SparseMatrix< number > Class Template Reference
[Basic matrices]

Inheritance diagram for SparseMatrix< number >:

Inheritance graph
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List of all members.

Classes

class  ExcDifferentSparsityPatterns
class  ExcInvalidIndex
class  ExcInvalidIndex1
class  ExcIteratorRange
class  ExcSourceEqualsDestination
struct  Traits

Public Types

typedef number value_type
typedef numbers::NumberTraits
< number >::real_type 
real_type
typedef
internals::SparseMatrixIterators::Iterator
< number, true
const_iterator
typedef
internals::SparseMatrixIterators::Iterator
< number, false
iterator

Public Member Functions

Constructors and initalization.
 SparseMatrix ()
 SparseMatrix (const SparseMatrix &)
 SparseMatrix (const SparsityPattern &sparsity)
 SparseMatrix (const SparsityPattern &sparsity, const IdentityMatrix &id)
virtual ~SparseMatrix ()
SparseMatrix< number > & operator= (const SparseMatrix< number > &)
SparseMatrix< number > & operator= (const IdentityMatrix &id)
SparseMatrixoperator= (const double d)
virtual void reinit (const SparsityPattern &sparsity)
virtual void clear ()
Information on the matrix
bool empty () const
unsigned int m () const
unsigned int n () const
unsigned int n_nonzero_elements () const
unsigned int n_actually_nonzero_elements (const double threshold=0.) const
const SparsityPatternget_sparsity_pattern () const
unsigned int memory_consumption () const
Modifying entries
void set (const unsigned int i, const unsigned int j, const number value)
template<typename number2 >
void set (const std::vector< unsigned int > &indices, const FullMatrix< number2 > &full_matrix, const bool elide_zero_values=false)
template<typename number2 >
void set (const std::vector< unsigned int > &row_indices, const std::vector< unsigned int > &col_indices, const FullMatrix< number2 > &full_matrix, const bool elide_zero_values=false)
template<typename number2 >
void set (const unsigned int row, const std::vector< unsigned int > &col_indices, const std::vector< number2 > &values, const bool elide_zero_values=false)
template<typename number2 >
void set (const unsigned int row, const unsigned int n_cols, const unsigned int *col_indices, const number2 *values, const bool elide_zero_values=false)
void add (const unsigned int i, const unsigned int j, const number value)
template<typename number2 >
void add (const std::vector< unsigned int > &indices, const FullMatrix< number2 > &full_matrix, const bool elide_zero_values=true)
template<typename number2 >
void add (const std::vector< unsigned int > &row_indices, const std::vector< unsigned int > &col_indices, const FullMatrix< number2 > &full_matrix, const bool elide_zero_values=true)
template<typename number2 >
void add (const unsigned int row, const std::vector< unsigned int > &col_indices, const std::vector< number2 > &values, const bool elide_zero_values=true)
template<typename number2 >
void add (const unsigned int row, const unsigned int n_cols, const unsigned int *col_indices, const number2 *values, const bool elide_zero_values=true, const bool col_indices_are_sorted=false)
SparseMatrixoperator*= (const number factor)
SparseMatrixoperator/= (const number factor)
void symmetrize ()
template<typename somenumber >
SparseMatrix< number > & copy_from (const SparseMatrix< somenumber > &source)
template<typename ForwardIterator >
void copy_from (const ForwardIterator begin, const ForwardIterator end)
template<typename somenumber >
void copy_from (const FullMatrix< somenumber > &matrix)
template<typename somenumber >
void add (const number factor, const SparseMatrix< somenumber > &matrix)
Entry Access
number operator() (const unsigned int i, const unsigned int j) const
number el (const unsigned int i, const unsigned int j) const
number diag_element (const unsigned int i) const
number & diag_element (const unsigned int i)
number raw_entry (const unsigned int row, const unsigned int index) const
number global_entry (const unsigned int i) const
number & global_entry (const unsigned int i)
Matrix vector multiplications
template<class OutVector , class InVector >
void vmult (OutVector &dst, const InVector &src) const
template<class OutVector , class InVector >
void Tvmult (OutVector &dst, const InVector &src) const
template<class OutVector , class InVector >
void vmult_add (OutVector &dst, const InVector &src) const
template<class OutVector , class InVector >
void Tvmult_add (OutVector &dst, const InVector &src) const
template<typename somenumber >
somenumber matrix_norm_square (const Vector< somenumber > &v) const
template<typename somenumber >
somenumber matrix_scalar_product (const Vector< somenumber > &u, const Vector< somenumber > &v) const
template<typename somenumber >
somenumber residual (Vector< somenumber > &dst, const Vector< somenumber > &x, const Vector< somenumber > &b) const
Matrix norms
real_type l1_norm () const
real_type linfty_norm () const
real_type frobenius_norm () const
Preconditioning methods
template<typename somenumber >
void precondition_Jacobi (Vector< somenumber > &dst, const Vector< somenumber > &src, const number omega=1.) const
template<typename somenumber >
void precondition_SSOR (Vector< somenumber > &dst, const Vector< somenumber > &src, const number om=1., const std::vector< unsigned int > &pos_right_of_diagonal=std::vector< unsigned int >()) const
template<typename somenumber >
void precondition_SOR (Vector< somenumber > &dst, const Vector< somenumber > &src, const number om=1.) const
template<typename somenumber >
void precondition_TSOR (Vector< somenumber > &dst, const Vector< somenumber > &src, const number om=1.) const
template<typename somenumber >
void SSOR (Vector< somenumber > &v, const number omega=1.) const
template<typename somenumber >
void SOR (Vector< somenumber > &v, const number om=1.) const
template<typename somenumber >
void TSOR (Vector< somenumber > &v, const number om=1.) const
template<typename somenumber >
void PSOR (Vector< somenumber > &v, const std::vector< unsigned int > &permutation, const std::vector< unsigned int > &inverse_permutation, const number om=1.) const
template<typename somenumber >
void TPSOR (Vector< somenumber > &v, const std::vector< unsigned int > &permutation, const std::vector< unsigned int > &inverse_permutation, const number om=1.) const
template<typename somenumber >
void SOR_step (Vector< somenumber > &v, const Vector< somenumber > &b, const number om=1.) const
template<typename somenumber >
void TSOR_step (Vector< somenumber > &v, const Vector< somenumber > &b, const number om=1.) const
template<typename somenumber >
void SSOR_step (Vector< somenumber > &v, const Vector< somenumber > &b, const number om=1.) const
Iterators
const_iterator begin () const
const_iterator end () const
iterator begin ()
iterator end ()
const_iterator begin (const unsigned int r) const
const_iterator end (const unsigned int r) const
iterator begin (const unsigned int r)
iterator end (const unsigned int r)
Input/Output
void print (std::ostream &out) const
void print_formatted (std::ostream &out, const unsigned int precision=3, const bool scientific=true, const unsigned int width=0, const char *zero_string=" ", const double denominator=1.) const
void print_pattern (std::ostream &out, const double threshold=0.) const
void block_write (std::ostream &out) const
void block_read (std::istream &in)

Private Member Functions

template<class OutVector , class InVector >
void threaded_vmult (OutVector &dst, const InVector &src, const unsigned int begin_row, const unsigned int end_row) const
template<typename somenumber >
void threaded_matrix_norm_square (const Vector< somenumber > &v, const unsigned int begin_row, const unsigned int end_row, somenumber *partial_sum) const
template<typename somenumber >
void threaded_matrix_scalar_product (const Vector< somenumber > &u, const Vector< somenumber > &v, const unsigned int begin_row, const unsigned int end_row, somenumber *partial_sum) const
template<typename somenumber >
void threaded_residual (Vector< somenumber > &dst, const Vector< somenumber > &u, const Vector< somenumber > &b, const std::pair< unsigned int, unsigned int > interval, somenumber *partial_norm) const

Private Attributes

SmartPointer< const
SparsityPattern
cols
number * val
unsigned int max_len

Friends

class SparseMatrix
class SparseLUDecomposition
class SparseILU


Detailed Description

template<typename number>
class SparseMatrix< number >

Sparse matrix. This class implements the function to store values in the locations of a sparse matrix denoted by a SparsityPattern. The separation of sparsity pattern and values is done since one can store data elements of different type in these locations without the SparsityPattern having to know this, and more importantly one can associate more than one matrix with the same sparsity pattern.

Note:
Instantiations for this template are provided for <float> and <double>; others can be generated in application programs (see the section on Template instantiations in the manual).
Author:
Essentially everyone who has ever worked on deal.II, 1994-2007

Member Typedef Documentation

template<typename number>
typedef number SparseMatrix< number >::value_type

Type of matrix entries. In analogy to the STL container classes.

template<typename number>
typedef numbers::NumberTraits<number>::real_type SparseMatrix< number >::real_type

Declare a type that has holds real-valued numbers with the same precision as the template argument to this class. If the template argument of this class is a real data type, then real_type equals the template argument. If the template argument is a std::complex type then real_type equals the type underlying the complex numbers.

This typedef is used to represent the return type of norms.

template<typename number>
typedef internals::SparseMatrixIterators::Iterator<number,true> SparseMatrix< number >::const_iterator

Typedef of an STL conforming iterator class walking over all the nonzero entries of this matrix. This iterator cannot change the values of the matrix.

template<typename number>
typedef internals::SparseMatrixIterators::Iterator<number,false> SparseMatrix< number >::iterator

Typedef of an STL conforming iterator class walking over all the nonzero entries of this matrix. This iterator can change the values of the matrix, but of course can't change the sparsity pattern as this is fixed once a sparse matrix is attached to it.


Constructor & Destructor Documentation

template<typename number>
SparseMatrix< number >::SparseMatrix (  ) 

Constructor; initializes the matrix to be empty, without any structure, i.e. the matrix is not usable at all. This constructor is therefore only useful for matrices which are members of a class. All other matrices should be created at a point in the data flow where all necessary information is available.

You have to initialize the matrix before usage with reinit(const SparsityPattern&).

template<typename number>
SparseMatrix< number >::SparseMatrix ( const SparseMatrix< number > &   ) 

Copy constructor. This constructor is only allowed to be called if the matrix to be copied is empty. This is for the same reason as for the SparsityPattern, see there for the details.

If you really want to copy a whole matrix, you can do so by using the copy_from() function.

template<typename number>
SparseMatrix< number >::SparseMatrix ( const SparsityPattern sparsity  )  [explicit]

Constructor. Takes the given matrix sparsity structure to represent the sparsity pattern of this matrix. You can change the sparsity pattern later on by calling the reinit(const SparsityPattern&) function.

You have to make sure that the lifetime of the sparsity structure is at least as long as that of this matrix or as long as reinit(const SparsityPattern&) is not called with a new sparsity pattern.

The constructor is marked explicit so as to disallow that someone passes a sparsity pattern in place of a sparse matrix to some function, where an empty matrix would be generated then.

template<typename number>
SparseMatrix< number >::SparseMatrix ( const SparsityPattern sparsity,
const IdentityMatrix id 
)

Copy constructor: initialize the matrix with the identity matrix. This constructor will throw an exception if the sizes of the sparsity pattern and the identity matrix do not coincide, or if the sparsity pattern does not provide for nonzero entries on the entire diagonal.

template<typename number>
virtual SparseMatrix< number >::~SparseMatrix (  )  [virtual]

Destructor. Free all memory, but do not release the memory of the sparsity structure.


Member Function Documentation

template<typename number>
SparseMatrix<number>& SparseMatrix< number >::operator= ( const SparseMatrix< number > &   ) 

Copy operator. Since copying entire sparse matrices is a very expensive operation, we disallow doing so except for the special case of empty matrices of size zero. This doesn't seem particularly useful, but is exactly what one needs if one wanted to have a std::vector<SparseMatrix<double> >: in that case, one can create a vector (which needs the ability to copy objects) of empty matrices that are then later filled with something useful.

template<typename number>
SparseMatrix<number>& SparseMatrix< number >::operator= ( const IdentityMatrix id  ) 

Copy operator: initialize the matrix with the identity matrix. This operator will throw an exception if the sizes of the sparsity pattern and the identity matrix do not coincide, or if the sparsity pattern does not provide for nonzero entries on the entire diagonal.

template<typename number>
SparseMatrix& SparseMatrix< number >::operator= ( const double  d  ) 

This operator assigns a scalar to a matrix. Since this does usually not make much sense (should we set all matrix entries to this value? Only the nonzero entries of the sparsity pattern?), this operation is only allowed if the actual value to be assigned is zero. This operator only exists to allow for the obvious notation matrix=0, which sets all elements of the matrix to zero, but keep the sparsity pattern previously used.

template<typename number>
virtual void SparseMatrix< number >::reinit ( const SparsityPattern sparsity  )  [virtual]

Reinitialize the sparse matrix with the given sparsity pattern. The latter tells the matrix how many nonzero elements there need to be reserved.

Regarding memory allocation, the same applies as said above.

You have to make sure that the lifetime of the sparsity structure is at least as long as that of this matrix or as long as reinit(const SparsityPattern &) is not called with a new sparsity structure.

The elements of the matrix are set to zero by this function.

Reimplemented in SparseLUDecomposition< number >, SparseMIC< number >, and SparseLUDecomposition< number >.

template<typename number>
virtual void SparseMatrix< number >::clear (  )  [virtual]

Release all memory and return to a state just like after having called the default constructor. It also forgets the sparsity pattern it was previously tied to.

Reimplemented in SparseLUDecomposition< number >, SparseMIC< number >, and SparseLUDecomposition< number >.

template<typename number>
bool SparseMatrix< number >::empty (  )  const

Return whether the object is empty. It is empty if either both dimensions are zero or no SparsityPattern is associated.

Reimplemented in SparseLUDecomposition< number >, and SparseLUDecomposition< number >.

template<typename number>
unsigned int SparseMatrix< number >::m (  )  const

Return the dimension of the image space. To remember: the matrix is of dimension $m \times n$.

template<typename number>
unsigned int SparseMatrix< number >::n (  )  const

Return the dimension of the range space. To remember: the matrix is of dimension $m \times n$.

template<typename number>
unsigned int SparseMatrix< number >::n_nonzero_elements (  )  const

Return the number of nonzero elements of this matrix. Actually, it returns the number of entries in the sparsity pattern; if any of the entries should happen to be zero, it is counted anyway.

template<typename number>
unsigned int SparseMatrix< number >::n_actually_nonzero_elements ( const double  threshold = 0.  )  const

Return the number of actually nonzero elements of this matrix. It is possible to specify the parameter threshold in order to count only the elements that have absolute value greater than the threshold.

Note, that this function does (in contrary to n_nonzero_elements()) not count all entries of the sparsity pattern but only the ones that are nonzero (or whose absolute value is greater than threshold).

template<typename number>
const SparsityPattern& SparseMatrix< number >::get_sparsity_pattern (  )  const

Return a (constant) reference to the underlying sparsity pattern of this matrix.

Though the return value is declared const, you should be aware that it may change if you call any nonconstant function of objects which operate on it.

template<typename number>
unsigned int SparseMatrix< number >::memory_consumption (  )  const

Determine an estimate for the memory consumption (in bytes) of this object. See MemoryConsumption.

Reimplemented in SparseLUDecomposition< number >, SparseILU< number >, SparseMIC< number >, and SparseLUDecomposition< number >.

template<typename number>
void SparseMatrix< number >::set ( const unsigned int  i,
const unsigned int  j,
const number  value 
)

Set the element (i,j) to value. Throws an error if the entry does not exist or if value is not a finite number. Still, it is allowed to store zero values in non-existent fields.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::set ( const std::vector< unsigned int > &  indices,
const FullMatrix< number2 > &  full_matrix,
const bool  elide_zero_values = false 
) [inline]

Set all elements given in a FullMatrix into the sparse matrix locations given by indices. In other words, this function writes the elements in full_matrix into the calling matrix, using the local-to-global indexing specified by indices for both the rows and the columns of the matrix. This function assumes a quadratic sparse matrix and a quadratic full_matrix, the usual situation in FE calculations.

The optional parameter elide_zero_values can be used to specify whether zero values should be set anyway or they should be filtered away (and not change the previous content in the respective element if it exists). The default value is false, i.e., even zero values are treated.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::set ( const std::vector< unsigned int > &  row_indices,
const std::vector< unsigned int > &  col_indices,
const FullMatrix< number2 > &  full_matrix,
const bool  elide_zero_values = false 
) [inline]

Same function as before, but now including the possibility to use rectangular full_matrices and different local-to-global indexing on rows and columns, respectively.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::set ( const unsigned int  row,
const std::vector< unsigned int > &  col_indices,
const std::vector< number2 > &  values,
const bool  elide_zero_values = false 
) [inline]

Set several elements in the specified row of the matrix with column indices as given by col_indices to the respective value.

The optional parameter elide_zero_values can be used to specify whether zero values should be set anyway or they should be filtered away (and not change the previous content in the respective element if it exists). The default value is false, i.e., even zero values are treated.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::set ( const unsigned int  row,
const unsigned int  n_cols,
const unsigned int col_indices,
const number2 *  values,
const bool  elide_zero_values = false 
) [inline]

Set several elements to values given by values in a given row in columns given by col_indices into the sparse matrix.

The optional parameter elide_zero_values can be used to specify whether zero values should be inserted anyway or they should be filtered away. The default value is false, i.e., even zero values are inserted/replaced.

template<typename number>
void SparseMatrix< number >::add ( const unsigned int  i,
const unsigned int  j,
const number  value 
)

Add value to the element (i,j). Throws an error if the entry does not exist or if value is not a finite number. Still, it is allowed to store zero values in non-existent fields.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::add ( const std::vector< unsigned int > &  indices,
const FullMatrix< number2 > &  full_matrix,
const bool  elide_zero_values = true 
) [inline]

Add all elements given in a FullMatrix<double> into sparse matrix locations given by indices. In other words, this function adds the elements in full_matrix to the respective entries in calling matrix, using the local-to-global indexing specified by indices for both the rows and the columns of the matrix. This function assumes a quadratic sparse matrix and a quadratic full_matrix, the usual situation in FE calculations.

The optional parameter elide_zero_values can be used to specify whether zero values should be added anyway or these should be filtered away and only non-zero data is added. The default value is true, i.e., zero values won't be added into the matrix.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::add ( const std::vector< unsigned int > &  row_indices,
const std::vector< unsigned int > &  col_indices,
const FullMatrix< number2 > &  full_matrix,
const bool  elide_zero_values = true 
) [inline]

Same function as before, but now including the possibility to use rectangular full_matrices and different local-to-global indexing on rows and columns, respectively.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::add ( const unsigned int  row,
const std::vector< unsigned int > &  col_indices,
const std::vector< number2 > &  values,
const bool  elide_zero_values = true 
) [inline]

Set several elements in the specified row of the matrix with column indices as given by col_indices to the respective value.

The optional parameter elide_zero_values can be used to specify whether zero values should be added anyway or these should be filtered away and only non-zero data is added. The default value is true, i.e., zero values won't be added into the matrix.

template<typename number>
template<typename number2 >
void SparseMatrix< number >::add ( const unsigned int  row,
const unsigned int  n_cols,
const unsigned int col_indices,
const number2 *  values,
const bool  elide_zero_values = true,
const bool  col_indices_are_sorted = false 
) [inline]

Add an array of values given by values in the given global matrix row at columns specified by col_indices in the sparse matrix.

The optional parameter elide_zero_values can be used to specify whether zero values should be added anyway or these should be filtered away and only non-zero data is added. The default value is true, i.e., zero values won't be added into the matrix.

template<typename number>
SparseMatrix& SparseMatrix< number >::operator*= ( const number  factor  ) 

Multiply the entire matrix by a fixed factor.

template<typename number>
SparseMatrix& SparseMatrix< number >::operator/= ( const number  factor  ) 

Divide the entire matrix by a fixed factor.

template<typename number>
void SparseMatrix< number >::symmetrize (  ) 

Symmetrize the matrix by forming the mean value between the existing matrix and its transpose, $A = \frac 12(A+A^T)$.

This operation assumes that the underlying sparsity pattern represents a symmetric object. If this is not the case, then the result of this operation will not be a symmetric matrix, since it only explicitly symmetrizes by looping over the lower left triangular part for efficiency reasons; if there are entries in the upper right triangle, then these elements are missed in the symmetrization. Symmetrization of the sparsity pattern can be obtain by SparsityPattern::symmetrize().

template<typename number>
template<typename somenumber >
SparseMatrix<number>& SparseMatrix< number >::copy_from ( const SparseMatrix< somenumber > &  source  )  [inline]

Copy the given matrix to this one. The operation throws an error if the sparsity patterns of the two involved matrices do not point to the same object, since in this case the copy operation is cheaper. Since this operation is notheless not for free, we do not make it available through operator =, since this may lead to unwanted usage, e.g. in copy arguments to functions, which should really be arguments by reference.

The source matrix may be a matrix of arbitrary type, as long as its data type is convertible to the data type of this matrix.

The function returns a reference to *this.

Reimplemented in SparseLUDecomposition< number >, and SparseLUDecomposition< number >.

template<typename number>
template<typename ForwardIterator >
void SparseMatrix< number >::copy_from ( const ForwardIterator  begin,
const ForwardIterator  end 
) [inline]

This function is complete analogous to the SparsityPattern::copy_from() function in that it allows to initialize a whole matrix in one step. See there for more information on argument types and their meaning. You can also find a small example on how to use this function there.

The only difference to the cited function is that the objects which the inner iterator points to need to be of type std::pair<unsigned int, value, where value needs to be convertible to the element type of this class, as specified by the number template argument.

Previous content of the matrix is overwritten. Note that the entries specified by the input parameters need not necessarily cover all elements of the matrix. Elements not covered remain untouched.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::copy_from ( const FullMatrix< somenumber > &  matrix  )  [inline]

Copy the nonzero entries of a full matrix into this object. Previous content is deleted. Note that the underlying sparsity pattern must be appropriate to hold the nonzero entries of the full matrix.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::add ( const number  factor,
const SparseMatrix< somenumber > &  matrix 
) [inline]

Add matrix scaled by factor to this matrix, i.e. the matrix factor*matrix is added to this. This function throws an error if the sparsity patterns of the two involved matrices do not point to the same object, since in this case the operation is cheaper.

The source matrix may be a sparse matrix over an arbitrary underlying scalar type, as long as its data type is convertible to the data type of this matrix.

template<typename number>
number SparseMatrix< number >::operator() ( const unsigned int  i,
const unsigned int  j 
) const

Return the value of the entry (i,j). This may be an expensive operation and you should always take care where to call this function. In order to avoid abuse, this function throws an exception if the required element does not exist in the matrix.

In case you want a function that returns zero instead (for entries that are not in the sparsity pattern of the matrix), use the el() function.

If you are looping over all elements, consider using one of the iterator classes instead, since they are tailored better to a sparse matrix structure.

template<typename number>
number SparseMatrix< number >::el ( const unsigned int  i,
const unsigned int  j 
) const

This function is mostly like operator()() in that it returns the value of the matrix entry (i,j). The only difference is that if this entry does not exist in the sparsity pattern, then instead of raising an exception, zero is returned. While this may be convenient in some cases, note that it is simple to write algorithms that are slow compared to an optimal solution, since the sparsity of the matrix is not used.

If you are looping over all elements, consider using one of the iterator classes instead, since they are tailored better to a sparse matrix structure.

Referenced by MatrixOut::get_element().

template<typename number>
number SparseMatrix< number >::diag_element ( const unsigned int  i  )  const

Return the main diagonal element in the ith row. This function throws an error if the matrix is not quadratic (see SparsityPattern::optimize_diagonal()).

This function is considerably faster than the operator()(), since for quadratic matrices, the diagonal entry may be the first to be stored in each row and access therefore does not involve searching for the right column number.

template<typename number>
number& SparseMatrix< number >::diag_element ( const unsigned int  i  ) 

Same as above, but return a writeable reference. You're sure you know what you do?

template<typename number>
number SparseMatrix< number >::raw_entry ( const unsigned int  row,
const unsigned int  index 
) const

Access to values in internal mode. Returns the value of the indexth entry in row. Here, index refers to the internal representation of the matrix, not the column. Be sure to understand what you are doing here.

Deprecated:
Use iterator or const_iterator instead!

template<typename number>
number SparseMatrix< number >::global_entry ( const unsigned int  i  )  const

template<typename number>
number& SparseMatrix< number >::global_entry ( const unsigned int  i  ) 

template<typename number>
template<class OutVector , class InVector >
void SparseMatrix< number >::vmult ( OutVector &  dst,
const InVector &  src 
) const [inline]

Matrix-vector multiplication: let dst = M*src with M being this matrix.

Note that while this function can operate on all vectors that offer iterator classes, it is only really effective for objects of type Vector. For all classes for which iterating over elements, or random member access is expensive, this function is not efficient. In particular, if you want to multiply with BlockVector objects, you should consider using a BlockSparseMatrix as well.

Source and destination must not be the same vector.

template<typename number>
template<class OutVector , class InVector >
void SparseMatrix< number >::Tvmult ( OutVector &  dst,
const InVector &  src 
) const [inline]

Matrix-vector multiplication: let dst = MT*src with M being this matrix. This function does the same as vmult() but takes the transposed matrix.

Note that while this function can operate on all vectors that offer iterator classes, it is only really effective for objects of type Vector. For all classes for which iterating over elements, or random member access is expensive, this function is not efficient. In particular, if you want to multiply with BlockVector objects, you should consider using a BlockSparseMatrix as well.

Source and destination must not be the same vector.

template<typename number>
template<class OutVector , class InVector >
void SparseMatrix< number >::vmult_add ( OutVector &  dst,
const InVector &  src 
) const [inline]

Adding Matrix-vector multiplication. Add M*src on dst with M being this matrix.

Note that while this function can operate on all vectors that offer iterator classes, it is only really effective for objects of type Vector. For all classes for which iterating over elements, or random member access is expensive, this function is not efficient. In particular, if you want to multiply with BlockVector objects, you should consider using a BlockSparseMatrix as well.

Source and destination must not be the same vector.

template<typename number>
template<class OutVector , class InVector >
void SparseMatrix< number >::Tvmult_add ( OutVector &  dst,
const InVector &  src 
) const [inline]

Adding Matrix-vector multiplication. Add MT*src to dst with M being this matrix. This function does the same as vmult_add() but takes the transposed matrix.

Note that while this function can operate on all vectors that offer iterator classes, it is only really effective for objects of type Vector. For all classes for which iterating over elements, or random member access is expensive, this function is not efficient. In particular, if you want to multiply with BlockVector objects, you should consider using a BlockSparseMatrix as well.

Source and destination must not be the same vector.

template<typename number>
template<typename somenumber >
somenumber SparseMatrix< number >::matrix_norm_square ( const Vector< somenumber > &  v  )  const [inline]

Return the square of the norm of the vector $v$ with respect to the norm induced by this matrix, i.e. $\left(v,Mv\right)$. This is useful, e.g. in the finite element context, where the $L_2$ norm of a function equals the matrix norm with respect to the mass matrix of the vector representing the nodal values of the finite element function.

Obviously, the matrix needs to be quadratic for this operation, and for the result to actually be a norm it also needs to be either real symmetric or complex hermitian.

The underlying template types of both this matrix and the given vector should either both be real or complex-valued, but not mixed, for this function to make sense.

template<typename number>
template<typename somenumber >
somenumber SparseMatrix< number >::matrix_scalar_product ( const Vector< somenumber > &  u,
const Vector< somenumber > &  v 
) const [inline]

Compute the matrix scalar product $\left(u,Mv\right)$.

template<typename number>
template<typename somenumber >
somenumber SparseMatrix< number >::residual ( Vector< somenumber > &  dst,
const Vector< somenumber > &  x,
const Vector< somenumber > &  b 
) const [inline]

Compute the residual of an equation Mx=b, where the residual is defined to be r=b-Mx. Write the residual into dst. The l2 norm of the residual vector is returned.

Source x and destination dst must not be the same vector.

template<typename number>
real_type SparseMatrix< number >::l1_norm (  )  const

Return the l1-norm of the matrix, that is $|M|_1=max_{all columns j}\sum_{all rows i} |M_ij|$, (max. sum of columns). This is the natural matrix norm that is compatible to the l1-norm for vectors, i.e. $|Mv|_1\leq |M|_1 |v|_1$. (cf. Haemmerlin-Hoffmann : Numerische Mathematik)

template<typename number>
real_type SparseMatrix< number >::linfty_norm (  )  const

Return the linfty-norm of the matrix, that is $|M|_infty=max_{all rows i}\sum_{all columns j} |M_ij|$, (max. sum of rows). This is the natural matrix norm that is compatible to the linfty-norm of vectors, i.e. $|Mv|_infty \leq |M|_infty |v|_infty$. (cf. Haemmerlin-Hoffmann : Numerische Mathematik)

template<typename number>
real_type SparseMatrix< number >::frobenius_norm (  )  const

Return the frobenius norm of the matrix, i.e. the square root of the sum of squares of all entries in the matrix.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::precondition_Jacobi ( Vector< somenumber > &  dst,
const Vector< somenumber > &  src,
const number  omega = 1. 
) const [inline]

Apply the Jacobi preconditioner, which multiplies every element of the src vector by the inverse of the respective diagonal element and multiplies the result with the relaxation factor omega.

Referenced by BlockSparseMatrix< number >::precondition_Jacobi().

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::precondition_SSOR ( Vector< somenumber > &  dst,
const Vector< somenumber > &  src,
const number  om = 1.,
const std::vector< unsigned int > &  pos_right_of_diagonal = std::vector< unsigned int >() 
) const [inline]

Apply SSOR preconditioning to src.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::precondition_SOR ( Vector< somenumber > &  dst,
const Vector< somenumber > &  src,
const number  om = 1. 
) const [inline]

Apply SOR preconditioning matrix to src.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::precondition_TSOR ( Vector< somenumber > &  dst,
const Vector< somenumber > &  src,
const number  om = 1. 
) const [inline]

Apply transpose SOR preconditioning matrix to src.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::SSOR ( Vector< somenumber > &  v,
const number  omega = 1. 
) const [inline]

Perform SSOR preconditioning in-place. Apply the preconditioner matrix without copying to a second vector. omega is the relaxation parameter.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::SOR ( Vector< somenumber > &  v,
const number  om = 1. 
) const [inline]

Perform an SOR preconditioning in-place. omega is the relaxation parameter.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::TSOR ( Vector< somenumber > &  v,
const number  om = 1. 
) const [inline]

Perform a transpose SOR preconditioning in-place. omega is the relaxation parameter.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::PSOR ( Vector< somenumber > &  v,
const std::vector< unsigned int > &  permutation,
const std::vector< unsigned int > &  inverse_permutation,
const number  om = 1. 
) const [inline]

Perform a permuted SOR preconditioning in-place.

The standard SOR method is applied in the order prescribed by permutation, that is, first the row permutation[0], then permutation[1] and so on. For efficiency reasons, the permutation as well as its inverse are required.

omega is the relaxation parameter.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::TPSOR ( Vector< somenumber > &  v,
const std::vector< unsigned int > &  permutation,
const std::vector< unsigned int > &  inverse_permutation,
const number  om = 1. 
) const [inline]

Perform a transposed permuted SOR preconditioning in-place.

The transposed SOR method is applied in the order prescribed by permutation, that is, first the row permutation[m()-1], then permutation[m()-2] and so on. For efficiency reasons, the permutation as well as its inverse are required.

omega is the relaxation parameter.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::SOR_step ( Vector< somenumber > &  v,
const Vector< somenumber > &  b,
const number  om = 1. 
) const [inline]

Do one SOR step on v. Performs a direct SOR step with right hand side b.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::TSOR_step ( Vector< somenumber > &  v,
const Vector< somenumber > &  b,
const number  om = 1. 
) const [inline]

Do one adjoint SOR step on v. Performs a direct TSOR step with right hand side b.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::SSOR_step ( Vector< somenumber > &  v,
const Vector< somenumber > &  b,
const number  om = 1. 
) const [inline]

Do one SSOR step on v. Performs a direct SSOR step with right hand side b by performing TSOR after SOR.

template<typename number>
const_iterator SparseMatrix< number >::begin (  )  const

STL-like iterator with the first entry of the matrix. This is the version for constant matrices.

template<typename number>
const_iterator SparseMatrix< number >::end (  )  const

Final iterator. This is the version for constant matrices.

template<typename number>
iterator SparseMatrix< number >::begin (  ) 

STL-like iterator with the first entry of the matrix. This is the version for non-constant matrices.

template<typename number>
iterator SparseMatrix< number >::end (  ) 

Final iterator. This is the version for non-constant matrices.

template<typename number>
const_iterator SparseMatrix< number >::begin ( const unsigned int  r  )  const

STL-like iterator with the first entry of row r. This is the version for constant matrices.

Note that if the given row is empty, i.e. does not contain any nonzero entries, then the iterator returned by this function equals end(r). Note also that the iterator may not be dereferencable in that case.

template<typename number>
const_iterator SparseMatrix< number >::end ( const unsigned int  r  )  const

Final iterator of row r. It points to the first element past the end of line r, or past the end of the entire sparsity pattern. This is the version for constant matrices.

Note that the end iterator is not necessarily dereferencable. This is in particular the case if it is the end iterator for the last row of a matrix.

template<typename number>
iterator SparseMatrix< number >::begin ( const unsigned int  r  ) 

STL-like iterator with the first entry of row r. This is the version for non-constant matrices.

Note that if the given row is empty, i.e. does not contain any nonzero entries, then the iterator returned by this function equals end(r). Note also that the iterator may not be dereferencable in that case.

template<typename number>
iterator SparseMatrix< number >::end ( const unsigned int  r  ) 

Final iterator of row r. It points to the first element past the end of line r, or past the end of the entire sparsity pattern. This is the version for non-constant matrices.

Note that the end iterator is not necessarily dereferencable. This is in particular the case if it is the end iterator for the last row of a matrix.

template<typename number>
void SparseMatrix< number >::print ( std::ostream &  out  )  const

Print the matrix to the given stream, using the format (line,col) value, i.e. one nonzero entry of the matrix per line.

template<typename number>
void SparseMatrix< number >::print_formatted ( std::ostream &  out,
const unsigned int  precision = 3,
const bool  scientific = true,
const unsigned int  width = 0,
const char *  zero_string = " ",
const double  denominator = 1. 
) const

Print the matrix in the usual format, i.e. as a matrix and not as a list of nonzero elements. For better readability, elements not in the matrix are displayed as empty space, while matrix elements which are explicitly set to zero are displayed as such.

The parameters allow for a flexible setting of the output format: precision and scientific are used to determine the number format, where scientific = false means fixed point notation. A zero entry for width makes the function compute a width, but it may be changed to a positive value, if output is crude.

Additionally, a character for an empty value may be specified.

Finally, the whole matrix can be multiplied with a common denominator to produce more readable output, even integers.

Attention:
This function may produce large amounts of output if applied to a large matrix!

template<typename number>
void SparseMatrix< number >::print_pattern ( std::ostream &  out,
const double  threshold = 0. 
) const

Print the actual pattern of the matrix. For each entry with an absolute value larger than threshold, a '*' is printed, a ':' for every value smaller and a '.' for every entry not allocated.

template<typename number>
void SparseMatrix< number >::block_write ( std::ostream &  out  )  const

Write the data of this object en bloc to a file. This is done in a binary mode, so the output is neither readable by humans nor (probably) by other computers using a different operating system of number format.

The purpose of this function is that you can swap out matrices and sparsity pattern if you are short of memory, want to communicate between different programs, or allow objects to be persistent across different runs of the program.

template<typename number>
void SparseMatrix< number >::block_read ( std::istream &  in  ) 

Read data that has previously been written by block_write() from a file. This is done using the inverse operations to the above function, so it is reasonably fast because the bitstream is not interpreted except for a few numbers up front.

The object is resized on this operation, and all previous contents are lost. Note, however, that no checks are performed whether new data and the underlying SparsityPattern object fit together. It is your responsibility to make sure that the sparsity pattern and the data to be read match.

A primitive form of error checking is performed which will recognize the bluntest attempts to interpret some data as a matrix stored bitwise to a file that wasn't actually created that way, but not more.

template<typename number>
template<class OutVector , class InVector >
void SparseMatrix< number >::threaded_vmult ( OutVector &  dst,
const InVector &  src,
const unsigned int  begin_row,
const unsigned int  end_row 
) const [inline, private]

Version of vmult() which only performs its actions on the region defined by [begin_row,end_row). This function is called by vmult() in the case of enabled multithreading.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::threaded_matrix_norm_square ( const Vector< somenumber > &  v,
const unsigned int  begin_row,
const unsigned int  end_row,
somenumber *  partial_sum 
) const [inline, private]

Version of matrix_norm_square() which only performs its actions on the region defined by [begin_row,end_row). This function is called by matrix_norm_square() in the case of enabled multithreading.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::threaded_matrix_scalar_product ( const Vector< somenumber > &  u,
const Vector< somenumber > &  v,
const unsigned int  begin_row,
const unsigned int  end_row,
somenumber *  partial_sum 
) const [inline, private]

Version of matrix_scalar_product() which only performs its actions on the region defined by [begin_row,end_row). This function is called by matrix_scalar_product() in the case of enabled multithreading.

template<typename number>
template<typename somenumber >
void SparseMatrix< number >::threaded_residual ( Vector< somenumber > &  dst,
const Vector< somenumber > &  u,
const Vector< somenumber > &  b,
const std::pair< unsigned int, unsigned int interval,
somenumber *  partial_norm 
) const [inline, private]

Version of residual() which only performs its actions on the region defined by [begin_row,end_row) (these numbers are the components of interval). This function is called by residual() in the case of enabled multithreading.


Friends And Related Function Documentation

template<typename number>
friend class SparseMatrix [friend]

template<typename number>
friend class SparseLUDecomposition [friend]

template<typename number>
friend class SparseILU [friend]


Member Data Documentation

template<typename number>
SmartPointer<const SparsityPattern> SparseMatrix< number >::cols [private]

Pointer to the sparsity pattern used for this matrix. In order to guarantee that it is not deleted while still in use, we subscribe to it using the SmartPointer class.

template<typename number>
number* SparseMatrix< number >::val [private]

Array of values for all the nonzero entries. The position within the matrix, i.e. the row and column number for a given entry can only be deduced using the sparsity pattern. The same holds for the more common operation of finding an entry by its coordinates.

template<typename number>
unsigned int SparseMatrix< number >::max_len [private]

Allocated size of val. This can be larger than the actually used part if the size of the matrix was reduced somewhen in the past by associating a sparsity pattern with a smaller size to this object, using the reinit() function.


The documentation for this class was generated from the following file:

deal.II documentation generated on Sat Aug 15 16:52:21 2009 by doxygen 1.5.9