# Triangular Matrix

#### Description

The templated class `triangular_matrix<T, F1, F2, A>` is the base container adaptor for triangular matrices. For a (n x n )-dimensional lower triangular matrix and 0 <= i < n,0 <= j < n holds ti, j = 0 , if i > j. If furthermore holds ti, i= 1 the matrix is called unit lower triangular. For a (n x n )-dimensional upper triangular matrix and 0 <= i < n,0 <= j < n holds ti, j = 0 , if i < j. If furthermore holds ti, i= 1 the matrix is called unit lower triangular. The storage of triangular matrices is packed.

#### Example

    #include <boost/numeric/ublas/triangular.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        triangular_matrix<double, lower> ml (3, 3);
        for (unsigned i = 0; i < ml.size1 (); ++ i)
            for (unsigned j = 0; j <= i; ++ j)
                ml (i, j) = 3 * i + j;
        std::cout << ml << std::endl;
        triangular_matrix<double, upper> mu (3, 3);
        for (unsigned i = 0; i < mu.size1 (); ++ i)
            for (unsigned j = i; j < mu.size2 (); ++ j)
                mu (i, j) = 3 * i + j;
        std::cout << mu << std::endl;
    }

#### Definition

Defined in the header triangular.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`T` | The type of object stored in the matrix. |
`F1` | Functor describing the type of the triangular matrix. [1] | `lower`
`F2` | Functor describing the storage organization. [2] | `row_major`
`A` | The type of the adapted array. [3] | `unbounded_array<T>`

#### Model of

Matrix .

#### Type requirements

None, except for those imposed by the requirements of Matrix .

#### Public base classes

`matrix_expression<triangular_matrix<T, F1, F2, A> >`

#### Members

Member | Description
--- | ---
`triangular_matrix ()` | Allocates an uninitialized `triangular_matrix` that holds zero rows of zero elements.
`triangular_matrix (size_type size1, size_type size2)` | Allocates an uninitialized `triangular_matrix` that holds `size1` rows of `size2` elements.
`triangular_matrix (const triangular_matrix &m)` | The copy constructor.
`template<class AE> triangular_matrix (const matrix_expression<AE> &ae)` | The extended copy constructor.
`void resize (size_type size1, size_type size2, bool preserve = true)` | Reallocates a `triangular_matrix` to hold `size1` rows of `size2` elements. The existing elements of the `triangular_matrix` are preseved when specified.
`size_type size1 () const` | Returns the number of rows.
`size_type size2 () const` | Returns the number of columns.
`const_reference operator () (size_type i, size_type j) const` | Returns a `const` reference of the `j` -th element in the `i`-th row.
`reference operator () (size_type i, size_type j)` | Returns a reference of the `j`-th element in the `i`-th row.
`triangular_matrix &operator = (const triangular_matrix &m)` | The assignment operator.
`triangular_matrix &assign_temporary (triangular_matrix &m)` | Assigns a temporary. May change the triangular matrix `m`.
`template<class AE> triangular_matrix &operator = (const matrix_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> triangular_matrix &assign (const matrix_expression<AE> &ae)` | Assigns a matrix expression to the triangular matrix. Left and right hand side of the assignment should be independent.
`template<class AE> triangular_matrix &operator += (const matrix_expression<AE> &ae)` | A computed assignment operator. Adds the matrix expression to the triangular matrix.
`template<class AE> triangular_matrix &plus_assign (const matrix_expression<AE> &ae)` | Adds a matrix expression to the triangular matrix. Left and right hand side of the assignment should be independent.
`template<class AE> triangular_matrix &operator -= (const matrix_expression<AE> &ae)` | A computed assignment operator. Subtracts the matrix expression from the triangular matrix.
`template<class AE> triangular_matrix &minus_assign (const matrix_expression<AE> &ae)` | Subtracts a matrix expression from the triangular matrix. Left and right hand side of the assignment should be independent.
`template<class AT> triangular_matrix &operator *= (const AT &at)` | A computed assignment operator. Multiplies the triangular matrix with a scalar.
`template<class AT> triangular_matrix &operator /= (const AT &at)` | A computed assignment operator. Divides the triangular matrix through a scalar.
`void swap (triangular_matrix &m)` | Swaps the contents of the triangular matrices.
`void insert (size_type i, size_type j, const_reference t)` | Inserts the value `t` at the `j`-th element of the `i`-th row.
`void erase (size_type i, size_type j)` | Erases the value at the `j`-th elemenst of the `i`-th row.
`void clear ()` | Clears the matrix.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the `triangular_matrix`.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the `triangular_matrix`.
`iterator1 begin1 ()` | Returns a `iterator1` pointing to the beginning of the `triangular_matrix`.
`iterator1 end1 ()` | Returns a `iterator1` pointing to the end of the `triangular_matrix`.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the `triangular_matrix`.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the `triangular_matrix`.
`iterator2 begin2 ()` | Returns a `iterator2` pointing to the beginning of the `triangular_matrix`.
`iterator2 end2 ()` | Returns a `iterator2` pointing to the end of the `triangular_matrix`.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed `triangular_matrix`.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed `triangular_matrix`.
`reverse_iterator1 rbegin1 ()` | Returns a `reverse_iterator1` pointing to the beginning of the reversed `triangular_matrix`.
`reverse_iterator1 rend1 ()` | Returns a `reverse_iterator1` pointing to the end of the reversed `triangular_matrix`.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed `triangular_matrix`.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed `triangular_matrix`.
`reverse_iterator2 rbegin2 ()` | Returns a `reverse_iterator2` pointing to the beginning of the reversed `triangular_matrix`.
`reverse_iterator2 rend2 ()` | Returns a `reverse_iterator2` pointing to the end of the reversed `triangular_matrix`.

#### Notes

[1] Supported parameters for the type of the triangular matrix are `lower` , `unit_lower`, `upper` and `unit_upper` .

[2] Supported parameters for the storage organization are `row_major` and `column_major`.

[3] Supported parameters for the adapted array are `unbounded_array<T>` , `bounded_array<T>` and `std::vector<T>` .

## Triangular Adaptor

#### Description

The templated class `triangular_adaptor<M, F>` is a triangular matrix adaptor for other matrices.

#### Example

    #include <boost/numeric/ublas/triangular.hpp>
    #include <boost/numeric/ublas/io.hpp>

    int main () {
        using namespace boost::numeric::ublas;
        matrix<double> m (3, 3);
        triangular_adaptor<matrix<double>, lower> tal (m);
        for (unsigned i = 0; i < tal.size1 (); ++ i)
            for (unsigned j = 0; j <= i; ++ j)
                tal (i, j) = 3 * i + j;
        std::cout << tal << std::endl;
        triangular_adaptor<matrix<double>, upper> tau (m);
        for (unsigned i = 0; i < tau.size1 (); ++ i)
            for (unsigned j = i; j < tau.size2 (); ++ j)
                tau (i, j) = 3 * i + j;
        std::cout << tau << std::endl;
    }

#### Definition

Defined in the header triangular.hpp.

#### Template parameters

Parameter | Description | Default
--- | --- | ---
`M` | The type of the adapted matrix. |
`F` | Functor describing the type of the triangular adaptor. [1] | `lower`

#### Model of

Matrix Expression .

#### Type requirements

None, except for those imposed by the requirements of Matrix Expression .

#### Public base classes

`matrix_expression<triangular_adaptor<M, F> >`

#### Members

Member | Description
--- | ---
`triangular_adaptor ()` | Constructs a `triangular_adaptor` that holds zero rows of zero elements.
`triangular_adaptor (matrix_type &data)` | Constructs a `triangular_adaptor` of a matrix.
`triangular_adaptor (const triangular_adaptor &m)` | The copy constructor.
`template<class AE> triangular_adaptor (const matrix_expression<AE> &ae)` | The extended copy constructor.
`size_type size1 () const` | Returns the number of rows.
`size_type size2 () const` | Returns the number of columns.
`const_reference operator () (size_type i, size_type j) const` | Returns a `const` reference of the `j` -th element in the `i`-th row.
`reference operator () (size_type i, size_type j)` | Returns a reference of the `j`-th element in the `i`-th row.
`triangular_adaptor &operator = (const triangular_adaptor &m)` | The assignment operator.
`triangular_adaptor &assign_temporary (triangular_adaptor &m)` | Assigns a temporary. May change the triangular adaptor `m`.
`template<class AE> triangular_adaptor &operator = (const matrix_expression<AE> &ae)` | The extended assignment operator.
`template<class AE> triangular_adaptor &assign (const matrix_expression<AE> &ae)` | Assigns a matrix expression to the triangular adaptor. Left and right hand side of the assignment should be independent.
`template<class AE> triangular_adaptor &operator += (const matrix_expression<AE> &ae)` | A computed assignment operator. Adds the matrix expression to the triangular adaptor.
`template<class AE> triangular_adaptor &plus_assign (const matrix_expression<AE> &ae)` | Adds a matrix expression to the triangular adaptor. Left and right hand side of the assignment should be independent.
`template<class AE> triangular_adaptor &operator -= (const matrix_expression<AE> &ae)` | A computed assignment operator. Subtracts the matrix expression from the triangular adaptor.
`template<class AE> triangular_adaptor &minus_assign (const matrix_expression<AE> &ae)` | Subtracts a matrix expression from the triangular adaptor. Left and right hand side of the assignment should be independent.
`template<class AT> triangular_adaptor &operator *= (const AT &at)` | A computed assignment operator. Multiplies the triangular adaptor with a scalar.
`template<class AT> triangular_adaptor &operator /= (const AT &at)` | A computed assignment operator. Divides the triangular adaptor through a scalar.
`void swap (triangular_adaptor &m)` | Swaps the contents of the triangular adaptors.
`const_iterator1 begin1 () const` | Returns a `const_iterator1` pointing to the beginning of the `triangular_adaptor`.
`const_iterator1 end1 () const` | Returns a `const_iterator1` pointing to the end of the `triangular_adaptor`.
`iterator1 begin1 ()` | Returns a `iterator1` pointing to the beginning of the `triangular_adaptor`.
`iterator1 end1 ()` | Returns a `iterator1` pointing to the end of the `triangular_adaptor`.
`const_iterator2 begin2 () const` | Returns a `const_iterator2` pointing to the beginning of the `triangular_adaptor`.
`const_iterator2 end2 () const` | Returns a `const_iterator2` pointing to the end of the `triangular_adaptor`.
`iterator2 begin2 ()` | Returns a `iterator2` pointing to the beginning of the `triangular_adaptor`.
`iterator2 end2 ()` | Returns a `iterator2` pointing to the end of the `triangular_adaptor`.
`const_reverse_iterator1 rbegin1 () const` | Returns a `const_reverse_iterator1` pointing to the beginning of the reversed `triangular_adaptor`.
`const_reverse_iterator1 rend1 () const` | Returns a `const_reverse_iterator1` pointing to the end of the reversed `triangular_adaptor`.
`reverse_iterator1 rbegin1 ()` | Returns a `reverse_iterator1` pointing to the beginning of the reversed `triangular_adaptor`.
`reverse_iterator1 rend1 ()` | Returns a `reverse_iterator1` pointing to the end of the reversed `triangular_adaptor`.
`const_reverse_iterator2 rbegin2 () const` | Returns a `const_reverse_iterator2` pointing to the beginning of the reversed `triangular_adaptor`.
`const_reverse_iterator2 rend2 () const` | Returns a `const_reverse_iterator2` pointing to the end of the reversed `triangular_adaptor`.
`reverse_iterator2 rbegin2 ()` | Returns a `reverse_iterator2` pointing to the beginning of the reversed `triangular_adaptor`.
`reverse_iterator2 rend2 ()` | Returns a `reverse_iterator2` pointing to the end of the reversed `triangular_adaptor`.

#### Notes

[1] Supported parameters for the type of the triangular adaptor are `lower` , `unit_lower`, `upper` and `unit_upper` .

Copyright (©) 2000-2002 Joerg Walter, Mathias Koch
Permission to copy, use, modify, sell and distribute this document is granted provided this copyright notice appears in all copies. This document is provided ``as is'' without express or implied warranty, and with no claim as to its suitability for any purpose.

Last revised: 1/15/2003