() [1/9]

template<typename Scalar , typename Exponent , int M, int dim>
PolynomialBound< Scalar, Exponent, M > capd::pdes::operator* ( const capd::vectalg::Matrix< Scalar, dim, dim > &  A,
const PolynomialBound< Scalar, Exponent, M > &  p 
)

this operator realizes change of coordiante on some PolynomialBound

if A is m\times m martix then the first m coordinates of result are given by result_i = $ \sum_{k=1}^m A_{i,k} p_k $ on the other coefficients change of coordinates is identity

Parameters
[in]Asquare matrix of dimension smaller than p.dimension()
[in]pobject of class PolynomialBound
Returns
PolynomialBound p in new coordinate system

◆ operator*() [2/9]

template<class ScalarT >
GeometricBound< ScalarT > capd::pdes::operator* ( const capd::vectalg::Matrix< ScalarT, 0, 0 > &  A,
const GeometricBound< ScalarT > &  x 
)

operator*

This operator realizes the following Matrix by vector multiplication (A*mainVariables,Id*tail)

this operator realizes linear transformation of a GeometricBound

if A is square matrix m\times m then the first m coordinates of result are given by $ result_i = \sum_{k=1}^m A_{i,k} x_k $ The remaining (infinite number) of coordinates remain unchanged.

Parameters
[in]Asquare matrix of dimension smaller than p.dimension()
[in]xobject of class GeometricBound
Returns
GeometricBound x in transformed by (A,Id)
Note
The dimension m of the matrix A must be less or equal than number of exactly represented coefficients in x. Exception is thrown if this requirement is violated.

◆ operator*() [3/9]

template<class ScalarT >
GeometricBound< ScalarT > capd::pdes::operator* ( const GeometricBound< ScalarT > &  x,
const ScalarT &  s 
)

operator*

this operator realizes multiplication of any coefficient in a GeometricBound by some scalar

using the following formula result_i = s * x_i

Parameters
[in]sobject of type Scalar
[in]xobject of class GeometricBound
Returns
GeometricBound x multiplied by s

◆ operator*() [4/9]

PolyLogBound capd::pdes::operator* ( const IMatrix &  A,
const PolyLogBound &  x 
)

operator*

this operator realizes linear transformation of a PolyLogBound

if A is square matrix m\times m then the first m coordinates of result are given by $ result_i = \sum_{k=1}^m A_{i,k} x_k $ The remaining (infinite number) of coordinates remain unchanged.

Parameters
[in]Asquare matrix of dimension smaller than p.dimension()
[in]xobject of class PolyLogBound
Returns
PolyLogBound x in transformed by (A,Id)
Note
The dimension m of the matrix A must be less or equal than number of exactly represented coefficients in x. Exception is thrown if this requirement is violated.

◆ operator*() [5/9]

PolyLogBound capd::pdes::operator* ( const interval &  s,
const PolyLogBound &  x 
)

operator*

this operator realizes multiplication of any coefficient in a PolyLogBound by some scalar

using the following formula result_i = s * x

Parameters
[in]sobject of type Scalar
[in]xobject of class PolyLogBound
Returns
PolyLogBound p multiplied by s

◆ operator*() [6/9]

PolyLogBound capd::pdes::operator* ( const PolyLogBound &  x,
const interval &  s 
)

operator*

this operator realizes multiplication of any coefficient in a PolyLogBound by some scalar

using the following formula result_i = s * x_i

Parameters
[in]sobject of type Scalar
[in]xobject of class PolyLogBound
Returns
PolyLogBound x multiplied by s

◆ operator*() [7/9]

template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > capd::pdes::operator* ( const PolynomialBound< Scalar, Exponent, M > &  p,
const Scalar &  s 
)

this operator realizes multiplication of any coefficient in PolynomialBound by some scalar

using the following formula result_i = s * p_i

Parameters
[in]pobject of class PolynomialBound
[in]sobject of type Scalar
Returns
PolynomialBound p multiplied by s

◆ operator*() [8/9]

template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > capd::pdes::operator* ( const Scalar &  s,
const PolynomialBound< Scalar, Exponent, M > &  v 
)

this operator realizes multiplication of any coefficient in PolynomialBound by some scalar

using the following formula result_i = s * p_i

Parameters
[in]sobject of type Scalar
[in]pobject of class PolynomialBound
Returns
PolynomialBound p multiplied by s

◆ operator*() [9/9]

template<class ScalarT >
GeometricBound< ScalarT > capd::pdes::operator* ( const ScalarT &  s,
const GeometricBound< ScalarT > &  x 
)

operator*

this operator realizes multiplication of any coefficient in a GeometricBound by some scalar

using the following formula result_i = s * x

Parameters
[in]sobject of type Scalar
[in]xobject of class GeometricBound
Returns
GeometricBound p multiplied by s

◆ operator*<interval >() [1/3]

template GeometricBound<interval> capd::pdes::operator*<interval > ( const capd::vectalg::Matrix< interval, 0, 0 > &  A,
const GeometricBound< interval > &  v 
)

◆ operator*<interval >() [2/3]

template GeometricBound<interval> capd::pdes::operator*<interval > ( const GeometricBound< interval > &  x,
const interval &  s 
)

◆ operator*<interval >() [3/3]

template GeometricBound<interval> capd::pdes::operator*<interval > ( const interval &  s,
const GeometricBound< interval > &  x 
)

◆ operator+() [1/3]

template<class ScalarT >
GeometricBound< ScalarT > capd::pdes::operator+ ( const GeometricBound< ScalarT > &  x,
const GeometricBound< ScalarT > &  y 
)

operator+

this operator realizes addition of two GeometricBound using the following formula result_i = x_i + y_i Since object is represented as a finite dimensional vector and a tail we do explicit summation on main variables only. Exponent of result is computed as t = min(x.exponent(),y.exponent())

The constant used in representation of tail in result is computed as M = M_result = min(M_x,M_y) - number of exactly represented coefficients C_result = C_x/(M+1)^{x.exponent()-t} + C_y/(M+1)^{y.exponent()-t}

Parameters
[in]xobject of class GeometricBound
[in]yobject of class GeometricBound
Returns
sum of x and y

◆ operator+() [2/3]

PolyLogBound capd::pdes::operator+ ( const PolyLogBound &  x,
const PolyLogBound &  y 
)

operator+

this operator realizes addition of two PolyLogBound using the following formula result_i = x_i + y_i Since object is represented as a finite dimensional vector and a tail we do explicit summation on main variables only. Exponent of result is computed as t = min(x.exponent(),y.exponent())

The constant used in representation of tail in result is computed as M = M_result = min(M_x,M_y) - number of exactly represented coefficients C_result = C_x/(M+1)^{x.exponent()-t} + C_y/(M+1)^{y.exponent()-t}

Parameters
[in]xobject of class PolyLogBound
[in]yobject of class PolyLogBound
Returns
sum of x and y

◆ operator+() [3/3]

template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > capd::pdes::operator+ ( const PolynomialBound< Scalar, Exponent, M > &  x,
const PolynomialBound< Scalar, Exponent, M > &  y 
)

this operator realizes addition of two PolynomialBounds

using the following formula result_i = x_i + y_i Since object is represented as a finite dimensional vector and a tail we do explicit summation on main variables only. Exponent of result is computed as t = min(x.exponent(),y.exponent()) The constant used in representation of tail in result is computed as C_result = C_x/(M+1)^{x.exponent()-t} + C_y/(M+1)^{y.exponent()-t}

Parameters
[in]xobject of class PolynomialBound
[in]yobject of class PolynomialBound
Returns
sum of x and y

◆ operator+<interval >()

template GeometricBound<interval> capd::pdes::operator+<interval > ( const GeometricBound< interval > &  x,
const GeometricBound< interval > &  y 
)

◆ operator-() [1/3]

template<class ScalarT >
GeometricBound< ScalarT > capd::pdes::operator- ( const GeometricBound< ScalarT > &  x,
const GeometricBound< ScalarT > &  y 
)

operator-

this operator realizes subtraction of two GeometricBound using the following formula result_i = x_i - y_i Since object is represented as a finite dimensional vector and a tail we do explicit summation on main variables only. Exponent of result is computed as t = min(x.exponent(),y.exponent())

The constant used in representation of tail in result is computed as M = M_result = min(M_x,M_y) - number of exactly represented coefficients C_result = C_x/(M+1)^{x.exponent()-t} + C_y/(M+1)^{y.exponent()-t}

Parameters
[in]xobject of class GeometricBound
[in]yobject of class GeometricBound
Returns
subtraction of x and y

◆ operator-() [2/3]

PolyLogBound capd::pdes::operator- ( const PolyLogBound &  x,
const PolyLogBound &  y 
)

operator-

this operator realizes subtraction of two PolyLogBound using the following formula result_i = x_i - y_i Since object is represented as a finite dimensional vector and a tail we do explicit summation on main variables only. Exponent of result is computed as t = min(x.exponent(),y.exponent())

The constant used in representation of tail in result is computed as M = M_result = min(M_x,M_y) - number of exactly represented coefficients C_result = C_x/(M+1)^{x.exponent()-t} + C_y/(M+1)^{y.exponent()-t}

Parameters
[in]xobject of class PolyLogBound
[in]yobject of class PolyLogBound
Returns
subtraction of x and y

◆ operator-() [3/3]

template<typename Scalar , typename Exponent , int M>
PolynomialBound< Scalar, Exponent, M > capd::pdes::operator- ( const PolynomialBound< Scalar, Exponent, M > &  v1,
const PolynomialBound< Scalar, Exponent, M > &  v2 
)

this operator realizes subtraction of two PolynomialBounds

using the following formula result_i = x_i - y_i Since object is represented as a finite dimensional vector and a tail we do explicit subtraction on main variables only. Exponent of result is computed as t = min(x.exponent(),y.exponent()) The constant used in representation of tail in result is computed as C_result = C_x/(M+1)^{x.exponent()-t} + C_y/(M+1)^{y.exponent()-t}

Parameters
[in]xobject of class PolynomialBound
[in]yobject of class PolynomialBound
Returns
difference of x and y

◆ operator-<interval >()

template GeometricBound<interval> capd::pdes::operator-<interval > ( const GeometricBound< interval > &  x,
const GeometricBound< interval > &  y 
)

◆ operator<<() [1/3]

template<class ScalarT >
std::ostream & capd::pdes::operator<< ( std::ostream &  s,
const GeometricBound< ScalarT > &  x 
)

operator<<

This operator writes a GeometricBound object to a given stream in the following form {{x_1,x_2,...,x_M},C,exponent} where M is the number of exactly represented coefficients of x.

Parameters
[in]out- a stream to which the object x is to be written
[in]x- an instance of class GeometricBound
Returns
a reference to stream out

◆ operator<<() [2/3]

std::ostream& capd::pdes::operator<< ( std::ostream &  s,
const PolyLogBound &  x 
)

operator<<

This operator writes a PolyLogBound object to a given stream in the following form {{x_1,x_2,...,x_M},C,exponent} where M is the number of exactly represented coefficients of x.

Parameters
[in]out- a stream to which the object x is to be written
[in]x- an instance of class PolyLogBound
Returns
a reference to stream out

◆ operator<<() [3/3]

template<typename Scalar , typename Exponent , int M>
std::ostream & capd::pdes::operator<< ( std::ostream &  s,
const PolynomialBound< Scalar, Exponent, M > &  p 
)

this operator writes a PolynomialBound object to a given stream in the following form {{p_1,p_2,...,p_M},C,exponent} where M is a number of main coefficients in an object p

Parameters
[in]sa stream to which object p is printed
[in]pobject of class PolynomialBound
Returns
a reference to stream s

◆ operator<<< interval >()

template std::ostream& capd::pdes::operator<<< interval > ( std::ostream &  out,
const GeometricBound< interval > &  x 
)

◆ split() [1/2]

template<class ScalarT >
void capd::pdes::split ( const GeometricBound< ScalarT > &  X,
GeometricBound< ScalarT > &  x,
GeometricBound< ScalarT > &  dx 
)

◆ split() [2/2]

void capd::pdes::split ( const PolyLogBound &  X,
PolyLogBound &  x,
PolyLogBound &  dx 
)

◆ split< interval >()

template void capd::pdes::split< interval > ( const GeometricBound< interval > &  X,
GeometricBound< interval > &  x,
GeometricBound< interval > &  dx 
)

◆ swap()

template<class ScalarT >
void capd::pdes::swap ( GeometricBound< ScalarT > &  a,
GeometricBound< ScalarT > &  b 
)
inline