Blaze  3.6
blaze::UniUpperMatrix< MT, SO, DF > Class Template Reference

Matrix adapter for upper unitriangular $ N \times N $ matrices. More...

#include <BaseTemplate.h>

Detailed Description

template<typename MT, bool SO = StorageOrder_v<MT>, bool DF = IsDenseMatrix_v<MT>>
class blaze::UniUpperMatrix< MT, SO, DF >

Matrix adapter for upper unitriangular $ N \times N $ matrices.

General

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

template< typename MT, bool SO, bool DF >
class UniUpperMatrix;

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

// Definition of a 3x3 row-major dense uniupper matrix with static memory
// Definition of a resizable column-major dense uniupper matrix based on HybridMatrix
// Definition of a resizable row-major dense uniupper matrix based on DynamicMatrix
// Definition of a fixed-size row-major dense uniupper matrix based on CustomMatrix
// Definition of a compressed row-major single precision uniupper matrix

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.


Special Properties of Upper Unitriangular Matrices

An upper unitriangular 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 upper unitriangular matrix constraint:

  1. Upper Unitriangular Matrices Must Always be Square!
  2. The Upper Unitriangular Matrix Property is Always Enforced!
  3. The Lower Elements of a Dense Upper Unitriangular Matrix are Always Default Initialized!
  4. Dense Upper Unitriangular Matrices Also Store the Lower Elements!
  5. Upper Unitriangular Matrices Cannot Be Scaled!


Upper Unitriangular 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):

// Default constructed, default initialized, row-major 3x3 uniupper dynamic matrix
UniUpperMatrix< 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:

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


The Upper Unitriangular Matrix Property is Always Enforced!

The diagonal elements of an upper unitriangular matrix are fixed to 1. This property has two implications. First, that means that the diagonal elements of a newly created uniupper matrix are pre-initialized to 1:

// Creating a default initialized dense uniupper matrix of size 3x3
//
// ( 1 0 0 )
// A = ( 0 1 0 )
// ( 0 0 1 )
UniUpperMatrix< DynamicMatrix<int,rowMajor> > A( 3UL );
// Creating a default initialized sparse uniupper matrix of size 3x3
//
// ( 1 0 0 )
// B = ( 0 1 0 )
// ( 0 0 1 )
UniUpperMatrix< CompressedMatrix<int,rowMajor> > B( 3UL );

Second, this means that it is only allowed to modify elements in the upper part of the matrix, but not the diagonal elements and not the elements in the lower part of the matrix. Also, it is only possible to assign matrices that are upper unitriangular matrices themselves:

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

The upper unitriangular matrix property is also enforced for uniupper custom matrices: In case the given array of elements does not represent an uniupper matrix, a std::invalid_argument exception is thrown:

using CustomUniUpper = UniUpperMatrix< CustomMatrix<double,unaligned,unpadded,rowMajor> >;
// Creating a 3x3 uniupper custom matrix from a properly initialized array
double array[9] = { 1.0, 2.0, 3.0,
0.0, 1.0, 4.0,
0.0, 0.0, 1.0 };
CustomUniUpper A( array, 3UL ); // OK
// Attempt to create a second 3x3 uniupper custom matrix from an uninitialized array
CustomUniUpper B( new double[9UL], 3UL, blaze::ArrayDelete() ); // Throws an exception

Finally, the upper unitriangular matrix property is enforced for views (rows, columns, submatrices, ...) on the uniupper matrix. The following example demonstrates that modifying the elements of an entire row and submatrix of an uniupper matrix only affects the upper matrix elements:

// Setup of the upper matrix
//
// ( 1 2 0 4 )
// A = ( 0 1 3 0 )
// ( 0 0 1 5 )
// ( 0 0 0 1 )
//
UniUpperMatrix< DynamicMatrix<int> > A( 4 );
A(0,1) = 2;
A(0,3) = 4;
A(1,2) = 3;
A(2,3) = 5;
// Setting the upper elements in the 1st row to 9 results in the matrix
//
// ( 1 1 0 4 )
// A = ( 0 1 9 9 )
// ( 0 0 1 5 )
// ( 0 0 0 1 )
//
row( A, 1 ) = 9;
// Setting the upper elements in the 1st and 2nd column to 7 results in
//
// ( 1 7 7 4 )
// A = ( 0 1 7 9 )
// ( 0 0 1 5 )
// ( 0 0 0 1 )
//
submatrix( A, 0, 1, 4, 2 ) = 7;

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

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


The Lower Elements of a Dense Upper Unitriangular Matrix are Always Default Initialized!

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

// Uninitialized, 5x5 row-major dynamic matrix
DynamicMatrix<int,rowMajor> A( 5, 5 );
// 5x5 row-major uniupper dynamic matrix with default initialized lower matrix
UniUpperMatrix< DynamicMatrix<int,rowMajor> > B( 5 );


Dense Upper Unitriangular Matrices Also Store the Lower Elements!

It is important to note that dense upper unitriangular matrices store all elements, including the elements in the lower part of the matrix, and therefore don't provide any kind of memory reduction! There are two main reasons for this: First, storing also the lower elements guarantees maximum performance for many algorithms that perform vectorized operations on the uniupper matrix, which is especially true for small dense matrices. Second, conceptually the UniUpperMatrix adaptor merely restricts the interface to the matrix type MT and does not change the data layout or the underlying matrix type.


Upper Unitriangular Matrices Cannot Be Scaled!

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

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


Arithmetic Operations

An UniUpperMatrix 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 UniUpperMatrix within arithmetic operations:

DynamicMatrix<double,rowMajor> A( 3, 3 );
CompressedMatrix<double,rowMajor> B( 3, 3 );
UniUpperMatrix< DynamicMatrix<double,rowMajor> > C( 3 );
UniUpperMatrix< CompressedMatrix<double,rowMajor> > D( 3 );
UniUpperMatrix< HybridMatrix<float,3UL,3UL,rowMajor> > E;
UniUpperMatrix< StaticMatrix<float,3UL,3UL,columnMajor> > F;
DynamicMatrix<double,rowMajor> G( 3, 3 ); // Initialized as strictly upper matrix
CompressedMatrix<double,rowMajor> H( 3, 3 ); // Initialized as strictly upper matrix
E = A + B; // Matrix addition and assignment to a row-major uniupper matrix (includes runtime check)
F = A - C; // Matrix subtraction and assignment to a column-major uniupper matrix (only compile time check)
F = A * D; // Matrix multiplication between a dense and a sparse matrix (includes runtime check)
E += G; // Addition assignment (note that G is a strictly upper matrix)
F -= H; // Subtraction assignment (note that H is a strictly upper matrix)
F *= A * D; // Multiplication assignment (includes runtime check)

Note that it is possible to assign any kind of matrix to an uniupper matrix. In case the matrix to be assigned is not uniupper at compile time, a runtime check is performed.


Performance Considerations

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


Positive Impact: Matrix/Matrix Multiplication

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

UniUpperMatrix< DynamicMatrix<double,rowMajor> > A;
UniUpperMatrix< 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 UniUpperMatrix adaptor when a matrix is known to be upper unitriangular. 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 an upper (uni-)triangular matrix in a matrix/vector multiplication:

UniUpperMatrix< 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 an upper (uni-)triangular matrix on the right-hand side of an assignment (i.e. for read access), which introduces absolutely no performance penalty, using an (uni-)upper 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 upper (uni-)triangular at compile time:

UniUpperMatrix< DynamicMatrix<double> > A, C;
DynamicMatrix<double> B;
B = A; // Only read-access to the upper matrix; no performance penalty
C = A; // Assignment of an uniupper matrix to another uniupper matrix; no runtime overhead
C = B; // Assignment of a general matrix to an uniupper matrix; some runtime overhead

When assigning a general, potentially not uniupper matrix to another uniupper matrix it is necessary to check whether the matrix is uniupper at runtime in order to guarantee the upper unitriangular property of the uniupper matrix. In case it turns out to be upper unitriangular, 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 uniupper matrices to other uniupper matrices.
In this context it is especially noteworthy that the multiplication of two upper unitriangular matrices always results in another uniupper matrix:

UniUpperMatrix< DynamicMatrix<double> > A, B, C;
C = A * B; // Results in an uniupper matrix; no runtime overhead

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