gsFunctionExpr< T > Class Template Reference

Detailed Description

template<typename T>
class gismo::gsFunctionExpr< T >

Class defining a multivariate (real or vector) function given by a string mathematical expression.

Numerous forms of functional and logic processing semantics are supported. The class is based on The C++ Mathematical Expression Toolkit Library (ExprTk), see

http://www.partow.net/programming/exprtk

and

https://github.com/ArashPartow/exprtk/blob/master/readme.txt

for more details.

The variables used are the ordered list:

x , y , z, w, u, v, t

Any function with domain dimension equal to 1 MUST use x as variable. the letters y...t are settable parameters. Any function with domain dimension equal to 2 MUST use x,y as variables. the letters z...t are settable parameters. Any function with domain dimension equal to 3 MUST use x,y,z as variables. the letters w...t are settable parameters. And so on.

Inheritance diagram for gsFunctionExpr< T >:

Collaboration diagram for gsFunctionExpr< T >:

Public Types
typedef memory::shared_ptr < gsFunctionExpr >	Ptr
	Shared pointer for gsFunctionExpr.
typedef memory::unique_ptr < gsFunctionExpr >	uPtr
	Unique pointer for gsFunctionExpr.

Public Member Functions
gsMatrix< index_t >	active (const gsMatrix< T > &u) const
	Returns the indices of active (nonzero) functions at points u, as a list of indices. More...
void	active_into (const gsMatrix< T > &u, gsMatrix< index_t > &result) const
	Indices of active (non-zero) function(s) for each point. More...
void	addComponent (const std::string &strExpression)
	Adds another component to this (vector) function.
const gsBasis< T > &	basis (const index_t k) const
	Helper which casts and returns the k-th piece of this function set as a gsBasis.
uPtr	clone ()
	Clone methode. Produceds a deep copy inside a uPtr.
virtual void	compute (const gsMatrix< T > &in, gsFuncData< T > &out) const
	Computes function data. More...
virtual void	computeMap (gsMapData< T > &InOut) const
	Computes map function data. More...
gsFuncCoordinate< T >	coord (const index_t c) const
	Returns the scalar function giving the i-th coordinate of this function.
gsMatrix< T >	deriv (const gsMatrix< T > &u) const
	Evaluate the derivatives,. More...
gsMatrix< T >	deriv2 (const gsMatrix< T > &u) const
	Evaluates the second derivatives of active (i.e., non-zero) functions at points u. More...
virtual void	deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate second derivatives of the function at points u into result. More...
virtual void	deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate derivatives of the function \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\) at points u into result. More...
virtual T	distanceL2 (gsFunction< T > const &) const
	Computes the L2-distance between this function and the field and a function func.
short_t	domainDim () const
	Dimension of the (source) domain. More...
gsMatrix< T >	eval (const gsMatrix< T > &u) const
	Evaluate the function,. More...
virtual void	eval_component_into (const gsMatrix< T > &u, const index_t comp, gsMatrix< T > &result) const
	Evaluate the function for component comp in the target dimension at points u into result.
virtual void	eval_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Evaluate the function at points u into result. More...
std::vector< gsMatrix< T > >	evalAllDers (const gsMatrix< T > &u, int n, bool sameElement=false) const
	Evaluate all derivatives upto order n,. More...
virtual void	evalAllDers_into (const gsMatrix< T > &u, int n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const
	Evaluate the nonzero functions and their derivatives up to order n at points u into result. More...
const gsFunction< T > &	function (const index_t k) const
	Helper which casts and returns the k-th piece of this function set as a gsFunction.
	gsFunctionExpr ()
	Default empty constructor.
	gsFunctionExpr (const std::string &expression_string, short_t ddim)
	Constructor taking an expression string and the domain dimension (scalar function in variable)
	gsFunctionExpr (const std::string &expression_string1, const std::string &expression_string2, short_t ddim)
	Constructor taking two expression strings (2D vector valued function)
	gsFunctionExpr (const std::string &expression_string1, const std::string &expression_string2, const std::string &expression_string3, short_t ddim)
	Constructor taking three expression strings (3D vector valued function)
	gsFunctionExpr (const std::string &expression_string1, const std::string &expression_string2, const std::string &expression_string3, const std::string &expression_string4, short_t ddim)
	Constructor taking four expression strings (4D vector valued function) used for matrix coefficients.
	gsFunctionExpr (const std::string &expression_string1, const std::string &expression_string2, const std::string &expression_string3, const std::string &expression_string4, const std::string &expression_string5, const std::string &expression_string6, const std::string &expression_string7, const std::string &expression_string8, const std::string &expression_string9, short_t ddim)
	Constructor taking nine expression strings (9D vector valued function) used for (3x3) matrix coefficients.
virtual void	invertPoints (const gsMatrix< T > &points, gsMatrix< T > &result, const T accuracy=1e-6, const bool useInitialPoint=false) const
gsMatrix< T >	laplacian (const gsMatrix< T > &u) const
	Evaluate the Laplacian at points u. More...
gsMatrix< T > *	mderiv (const gsMatrix< T > &u, const index_t k, const index_t j) const
	Mixed derivative wrt variables k and j.
int	newtonRaphson (const gsVector< T > &value, gsVector< T > &arg, bool withSupport=true, const T accuracy=1e-6, int max_loop=100, T damping_factor=1) const
virtual index_t	nPieces () const
	Number of pieces in the domain of definition.
virtual gsMatrix< T >	parameterCenter () const
	Returns a "central" point inside inside the parameter domain.
gsMatrix< T >	parameterCenter (const boxCorner &bc) const
	Get coordinates of the boxCorner bc in the parameter domain.
gsMatrix< T >	parameterCenter (const boxSide &bs) const
	Get coordinates of the midpoint of the boxSide bs in the parameter domain.
virtual const gsFunctionExpr &	piece (const index_t) const
	Returns the piece(s) of the function(s) at subdomain k.
std::ostream &	print (std::ostream &os) const
	Prints the object as a string.
void	recoverPoints (gsMatrix< T > &xyz, gsMatrix< T > &uv, index_t k, const T accuracy=1e-6) const
void	set_t (T const &t) const
	Sets the symbol "t" to a value.
void	set_u (T const &v) const
	Sets the symbol "u" to a value.
void	set_v (T const &v) const
	Sets the symbol "v" to a value.
void	set_w (T const &v) const
	Sets the symbol "w" to a value.
void	set_x (T const &v) const
	Sets the symbol "x" to a value.
void	set_y (T const &v) const
	Sets the symbol "y" to a value.
void	set_z (T const &v) const
	Sets the symbol "z" to a value.
index_t	size () const
	size More...
short_t	targetDim () const
	Dimension of the target space. More...
Evaluation functions
These functions allow one to evaluate the function as well as its derivatives at one or more points in the parameter space. See also Evaluation members.
virtual void	jacobian_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Computes for each point u a block of result containing the Jacobian matrix.
void	div_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
	Computes for each point u a block of result containing the divergence matrix.
gsMatrix< T >	jacobian (const gsMatrix< T > &u) const
virtual void	hessian_into (const gsMatrix< T > &u, gsMatrix< T > &result, index_t coord=0) const
virtual gsMatrix< T >	hessian (const gsMatrix< T > &u, index_t coord=0) const

Static Public Member Functions
static uPtr	make (const std::string &expression_string, short_t ddim)
	Make function taking an expression string and the domain dimension (scalar function)
static uPtr	make (const std::string &expression_string1, const std::string &expression_string2, short_t ddim)
	Make function taking two expression strings (2D vector valued function)
static uPtr	make (const std::string &expression_string1, const std::string &expression_string2, const std::string &expression_string3, short_t ddim)
	Make function taking two expression strings (3D vector valued function)
static uPtr	make (const std::string &expression_string1, const std::string &expression_string2, const std::string &expression_string3, const std::string &expression_string4, short_t ddim)
	Make function taking four expression strings (4D vector valued function) used for matrix coefficients.
static uPtr	make (const std::string &expression_string1, const std::string &expression_string2, const std::string &expression_string3, const std::string &expression_string4, const std::string &expression_string5, const std::string &expression_string6, const std::string &expression_string7, const std::string &expression_string8, const std::string &expression_string9, short_t ddim)
	Make function taking nine expression strings (9D vector valued function) used for (3x3) matrix coefficients.

Member Function Documentation

gsMatrix<index_t> active ( const gsMatrix< T > &  u ) const

inlineinherited

Returns the indices of active (nonzero) functions at points u, as a list of indices.

See Also

active_into()

void active_into ( const gsMatrix< T > &  u, gsMatrix< index_t > &  result  ) const

inlinevirtualinherited

Indices of active (non-zero) function(s) for each point.

The columns are sorted in increasing order, if on a point there are less active then the number of rows in the result matrix (some other point has more actives) then the rest of the column is filled with 0s.

Parameters

u
result

Reimplemented from gsFunctionSet< T >.

void compute ( const gsMatrix< T > &  in, gsFuncData< T > &  out  ) const

virtualinherited

Computes function data.

This function evaluates the functions and their derivatives at the points in and writes them in the corresponding fields of out. Which field to write (and what to compute) is controlled by the out.flags (see also gsFuncData).

The input points in are expected to be compatible with the implementation/representation of the function, i.e. they should be points inside the domain of definitition of the function

Parameters

[in]	in
[out]	out

Reimplemented in gsGeometry< T >, and gsConstantFunction< T >.

void computeMap ( gsMapData< T > &  InOut ) const

virtualinherited

Computes map function data.

This function evaluates the functions and their derivatives at the points InOut.points and writes them in the corresponding fields of InOut. Which field to write (and what to compute) is controlled by the InOut.flags (see also gsMapData). This is intended for parametrizations only and it works on functions sets of cardinality 1 only.

Parameters

[in,out]

InOut

gsMatrix< T > deriv ( const gsMatrix< T > &  u ) const

inherited

Evaluate the derivatives,.

See Also

deriv_into()

gsMatrix< T > deriv2 ( const gsMatrix< T > &  u ) const

inherited

Evaluates the second derivatives of active (i.e., non-zero) functions at points u.

See documentation for deriv2_into() (the one without input parameter coefs) for details.

See Also

deriv2_into()

Parameters

[in]

u

Evaluation points in columns.

Returns

For every column of u, a column containing the second derivatives. See documentation for deriv2_into() (the one without input parameter coefs) for details.

void deriv2_into ( const gsMatrix< T > &  u, gsMatrix< T > &  result  ) const

virtual

Evaluate second derivatives of the function at points u into result.

Let n be the dimension of the source space ( n = domainDim() ).
Let m be the dimension of the image/target space ( m = targetDim() ).
Let N denote the number of evaluation points.

Parameters

[in]	u	gsMatrix of size n x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (S*m) x N, where S=n*(n+1)/2. Each column in result corresponds to one point (i.e., one column in u) and contains the following values (for n=3, m=3): \( (\partial_{xx} f^{(1)}, \partial_{yy} f^{(1)}, \partial_{zz} f^{(1)}, \partial_{xy} f^{(1)}, \partial_{xz} f^{(1)}, \partial_{yz} f^{(1)}, \partial_{xx} f^{(2)},\ldots,\partial_{yz} f^{(3)} )^T\)

Warning

By default uses central finite differences with h=0.00001! One must override this function in derived classes to get proper results.

Reimplemented from gsFunction< T >.

void deriv_into ( const gsMatrix< T > &  u, gsMatrix< T > &  result  ) const

virtual

Evaluate derivatives of the function \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\) at points u into result.

Let n be the dimension of the source space ( n = domainDim() ).
Let m be the dimension of the image/target space ( m = targetDim() ).
Let N denote the number of evaluation points.

Let \( f:\mathbb R^2 \rightarrow \mathbb R^3 \), i.e., \( f(x,y) = ( f^{(1)}(x,y), f^{(2)}(x,y), f^{(3)}(x,y) )^T\),
and let \( u = ( u_1, \ldots, u_N) = ( (x_1,y_1)^T, \ldots, (x_N, y_N)^T )\).
Then, result is of the form

\[ \left[ \begin{array}{cccc} \partial_x f^{(1)}(u_1) & \partial_x f^{(1)}(u_2) & \ldots & \partial_x f^{(1)}(u_N) \\ \partial_y f^{(1)}(u_1) & \partial_y f^{(1)}(u_2) & \ldots & \partial_y f^{(1)}(u_N) \\ \partial_x f^{(2)}(u_1) & \partial_x f^{(2)}(u_2) & \ldots & \partial_x f^{(2)}(u_N) \\ \partial_y f^{(2)}(u_1) & \partial_y f^{(2)}(u_2) & \ldots & \partial_x f^{(2)}(u_N) \\ \partial_x f^{(3)}(u_1) & \partial_x f^{(3)}(u_2) & \ldots & \partial_x f^{(3)}(u_N)\\ \partial_y f^{(3)}(u_1) & \partial_y f^{(3)}(u_2) & \ldots & \partial_y f^{(3)}(u_N) \end{array} \right] \]

Parameters

[in]	u	gsMatrix of size n x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size (n * m) x N. Each row of result corresponds to one component in the target space and contains the gradients for each evaluation point, as row vectors, one after the other (see above for details on the format).

Warning

By default, gsFunction uses central finite differences with h=0.00001! One must override this function in derived classes to get proper results.

Reimplemented from gsFunction< T >.

short_t domainDim ( ) const

virtual

Dimension of the (source) domain.

Returns

For \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\), returns \(n\).

Implements gsFunctionSet< T >.

gsMatrix< T > eval ( const gsMatrix< T > &  u ) const

inherited

Evaluate the function,.

See Also

eval_into()

void eval_into ( const gsMatrix< T > &  u, gsMatrix< T > &  result  ) const

virtual

Evaluate the function at points u into result.

Let n be the dimension of the source space ( n = domainDim() ).
Let m be the dimension of the image/target space ( m = targetDim() ).
Let N denote the number of evaluation points.

Parameters

[in]	u	gsMatrix of size n x N, where each column of u represents one evaluation point.
[out]	result	gsMatrix of size m x N, where each column of u represents the result of the function at the respective valuation point.

Implements gsFunction< T >.

std::vector< gsMatrix< T > > evalAllDers ( const gsMatrix< T > &  u, int  n, bool  sameElement = false  ) const

inherited

Evaluate all derivatives upto order n,.

See Also

evalAllDers_into

void evalAllDers_into ( const gsMatrix< T > &  u, int  n, std::vector< gsMatrix< T > > &  result, bool  sameElement = false  ) const

virtualinherited

Evaluate the nonzero functions and their derivatives up to order n at points u into result.

The derivatives (the 0-th derivative is the function value) are stored in a result. result is a std::vector, where result[i] is a gsMatrix which contains the i-th derivatives.

The entries in result[0], result[1], and result[2] are ordered as in eval_into(), deriv_into(), and deriv2_into(), respectively. For i > 2, the derivatives are stored in lexicographical order, e.g. for order i = 3 and dimension 2 the derivatives are stored as follows: \( \partial_{xxx}, \, \partial_{xxy}, \, \partial_{xyy}, \, \partial_{yyy}.\, \)

Parameters

[in]	u	Evaluation points, each column corresponds to one evaluation point.
[in]	n	All derivatives up to order n are computed and stored in result.
[in,out]	result	See above for format.

Reimplemented in gsTensorBSplineBasis< 1, T >, gsTensorBasis< d, T >, gsTensorBasis< 1, T >, gsGeometry< T >, gsMappedSingleBasis< d, T >, gsConstantFunction< T >, gsTHBSplineBasis< d, T >, gsMappedSingleSpline< d, T >, and gsSquaredDistance< T >.

virtual gsMatrix<T> hessian ( const gsMatrix< T > &  u, index_t  coord = 0  ) const

inlinevirtualinherited

Evaluates the Hessian (matrix of second partial derivatives) of coordinate coord at points u.

void invertPoints ( const gsMatrix< T > &  points, gsMatrix< T > &  result, const T  accuracy = 1e-6, const bool  useInitialPoint = false  ) const

virtualinherited

Takes the physical points and computes the corresponding parameter values. If the point cannot be inverted (eg. is not part of the geometry) the corresponding parameter values will be undefined

gsMatrix< T > laplacian ( const gsMatrix< T > &  u ) const

virtual

Evaluate the Laplacian at points u.

By default uses central finite differences with h=0.00001

Reimplemented from gsFunction< T >.

int newtonRaphson ( const gsVector< T > &  value, gsVector< T > &  arg, bool  withSupport = true, const T  accuracy = 1e-6, int  max_loop = 100, T  damping_factor = 1  ) const

inherited

Newton-Raphson method to find a solution of the equation f(arg) = value with starting vector arg. If the point cannot be inverted the corresponding parameter values will be undefined

void recoverPoints ( gsMatrix< T > &  xyz, gsMatrix< T > &  uv, index_t  k, const T  accuracy = 1e-6  ) const

inherited

Recovers a point on the (geometry) together with its parameters uv, assuming that the k-th coordinate of the point xyz is not known (and has a random value as input argument).

index_t size ( ) const

inlinevirtualinherited

size

Warning

gsFunction and gsGeometry have size() == 1. This should not be confused with the size eg. of gsGeometry::basis(), which is the number of basis functions in the basis

Returns

the size of the function set: the total number of functions

Reimplemented from gsFunctionSet< T >.

Reimplemented in gsPiecewiseFunction< T >.

short_t targetDim ( ) const

virtual

Dimension of the target space.

Returns

For \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\), returns \(m\).

Reimplemented from gsFunctionSet< T >.