# solver(2rheolef) [debian man page]

```solver(2rheolef)						    rheolef-6.1 						  solver(2rheolef)

NAME
solver - direct or interative solver interface

DESCRIPTION
The class implements a matrix factorization:
LU factorization for an unsymmetric matrix and
Choleski fatorisation for a symmetric one.

Let a be a square invertible matrix in
csr format (see csr(2)).

csr<Float> a;

We get the factorization by:

solver<Float> sa (a);

Each call to the direct solver for a*x = b writes either:

vec<Float> x = sa.solve(b);

When the matrix is modified in a computation loop but
conserves its sparsity pattern, an efficient re-factorization
writes:

sa.update_values (new_a);
x = sa.solve(b);

This approach skip the long step of the symbolic factization step.

ITERATIVE SOLVER
The factorization can also be incomplete, i.e. a pseudo-inverse,
suitable for preconditionning iterative methods.
In that case, the sa.solve(b) call runs a conjugate gradient
when the matrix is symmetric, or a generalized minimum residual
algorithm when the matrix is unsymmetric.

AUTOMATIC CHOICE AND CUSTOMIZATION
The symmetry of the matrix is tested via the a.is_symmetric() property
(see csr(2)) while the choice between direct or iterative solver
is switched from the a.pattern_dimension() value. When the pattern
is 3D, an iterative method is faster and less memory consuming.
Otherwhise, for 1D or 2D problems, the direct method is prefered.

These default choices can be supersetted by using explicit options:

solver_option_type opt;
opt.iterative = true;
solver<Float> sa (a, opt);

See the solver.h header for the complete list of available options.

IMPLEMENTATION NOTE
The implementation bases on the pastix library.

IMPLEMENTATION
template <class T, class M = rheo_default_memory_model>
class solver_basic : public smart_pointer<solver_rep<T,M> > {
public:
// typedefs:

typedef solver_rep<T,M>    rep;
typedef smart_pointer<rep> base;

// allocator:

solver_basic ();
explicit solver_basic (const csr<T,M>& a, const solver_option_type& opt = solver_option_type());
void update_values (const csr<T,M>& a);

// accessors:

vec<T,M> trans_solve (const vec<T,M>& b) const;
vec<T,M> solve       (const vec<T,M>& b) const;
};
// factorizations:
template <class T, class M>
solver_basic<T,M> ldlt(const csr<T,M>& a, const solver_option_type& opt = solver_option_type());
template <class T, class M>
solver_basic<T,M> lu  (const csr<T,M>& a, const solver_option_type& opt = solver_option_type());
template <class T, class M>
solver_basic<T,M> ic0 (const csr<T,M>& a, const solver_option_type& opt = solver_option_type());
template <class T, class M>
solver_basic<T,M> ilu0(const csr<T,M>& a, const solver_option_type& opt = solver_option_type());

typedef solver_basic<Float> solver;

csr(2), csr(2)

rheolef-6.1							    rheolef-6.1 						  solver(2rheolef)```

## Check Out this Related Man Page

```form(2rheolef)							    rheolef-6.1 						    form(2rheolef)

NAME
form - representation of a finite element bilinear form

DESCRIPTION
The form class groups four sparse matrix, associated to a bilinear form on two finite element spaces:

a: U*V   ----> IR
(u,v)  |---> a(u,v)

The operator A associated to the bilinear form is defined by:

A: U  ----> V'
u  |---> A(u)

where  u and v are fields (see field(2)), and A(u) is such that a(u,v)=<A(u),v> for all u in U and v in V and where <.,.> denotes the dual-
ity product between V and V'.  Since V is a finite dimensional spaces, the duality product is the euclidian product in IR^dim(V).

Since both U and V are finite dimensional spaces, the linear operator can be represented by a matrix.  The form	class  is  represented	by
four  sparse  matrix in csr format (see csr(2)), associated to unknown and blocked degrees of freedom of origin and destination spaces (see
space(2)).

EXAMPLE
The operator A associated to a bilinear form a(.,.) by the relation (Au,v) = a(u,v) could be applied by using a sample matrix notation A*u,
as shown by the following code:

geo omega("square");
space V (omega,"P1");
field uh = interpolate (fct, V);
field vh = a*uh;
cout << v;

The form-field vh=a*uh operation is equivalent to the following matrix-vector operations:

vh.set_u() = a.uu()*uh.u() + a.ub()*uh.b();
vh.set_b() = a.bu()*uh.u() + a.bb()*uh.b();

ALGABRA
Forms, as matrices (see csr(2)), support linear algebra:
Adding or substracting two forms writes a+b and a-b, respectively,
and multiplying a form by a field uh writes a*uh.
Thus, any linear combination of forms is available.

IMPLEMENTATION
template<class T, class M>
class form_basic {
public :
// typedefs:

typedef typename csr<T,M>::size_type    size_type;
typedef T				   value_type;
typedef typename scalar_traits<T>::type float_type;
typedef geo_basic<float_type,M>	   geo_type;
typedef space_basic<float_type,M>	   space_type;

// allocator/deallocator:

form_basic ();
form_basic (const form_basic<T,M>&);

form_basic (const space_type& X, const space_type& Y,
const std::string& name = "",

form_basic (const space_type& X, const space_type& Y,
const std::string& name,
const geo_basic<T,M>& gamma,

form_basic (const space_type& X, const space_type& Y,
const std::string& name,
const field_basic<T,M>& weight,

form_basic (const space_type& X, const space_type& Y,
const std::string& name,
const band_basic<T,M>& bh,

// allocators from initializer list (c++ 2011):

#ifdef _RHEOLEF_HAVE_STD_INITIALIZER_LIST
form_basic (const std::initializer_list<form_concat_value<T,M> >& init_list);
form_basic (const std::initializer_list<form_concat_line <T,M> >& init_list);
#endif // _RHEOLEF_HAVE_STD_INITIALIZER_LIST

// accessors:

const space_type& get_first_space() const;
const space_type& get_second_space() const;
const geo_type&   get_geo() const;

const communicator& comm() const;

// linear algebra:

form_basic<T,M>  operator+  (const form_basic<T,M>& b) const;
form_basic<T,M>  operator-  (const form_basic<T,M>& b) const;
form_basic<T,M>& operator*= (const T& lambda);
field_basic<T,M> operator*  (const field_basic<T,M>& xh) const;
#ifdef TO_CLEAN
template <class Expr>
field_basic<T,M> operator*  (const field_expr<Expr>& xh) const;
#endif // TO_CLEAN
field_basic<T,M> trans_mult (const field_basic<T,M>& yh) const;

float_type operator () (const field_basic<T,M>& uh, const field_basic<T,M>& vh) const;

// io:

odiststream& put (odiststream& ops, bool show_partition = true) const;
void dump (std::string name) const;

// accessors & modifiers to unknown & blocked parts:

const csr<T,M>&     uu() const { return _uu; }
const csr<T,M>&     ub() const { return _ub; }
const csr<T,M>&     bu() const { return _bu; }
const csr<T,M>&     bb() const { return _bb; }
csr<T,M>& set_uu()	  { return _uu; }
csr<T,M>& set_ub()	  { return _ub; }
csr<T,M>& set_bu()	  { return _bu; }
csr<T,M>& set_bb()	  { return _bb; }

// data
protected:
space_type  _X;
space_type  _Y;
csr<T,M>    _uu;
csr<T,M>    _ub;
csr<T,M>    _bu;
csr<T,M>    _bb;

// internals:
void assembly (const form_element<T,M>&   form_e,
const geo_basic<T,M>&      X_geo,
const geo_basic<T,M>&      Y_geo,
bool	X_geo_is_background = true);
void form_init (
const std::string&	  name,
bool			  has_weight,
const field_basic<T,M>& weight,