# Wiki

Clone wiki# blaze / Triangular Matrices

## Triangular Matrices

Triangular matrices come in three flavors: Lower triangular matrices provide the compile time guarantee to be square matrices and that the upper part of the matrix contains only default elements that cannot be modified. Upper triangular matrices on the other hand provide the compile time guarantee to be square and that the lower part of the matrix contains only fixed default elements. Finally, diagonal matrices provide the compile time guarantee to be square and that both the lower and upper part of the matrix contain only immutable default elements. These properties can be exploited to gain higher performance and/or to save memory. Within the **Blaze** library, several kinds of lower and upper triangular and diagonal matrices are realized by the following class templates:

Lower triangular matrices:

Upper triangular matrices:

Diagonal matrices:

## LowerMatrix

The `blaze::LowerMatrix`

class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type `MT`

and extends it by enforcing the additional invariant that all matrix elements above the diagonal are 0 (lower triangular matrix). It can be included via the header files

```
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/LowerMatrix.h>
```

and forward declared via the header file

```
#include <blaze/Forward.h>
```

The type of the adapted matrix can be specified via the first template parameter:

```
namespace blaze {
template< typename MT >
class LowerMatrix;
} // namespace blaze
```

`MT`

specifies the type of the matrix to be adapted. `LowerMatrix`

can be used with any non-cv-qualified, non-reference, non-pointer, non-expression dense or sparse matrix type. Note that the given matrix type must be either resizable (as for instance `blaze::HybridMatrix`

or `blaze::DynamicMatrix`

) or must be square at compile time (as for instance `blaze::StaticMatrix`

).

The following examples give an impression of several possible lower matrices:

```
using blaze::unaligned;
using blaze::unpadded;
using blaze::rowMajor;
using blaze::columnMajor;
// Definition of a 3x3 row-major dense lower matrix with static memory
blaze::LowerMatrix< blaze::StaticMatrix<int,3UL,3UL,rowMajor> > A;
// Definition of a resizable column-major dense lower matrix based on HybridMatrix
blaze::LowerMatrix< blaze::HybridMatrix<float,4UL,4UL,columnMajor> B;
// Definition of a resizable row-major dense lower matrix based on DynamicMatrix
blaze::LowerMatrix< blaze::DynamicMatrix<double,rowMajor> > C;
// Definition of a fixed size row-major dense lower matrix based on CustomMatrix
blaze::LowerMatrix< blaze::CustomMatrix<double,unaligned,unpadded,rowMajor> > D;
// Definition of a compressed row-major single precision lower matrix
blaze::LowerMatrix< blaze::CompressedMatrix<float,rowMajor> > E;
```

The storage order of a lower matrix is depending on the storage order of the adapted matrix type `MT`

. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as `blaze::rowMajor`

), the lower matrix will also be a row-major matrix. Otherwise, if the adapted matrix is column-major (i.e. is specified as `blaze::columnMajor`

), the lower matrix will also be a column-major matrix.

## UniLowerMatrix

The `blaze::UniLowerMatrix`

class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type `MT`

and extends it by enforcing the additional invariant that all diagonal matrix elements are 1 and all matrix elements above the diagonal are 0 (lower unitriangular matrix). It can be included via the header files

```
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/UniLowerMatrix.h>
```

and forward declared via the header file

```
#include <blaze/Forward.h>
```

The type of the adapted matrix can be specified via the first template parameter:

```
namespace blaze {
template< typename MT >
class UniLowerMatrix;
} // namespace blaze
```

`MT`

specifies the type of the matrix to be adapted. `blaze::UniLowerMatrix`

can be used with any non-cv-qualified, non-reference, non-pointer, non-expression dense or sparse matrix type. Also, the given matrix type must have numeric element types (i.e. all integral types except `bool`

, floating point and complex types). Note that the given matrix type must be either resizable (as for instance `blaze::HybridMatrix`

or `blaze::DynamicMatrix`

) or must be square at compile time (as for instance `blaze::StaticMatrix`

).

The following examples give an impression of several possible lower unitriangular matrices:

```
// Definition of a 3x3 row-major dense unilower matrix with static memory
blaze::UniLowerMatrix< blaze::StaticMatrix<int,3UL,3UL,blaze::rowMajor> > A;
// Definition of a resizable column-major dense unilower matrix based on HybridMatrix
blaze::UniLowerMatrix< blaze::HybridMatrix<float,4UL,4UL,blaze::columnMajor> B;
// Definition of a resizable row-major dense unilower matrix based on DynamicMatrix
blaze::UniLowerMatrix< blaze::DynamicMatrix<double,blaze::rowMajor> > C;
// Definition of a compressed row-major single precision unilower matrix
blaze::UniLowerMatrix< blaze::CompressedMatrix<float,blaze::rowMajor> > D;
```

The storage order of a lower unitriangular matrix is depending on the storage order of the adapted matrix type `MT`

. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as `blaze::rowMajor`

), the unilower matrix will also be a row-major matrix. Otherwise if the adapted matrix is column-major (i.e. is specified as `blaze::columnMajor`

), the unilower matrix will also be a column-major matrix.

## StrictlyLowerMatrix

The `blaze::StrictlyLowerMatrix`

class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type `MT`

and extends it by enforcing the additional invariant that all diagonal matrix elements and all matrix elements above the diagonal are 0 (strictly lower triangular matrix). It can be included via the header files

```
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/StrictlyLowerMatrix.h>
```

and forward declared via the header file

```
#include <blaze/Forward.h>
```

The type of the adapted matrix can be specified via the first template parameter:

```
namespace blaze {
template< typename MT >
class StrictlyLowerMatrix;
} // namespace blaze
```

`MT`

specifies the type of the matrix to be adapted. `blaze::StrictlyLowerMatrix`

can be used with any non-cv-qualified, non-reference, non-pointer, non-expression dense or sparse matrix type. Note that the given matrix type must be either resizable (as for instance `blaze::HybridMatrix`

or `blaze::DynamicMatrix`

) or must be square at compile time (as for instance `blaze::StaticMatrix`

).

The following examples give an impression of several possible strictly lower triangular matrices:

```
// Definition of a 3x3 row-major dense strictly lower matrix with static memory
blaze::StrictlyLowerMatrix< blaze::StaticMatrix<int,3UL,3UL,blaze::rowMajor> > A;
// Definition of a resizable column-major dense strictly lower matrix based on HybridMatrix
blaze::StrictlyLowerMatrix< blaze::HybridMatrix<float,4UL,4UL,blaze::columnMajor> B;
// Definition of a resizable row-major dense strictly lower matrix based on DynamicMatrix
blaze::StrictlyLowerMatrix< blaze::DynamicMatrix<double,blaze::rowMajor> > C;
// Definition of a compressed row-major single precision strictly lower matrix
blaze::StrictlyLowerMatrix< blaze::CompressedMatrix<float,blaze::rowMajor> > D;
```

The storage order of a strictly lower triangular matrix is depending on the storage order of the adapted matrix type `MT`

. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as `blaze::rowMajor`

), the strictly lower matrix will also be a row-major matrix. Otherwise if the adapted matrix is column-major (i.e. is specified as `blaze::columnMajor`

), the strictly lower matrix will also be a column-major matrix.

## UpperMatrix

The `blaze::UpperMatrix`

class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type `MT`

and extends it by enforcing the additional invariant that all matrix elements below the diagonal are 0 (upper triangular matrix). It can be included via the header files

```
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/UpperMatrix.h>
```

and forward declared via the header file

```
#include <blaze/Forward.h>
```

The type of the adapted matrix can be specified via the first template parameter:

```
namespace blaze {
template< typename MT >
class UpperMatrix;
} // namespace blaze
```

`MT`

specifies the type of the matrix to be adapted. `UpperMatrix`

can be used with any non-cv-qualified, non-reference, non-pointer, non-expression dense or sparse matrix type. Note that the given matrix type must be either resizable (as for instance `blaze::HybridMatrix`

or `blaze::DynamicMatrix`

) or must be square at compile time (as for instance `blaze::StaticMatrix`

).

The following examples give an impression of several possible upper matrices:

```
// Definition of a 3x3 row-major dense upper matrix with static memory
blaze::UpperMatrix< blaze::StaticMatrix<int,3UL,3UL,blaze::rowMajor> > A;
// Definition of a resizable column-major dense upper matrix based on HybridMatrix
blaze::UpperMatrix< blaze::HybridMatrix<float,4UL,4UL,blaze::columnMajor> B;
// Definition of a resizable row-major dense upper matrix based on DynamicMatrix
blaze::UpperMatrix< blaze::DynamicMatrix<double,blaze::rowMajor> > C;
// Definition of a compressed row-major single precision upper matrix
blaze::UpperMatrix< blaze::CompressedMatrix<float,blaze::rowMajor> > D;
```

The storage order of an upper matrix is depending on the storage order of the adapted matrix type `MT`

. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as `blaze::rowMajor`

), the upper matrix will also be a row-major matrix. Otherwise, if the adapted matrix is column-major (i.e. is specified as `blaze::columnMajor`

), the upper matrix will also be a column-major matrix.

## UniUpperMatrix

The `blaze::UniUpperMatrix`

class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type `MT`

and extends it by enforcing the additional invariant that all diagonal matrix elements are 1 and all matrix elements below the diagonal are 0 (upper unitriangular matrix). It can be included via the header files

```
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/UniUpperMatrix.h>
```

and forward declared via the header file

```
#include <blaze/Forward.h>
```

The type of the adapted matrix can be specified via the first template parameter:

```
namespace blaze {
template< typename MT >
class UniUpperMatrix;
} // namespace blaze
```

`MT`

specifies the type of the matrix to be adapted. `blaze::UniUpperMatrix`

can be used with any non-cv-qualified, non-reference, non-pointer, non-expression dense or sparse matrix type. Also, the given matrix type must have numeric element types (i.e. all integral types except `bool`

, floating point and complex types). Note that the given matrix type must be either resizable (as for instance `blaze::HybridMatrix`

or `blaze::DynamicMatrix`

) or must be square at compile time (as for instance `blaze::StaticMatrix`

).

The following examples give an impression of several possible upper unitriangular matrices:

```
// Definition of a 3x3 row-major dense uniupper matrix with static memory
blaze::UniUpperMatrix< blaze::StaticMatrix<int,3UL,3UL,blaze::rowMajor> > A;
// Definition of a resizable column-major dense uniupper matrix based on HybridMatrix
blaze::UniUpperMatrix< blaze::HybridMatrix<float,4UL,4UL,blaze::columnMajor> B;
// Definition of a resizable row-major dense uniupper matrix based on DynamicMatrix
blaze::UniUpperMatrix< blaze::DynamicMatrix<double,blaze::rowMajor> > C;
// Definition of a compressed row-major single precision uniupper matrix
blaze::UniUpperMatrix< blaze::CompressedMatrix<float,blaze::rowMajor> > D;
```

The storage order of an upper unitriangular matrix is depending on the storage order of the adapted matrix type `MT`

. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as `blaze::rowMajor`

), the uniupper matrix will also be a row-major matrix. Otherwise, if the adapted matrix is column-major (i.e. is specified as `blaze::columnMajor`

), the uniupper matrix will also be a column-major matrix.

## StrictlyUpperMatrix

The `blaze::StrictlyUpperMatrix`

class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type `MT`

and extends it by enforcing the additional invariant that all diagonal matrix elements and all matrix elements below the diagonal are 0 (strictly upper triangular matrix). It can be included via the header files

```
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/StrictlyUpperMatrix.h>
```

and forward declared via the header file

```
#include <blaze/Forward.h>
```

The type of the adapted matrix can be specified via the first template parameter:

```
namespace blaze {
template< typename MT >
class StrictlyUpperMatrix;
} // namespace blaze
```

`MT`

specifies the type of the matrix to be adapted. `blaze::StrictlyUpperMatrix`

can be used with any non-cv-qualified, non-reference, non-pointer, non-expression dense or sparse matrix type. Note that the given matrix type must be either resizable (as for instance `blaze::HybridMatrix`

or `blaze::DynamicMatrix`

) or must be square at compile time (as for instance `blaze::StaticMatrix`

).

The following examples give an impression of several possible strictly upper triangular matrices:

```
// Definition of a 3x3 row-major dense strictly upper matrix with static memory
blaze::StrictlyUpperMatrix< blaze::StaticMatrix<int,3UL,3UL,blaze::rowMajor> > A;
// Definition of a resizable column-major dense strictly upper matrix based on HybridMatrix
blaze::StrictlyUpperMatrix< blaze::HybridMatrix<float,4UL,4UL,blaze::columnMajor> B;
// Definition of a resizable row-major dense strictly upper matrix based on DynamicMatrix
blaze::StrictlyUpperMatrix< blaze::DynamicMatrix<double,blaze::rowMajor> > C;
// Definition of a compressed row-major single precision strictly upper matrix
blaze::StrictlyUpperMatrix< blaze::CompressedMatrix<float,blaze::rowMajor> > D;
```

The storage order of a strictly upper triangular matrix is depending on the storage order of the adapted matrix type `MT`

. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as `blaze::rowMajor`

), the strictly upper matrix will also be a row-major matrix. Otherwise, if the adapted matrix is column-major (i.e. is specified as `blaze::columnMajor`

), the strictly upper matrix will also be a column-major matrix.

## DiagonalMatrix

The `blaze::DiagonalMatrix`

class template is an adapter for existing dense and sparse matrix types. It inherits the properties and the interface of the given matrix type `MT`

and extends it by enforcing the additional invariant that all matrix elements above and below the diagonal are 0 (diagonal matrix). It can be included via the header files

```
#include <blaze/Blaze.h>
// or
#include <blaze/Math.h>
// or
#include <blaze/math/DiagonalMatrix.h>
```

and forward declared via the header file

```
#include <blaze/Forward.h>
```

The type of the adapted matrix can be specified via the first template parameter:

```
namespace blaze {
template< typename MT >
class DiagonalMatrix;
} // namespace blaze
```

`MT`

specifies the type of the matrix to be adapted. blaze::DiagonalMatrix can be used with any non-cv-qualified, non-reference, non-pointer, non-expression dense or sparse matrix type. Note that the given matrix type must be either resizable (as for instance `blaze::HybridMatrix`

or `blaze::DynamicMatrix`

) or must be square at compile time (as for instance `blaze::StaticMatrix`

).

The following examples give an impression of several possible diagonal matrices:

```
// Definition of a 3x3 row-major dense diagonal matrix with static memory
blaze::DiagonalMatrix< blaze::StaticMatrix<int,3UL,3UL,blaze::rowMajor> > A;
// Definition of a resizable column-major dense diagonal matrix based on HybridMatrix
blaze::DiagonalMatrix< blaze::HybridMatrix<float,4UL,4UL,blaze::columnMajor> B;
// Definition of a resizable row-major dense diagonal matrix based on DynamicMatrix
blaze::DiagonalMatrix< blaze::DynamicMatrix<double,blaze::rowMajor> > C;
// Definition of a compressed row-major single precision diagonal matrix
blaze::DiagonalMatrix< blaze::CompressedMatrix<float,blaze::rowMajor> > D;
```

The storage order of a diagonal matrix is depending on the storage order of the adapted matrix type `MT`

. In case the adapted matrix is stored in a row-wise fashion (i.e. is specified as `blaze::rowMajor`

), the diagonal matrix will also be a row-major matrix. Otherwise, if the adapted matrix is column-major (i.e. is specified as `blaze::columnMajor`

), the diagonal matrix will also be a column-major matrix.

## Special Properties of Triangular Matrices

A triangular matrix is used exactly like a matrix of the underlying, adapted matrix type `MT`

. It also provides (nearly) the same interface as the underlying matrix type. However, there are some important exceptions resulting from the triangular matrix constraint:

- Triangular Matrices Must Always be Square
- The Triangular Property is Always Enforced
- The Elements of a Dense Triangular Matrix are Always Default Initialized
- Dense Triangular Matrices Store All Elements
- Unitriangular Matrices Cannot Be Scaled

#### Triangular Matrices Must Always be Square

In case a resizable matrix is used (as for instance `blaze::HybridMatrix`

, `blaze::DynamicMatrix`

, or `blaze::CompressedMatrix`

), this means that the according constructors, the `resize()`

and the `extend()`

functions only expect a single parameter, which specifies both the number of rows and columns, instead of two (one for the number of rows and one for the number of columns):

```
using blaze::DynamicMatrix;
using blaze::LowerMatrix;
using blaze::rowMajor;
// Default constructed, default initialized, row-major 3x3 lower dynamic matrix
LowerMatrix< DynamicMatrix<double,rowMajor> > A( 3 );
// Resizing the matrix to 5x5
A.resize( 5 );
// Extending the number of rows and columns by 2, resulting in a 7x7 matrix
A.extend( 2 );
```

In case a matrix with a fixed size is used (as for instance `blaze::StaticMatrix`

), the number of rows and number of columns must be specified equally:

```
using blaze::StaticMatrix;
using blaze::LowerMatrix;
using blaze::columnMajor;
// Correct setup of a fixed size column-major 3x3 lower static matrix
LowerMatrix< StaticMatrix<int,3UL,3UL,columnMajor> > A;
// Compilation error: the provided matrix type is not a square matrix type
LowerMatrix< StaticMatrix<int,3UL,4UL,columnMajor> > B;
```

#### The Triangular Property is Always Enforced

This means that it is only allowed to modify elements in the lower part or the diagonal of a lower triangular matrix and in the upper part or the diagonal of an upper triangular matrix. Unitriangular and strictly triangular matrices are even more restrictive and don't allow the modification of diagonal elements. Also, triangular matrices can only be assigned matrices that don't violate their triangular property. The following example demonstrates this restriction by means of the `blaze::LowerMatrix`

adaptor. For examples with other triangular matrix types see the according class documentations.

```
using blaze::CompressedMatrix;
using blaze::DynamicMatrix;
using blaze::StaticMatrix;
using blaze::LowerMatrix;
using blaze::rowMajor;
using CompressedLower = LowerMatrix< CompressedMatrix<double,rowMajor> >;
// Default constructed, row-major 3x3 lower compressed matrix
CompressedLower A( 3 );
// Initializing elements via the function call operator
A(0,0) = 1.0; // Initialization of the diagonal element (0,0)
A(2,0) = 2.0; // Initialization of the lower element (2,0)
A(1,2) = 9.0; // Throws an exception; invalid modification of upper element
// Inserting two more elements via the insert() function
A.insert( 1, 0, 3.0 ); // Inserting the lower element (1,0)
A.insert( 2, 1, 4.0 ); // Inserting the lower element (2,1)
A.insert( 0, 2, 9.0 ); // Throws an exception; invalid insertion of upper element
// Appending an element via the append() function
A.reserve( 1, 3 ); // Reserving enough capacity in row 1
A.append( 1, 1, 5.0 ); // Appending the diagonal element (1,1)
A.append( 1, 2, 9.0 ); // Throws an exception; appending an element in the upper part
// Access via a non-const iterator
CompressedLower::Iterator it = A.begin(1);
*it = 6.0; // Modifies the lower element (1,0)
++it;
*it = 9.0; // Modifies the diagonal element (1,1)
// Erasing elements via the erase() function
A.erase( 0, 0 ); // Erasing the diagonal element (0,0)
A.erase( 2, 0 ); // Erasing the lower element (2,0)
// Construction from a lower dense matrix
StaticMatrix<double,3UL,3UL> B{ { 3.0, 0.0, 0.0 },
{ 8.0, 0.0, 0.0 },
{ -2.0, -1.0, 4.0 } };
LowerMatrix< DynamicMatrix<double,rowMajor> > C( B ); // OK
// Assignment of a non-lower dense matrix
StaticMatrix<double,3UL,3UL> D{ { 3.0, 0.0, -2.0 },
{ 8.0, 0.0, 0.0 },
{ -2.0, -1.0, 4.0 } };
C = D; // Throws an exception; lower matrix invariant would be violated!
```

The triangular property is also enforced during the construction of triangular custom matrices: In case the given array of elements does not represent the according triangular matrix type, a `std::invalid_argument`

exception is thrown:

```
using blaze::CustomMatrix;
using blaze::LowerMatrix;
using blaze::unaligned;
using blaze::unpadded;
using blaze::rowMajor;
using CustomLower = LowerMatrix< CustomMatrix<double,unaligned,unpadded,rowMajor> >;
// Creating a 3x3 lower custom matrix from a properly initialized array
double array[9] = { 1.0, 0.0, 0.0,
2.0, 3.0, 0.0,
4.0, 5.0, 6.0 };
CustomLower A( array, 3UL ); // OK
// Attempt to create a second 3x3 lower custom matrix from an uninitialized array
std::unique_ptr<double[]> memory( new double[9UL] );
CustomLower B( memory.get(), 3UL ); // Throws an exception
```

Finally, the triangular matrix property is enforced for views (rows, columns, submatrices, ...) on the triangular matrix. The following example demonstrates that modifying the elements of an entire row and submatrix of a lower matrix only affects the lower and diagonal matrix elements. Again, this example uses `blaze::LowerMatrix`

, for examples with other triangular matrix types see the according class documentations.

```
using blaze::DynamicMatrix;
using blaze::LowerMatrix;
// Setup of the lower matrix
//
// ( 0 0 0 0 )
// A = ( 1 2 0 0 )
// ( 0 3 0 0 )
// ( 4 0 5 0 )
//
LowerMatrix< DynamicMatrix<int> > A( 4 );
A(1,0) = 1;
A(1,1) = 2;
A(2,1) = 3;
A(3,0) = 4;
A(3,2) = 5;
// Setting the lower and diagonal elements in the 2nd row to 9 results in the matrix
//
// ( 0 0 0 0 )
// A = ( 1 2 0 0 )
// ( 9 9 9 0 )
// ( 4 0 5 0 )
//
row( A, 2 ) = 9;
// Setting the lower and diagonal elements in the 1st and 2nd column to 7 results in
//
// ( 0 0 0 0 )
// A = ( 1 7 0 0 )
// ( 9 7 7 0 )
// ( 4 7 7 0 )
//
submatrix( A, 0, 1, 4, 2 ) = 7;
```

The next example demonstrates the (compound) assignment to rows/columns and submatrices of triangular matrices. Since only lower/upper and potentially diagonal elements may be modified the matrix to be assigned must be structured such that the triangular matrix invariant of the matrix is preserved. Otherwise a `std::invalid_argument`

exception is thrown:

```
using blaze::DynamicMatrix;
using blaze::DynamicVector;
using blaze::LowerMatrix;
using blaze::rowVector;
// Setup of two default 4x4 lower matrices
LowerMatrix< DynamicMatrix<int> > A1( 4 ), A2( 4 );
// Setup of a 4-dimensional vector
//
// v = ( 1 2 3 0 )
//
DynamicVector<int,rowVector> v{ 1, 2, 3, 0 };
// OK: Assigning v to the 2nd row of A1 preserves the lower matrix invariant
//
// ( 0 0 0 0 )
// A1 = ( 0 0 0 0 )
// ( 1 2 3 0 )
// ( 0 0 0 0 )
//
row( A1, 2 ) = v; // OK
// Error: Assigning v to the 1st row of A1 violates the lower matrix invariant! The element
// marked with X cannot be assigned and triggers an exception.
//
// ( 0 0 0 0 )
// A1 = ( 1 2 X 0 )
// ( 1 2 3 0 )
// ( 0 0 0 0 )
//
row( A1, 1 ) = v; // Assignment throws an exception!
// Setup of the 3x2 dynamic matrix
//
// ( 0 0 )
// B = ( 7 0 )
// ( 8 9 )
//
DynamicMatrix<int> B( 3UL, 2UL, 0 );
B(1,0) = 7;
B(2,0) = 8;
B(2,1) = 9;
// OK: Assigning B to a submatrix of A2 such that the lower matrix invariant can be preserved
//
// ( 0 0 0 0 )
// A2 = ( 0 7 0 0 )
// ( 0 8 9 0 )
// ( 0 0 0 0 )
//
submatrix( A2, 0UL, 1UL, 3UL, 2UL ) = B; // OK
// Error: Assigning B to a submatrix of A2 such that the lower matrix invariant cannot be
// preserved! The elements marked with X cannot be assigned without violating the invariant!
//
// ( 0 0 0 0 )
// A2 = ( 0 7 X 0 )
// ( 0 8 8 X )
// ( 0 0 0 0 )
//
submatrix( A2, 0UL, 2UL, 3UL, 2UL ) = B; // Assignment throws an exception!
```

#### The Elements of a Dense Triangular Matrix are Always Default Initialized

Although this results in a small loss of efficiency during the creation of a dense lower or upper matrix this initialization is important since otherwise the lower/upper matrix property of dense lower matrices would not be guaranteed:

```
using blaze::DynamicMatrix;
using blaze::LowerMatrix;
using blaze::UpperMatrix;
// Uninitialized, 5x5 row-major dynamic matrix
DynamicMatrix<int,rowMajor> A( 5, 5 );
// 5x5 row-major lower dynamic matrix with default initialized upper matrix
LowerMatrix< DynamicMatrix<int,rowMajor> > B( 5 );
// 7x7 column-major upper dynamic matrix with default initialized lower matrix
UpperMatrix< DynamicMatrix<int,columnMajor> > C( 7 );
// 3x3 row-major diagonal dynamic matrix with default initialized lower and upper matrix
DiagonalMatrix< DynamicMatrix<int,rowMajor> > D( 3 );
```

#### Dense Triangular Matrices Store All Elements

All dense triangular matrices store all NxN elements, including the immutable elements in the lower or upper part, respectively. Therefore dense triangular matrices don't provide any kind of memory reduction! There are two main reasons for this: First, storing also the zero elements guarantees maximum performance for many algorithms that perform vectorized operations on the triangular matrices, which is especially true for small dense matrices. Second, conceptually all triangular adaptors merely restrict the interface to the matrix type `MT`

and do not change the data layout or the underlying matrix type.

This property matters most for diagonal matrices. In order to achieve the perfect combination of performance and memory consumption for a diagonal matrix it is recommended to use dense matrices for small diagonal matrices and sparse matrices for large diagonal matrices:

```
// Recommendation 1: use dense matrices for small diagonal matrices
using SmallDiagonalMatrix = blaze::DiagonalMatrix< blaze::StaticMatrix<float,3UL,3UL> >;
// Recommendation 2: use sparse matrices for large diagonal matrices
using LargeDiagonalMatrix = blaze::DiagonalMatrix< blaze::CompressedMatrix<float> >;
```

#### Unitriangular Matrices Cannot Be Scaled

Since the diagonal elements of a unitriangular matrix have a fixed value of 1 it is not possible to self-scale such a matrix:

```
using blaze::DynamicMatrix;
using blaze::UniLowerMatrix;
UniLowerMatrix< DynamicMatrix<int> > A( 4 );
A *= 2; // Compilation error; Scale operation is not available on an unilower matrix
A /= 2; // Compilation error; Scale operation is not available on an unilower matrix
A.scale( 2 ); // Compilation error; Scale function is not available on an unilower matrix
A = A * 2; // Throws an exception; Invalid assignment of non-unilower matrix
A = A / 2; // Throws an exception; Invalid assignment of non-unilower matrix
```

## Arithmetic Operations

A lower and upper triangular matrix can participate in numerical operations in any way any other dense or sparse matrix can participate. It can also be combined with any other dense or sparse vector or matrix. The following code example gives an impression of the use of `blaze::LowerMatrix`

and `blaze::UpperMatrix`

within arithmetic operations:

```
using blaze::LowerMatrix;
using blaze::DynamicMatrix;
using blaze::HybridMatrix;
using blaze::StaticMatrix;
using blaze::CompressedMatrix;
using blaze::rowMajor;
using blaze::columnMajor;
DynamicMatrix<double,rowMajor> A( 3, 3 );
CompressedMatrix<double,rowMajor> B( 3, 3 );
LowerMatrix< DynamicMatrix<double,rowMajor> > C( 3 );
LowerMatrix< CompressedMatrix<double,rowMajor> > D( 3 );
LowerMatrix< HybridMatrix<float,3UL,3UL,rowMajor> > E;
LowerMatrix< StaticMatrix<float,3UL,3UL,columnMajor> > F;
E = A + B; // Matrix addition and assignment to a row-major lower matrix (includes runtime check)
F = C - D; // Matrix subtraction and assignment to a column-major lower matrix (only compile time check)
F = A * D; // Matrix multiplication between a dense and a sparse matrix (includes runtime check)
C *= 2.0; // In-place scaling of matrix C
E = 2.0 * B; // Scaling of matrix B (includes runtime check)
F = C * 2.0; // Scaling of matrix C (only compile time check)
E += A - B; // Addition assignment (includes runtime check)
F -= C + D; // Subtraction assignment (only compile time check)
F *= A * D; // Multiplication assignment (includes runtime check)
```

Note that it is possible to assign any kind of matrix to a triangular matrix. In case the matrix to be assigned does not satisfy the invariants of the triangular matrix at compile time, a runtime check is performed. Also note that upper triangular, diagonal, unitriangular and strictly triangular matrix types can be used in the same way, but may pose some additional restrictions (see the according class documentations).

## Triangular Block Matrices

It is also possible to use triangular block matrices:

```
using blaze::CompressedMatrix;
using blaze::DynamicMatrix;
using blaze::StaticMatrix;
using blaze::LowerMatrix;
using blaze::UpperMatrix;
// Definition of a 5x5 lower block matrix based on DynamicMatrix
LowerMatrix< DynamicMatrix< StaticMatrix<int,3UL,3UL> > > A( 5 );
// Definition of a 7x7 upper block matrix based on CompressedMatrix
UpperMatrix< CompressedMatrix< StaticMatrix<int,3UL,3UL> > > B( 7 );
```

Also in this case the triangular matrix invariant is enforced, i.e. it is not possible to manipulate elements in the upper part (lower triangular matrix) or the lower part (upper triangular matrix) of the matrix:

```
const StaticMatrix<int,3UL,3UL> C{ { 1, -4, 5 },
{ 6, 8, -3 },
{ 2, -1, 2 } };
A(2,4)(1,1) = -5; // Invalid manipulation of upper matrix element; Results in an exception
B.insert( 4, 2, C ); // Invalid insertion of the elements (4,2); Results in an exception
```

Note that unitriangular matrices are restricted to numeric element types and therefore cannot be used for block matrices:

```
using blaze::CompressedMatrix;
using blaze::DynamicMatrix;
using blaze::StaticMatrix;
using blaze::UniLowerMatrix;
using blaze::UniUpperMatrix;
// Compilation error: lower unitriangular matrices are restricted to numeric element types
UniLowerMatrix< DynamicMatrix< StaticMatrix<int,3UL,3UL> > > A( 5 );
// Compilation error: upper unitriangular matrices are restricted to numeric element types
UniUpperMatrix< CompressedMatrix< StaticMatrix<int,3UL,3UL> > > B( 7 );
```

For more information on block matrices, see the tutorial on Block Vectors and Matrices.

## Performance Considerations

The **Blaze** library tries to exploit the properties of lower and upper triangular matrices whenever and wherever possible. Therefore using triangular matrices instead of a general matrices can result in a considerable performance improvement. However, there are also situations when using a triangular matrix introduces some overhead. The following examples demonstrate several common situations where triangular matrices can positively or negatively impact performance.

#### Positive Impact: Matrix/Matrix Multiplication

When multiplying two matrices, at least one of which is triangular, **Blaze** can exploit the fact that either the lower or upper part of the matrix contains only default elements and restrict the algorithm to the non-zero elements. The following example demonstrates this by means of a dense matrix/dense matrix multiplication with lower triangular matrices:

```
using blaze::DynamicMatrix;
using blaze::LowerMatrix;
using blaze::rowMajor;
using blaze::columnMajor;
LowerMatrix< DynamicMatrix<double,rowMajor> > A;
LowerMatrix< DynamicMatrix<double,columnMajor> > B;
DynamicMatrix<double,columnMajor> C;
// ... Resizing and initialization
C = A * B;
```

In comparison to a general matrix multiplication, the performance advantage is significant, especially for large matrices. Therefore is it highly recommended to use the `blaze::LowerMatrix`

and `blaze::UpperMatrix`

adaptors when a matrix is known to be lower or upper triangular, respectively. Note however that the performance advantage is most pronounced for dense matrices and much less so for sparse matrices.

#### Positive Impact: Matrix/Vector Multiplication

A similar performance improvement can be gained when using a triangular matrix in a matrix/vector multiplication:

```
using blaze::DynamicMatrix;
using blaze::DynamicVector;
using blaze::rowMajor;
using blaze::columnVector;
LowerMatrix< DynamicMatrix<double,rowMajor> > A;
DynamicVector<double,columnVector> x, y;
// ... Resizing and initialization
y = A * x;
```

In this example, **Blaze** also exploits the structure of the matrix and approx. halves the runtime of the multiplication. Also in case of matrix/vector multiplications the performance improvement is most pronounced for dense matrices and much less so for sparse matrices.

#### Negative Impact: Assignment of a General Matrix

In contrast to using a triangular matrix on the right-hand side of an assignment (i.e. for read access), which introduces absolutely no performance penalty, using a triangular matrix on the left-hand side of an assignment (i.e. for write access) may introduce additional overhead when it is assigned a general matrix, which is not triangular at compile time:

```
using blaze::DynamicMatrix;
using blaze::LowerMatrix;
LowerMatrix< DynamicMatrix<double> > A, C;
DynamicMatrix<double> B;
B = A; // Only read-access to the lower matrix; no performance penalty
C = A; // Assignment of a lower matrix to another lower matrix; no runtime overhead
C = B; // Assignment of a general matrix to a lower matrix; some runtime overhead
```

When assigning a general (potentially not lower triangular) matrix to a lower matrix or a general (potentially not upper triangular) matrix to an upper matrix it is necessary to check whether the matrix is lower or upper at runtime in order to guarantee the triangular property of the matrix. In case it turns out to be lower or upper, respectively, it is assigned as efficiently as possible, if it is not, an exception is thrown. In order to prevent this runtime overhead it is therefore generally advisable to assign lower or upper triangular matrices to other lower or upper triangular matrices.

In this context it is especially noteworthy that the addition, subtraction, and multiplication of two triangular matrices of the same structure always results in another triangular matrix:

```
LowerMatrix< DynamicMatrix<double> > A, B, C;
C = A + B; // Results in a lower matrix; no runtime overhead
C = A - B; // Results in a lower matrix; no runtime overhead
C = A * B; // Results in a lower matrix; no runtime overhead
```

```
UpperMatrix< DynamicMatrix<double> > A, B, C;
C = A + B; // Results in an upper matrix; no runtime overhead
C = A - B; // Results in an upper matrix; no runtime overhead
C = A * B; // Results in an upper matrix; no runtime overhead
```

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