blaze::StrictlyUpperMatrix< MT, SO, DF > Class Template Reference

Matrix adapter for strictly upper triangular $ 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::StrictlyUpperMatrix< MT, SO, DF >

Matrix adapter for strictly upper triangular $ N \times N $ matrices.

General

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

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

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
// Definition of a resizable column-major dense strictly upper matrix based on HybridMatrix
// Definition of a resizable row-major dense strictly upper matrix based on DynamicMatrix
// Definition of a fixed-size row-major dense strictly upper matrix based on CustomMatrix
// Definition of a compressed row-major single precision strictly upper matrix

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.


Special Properties of Strictly Upper Triangular Matrices

A strictly upper 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 strictly upper triangular matrix constraint:

  1. Strictly Upper Triangular Matrices Must Always be Square!
  2. The Strictly Upper triangular Matrix Property is Always Enforced!
  3. The Diagonal and Lower Elements of a Dense Strictly Upper Triangular Matrix are Always Default Initialized!
  4. Dense Strictly Upper Matrices Also Store the Diagonal and Lower Elements!


Strictly Upper 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):

// Default constructed, default initialized, row-major 3x3 strictly upper dynamic matrix
StrictlyUpperMatrix< 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 strictly upper static matrix
StrictlyUpperMatrix< StaticMatrix<int,3UL,3UL,columnMajor> > A;
// Compilation error: the provided matrix type is not a square matrix type
StrictlyUpperMatrix< StaticMatrix<int,3UL,4UL,columnMajor> > B;


The Strictly Upper triangular Matrix Property is Always Enforced!

This means that it is only allowed to modify elements in the upper part of the matrix, but not the elements on the diagonal or in the lower part of the matrix. Also, it is only possible to assign matrices that are strictly upper triangular matrices themselves:

using CompressedStrictlyUpper = StrictlyUpperMatrix< CompressedMatrix<double,rowMajor> >;
// Default constructed, row-major 3x3 strictly upper compressed matrix
CompressedStrictlyUpper 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, 1 ); // 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
*it = 7.0; // Modifies the upper element (0,1)
++it;
*it = 8.0; // Modifies the upper element (0,2)
// 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 a strictly upper dense matrix
StaticMatrix<double,3UL,3UL> B( { { 0.0, 8.0, -2.0 },
{ 0.0, 0.0, -1.0 },
{ 0.0, 0.0, 0.0 } } );
StrictlyUpperMatrix< DynamicMatrix<double,rowMajor> > C( B ); // OK
// Assignment of a general 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; strictly upper triangular matrix invariant would be violated!

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

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

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

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

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

// Setup of two default 4x4 upper matrices
StrictlyUpperMatrix< DynamicMatrix<int> > A1( 4 ), A2( 4 );
// Setup of a 4-dimensional vector
//
// v = ( 0 0 2 3 )
//
DynamicVector<int,rowVector> v( 4, 0 );
v[2] = 2;
v[3] = 3;
// OK: Assigning v to the 1st row of A1 preserves the upper matrix invariant
//
// ( 0 0 0 0 )
// A1 = ( 0 0 2 3 )
// ( 0 0 0 0 )
// ( 0 0 0 0 )
//
row( A1, 1 ) = v; // OK
// Error: Assigning v to the 2nd row of A1 violates the strictly upper matrix invariant! The
// element marked with X cannot be assigned and triggers an exception.
//
// ( 0 0 0 0 )
// A1 = ( 0 0 2 3 )
// ( 0 0 X 3 )
// ( 0 0 0 0 )
//
row( A1, 2 ) = v; // Assignment throws an exception!
// Setup of the 3x2 dynamic matrix
//
// ( 7 8 )
// B = ( 0 9 )
// ( 0 0 )
//
DynamicMatrix<int> B( 3UL, 2UL, 0 );
B(0,0) = 7;
B(0,1) = 8;
B(1,1) = 9;
// OK: Assigning B to a submatrix of A2 such that the invariant can be preserved
//
// ( 0 7 8 0 )
// A2 = ( 0 0 9 0 )
// ( 0 0 0 0 )
// ( 0 0 0 0 )
//
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 = ( 0 X 9 0 )
// ( 0 0 0 0 )
// ( 0 0 0 0 )
//
submatrix( A2, 0UL, 0UL, 3UL, 2UL ) = B; // Assignment throws an exception!


The Diagonal and Lower Elements of a Dense Strictly Upper Triangular Matrix are Always Default Initialized!

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

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


Dense Strictly Upper Matrices Also Store the Diagonal and Lower Elements!

It is important to note that dense strictly upper matrices store all elements, including the elements on the diagonal and 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 diagonal and lower elements guarantees maximum performance for many algorithms that perform vectorized operations on the upper matrix, which is especially true for small dense matrices. Second, conceptually the StrictlyUpperMatrix adaptor merely restricts the interface to the matrix type MT and does not change the data layout or the underlying matrix type.


Arithmetic Operations

An StrictlyUpperMatrix 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 StrictlyUpperMatrix within arithmetic operations:

DynamicMatrix<double,rowMajor> A( 3, 3 );
CompressedMatrix<double,rowMajor> B( 3, 3 );
StrictlyUpperMatrix< DynamicMatrix<double,rowMajor> > C( 3 );
StrictlyUpperMatrix< CompressedMatrix<double,rowMajor> > D( 3 );
StrictlyUpperMatrix< HybridMatrix<float,3UL,3UL,rowMajor> > E;
StrictlyUpperMatrix< StaticMatrix<float,3UL,3UL,columnMajor> > F;
E = A + B; // Matrix addition and assignment to a row-major strictly upper matrix (includes runtime check)
F = A - C; // Matrix subtraction and assignment to a column-major strictly upper 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 strictly upper matrix. In case the matrix to be assigned is not strictly upper at compile time, a runtime check is performed.


Block-Structured Strictly Upper Matrices

It is also possible to use block-structured strictly upper matrices:

// Definition of a 5x5 block-structured strictly upper matrix based on CompressedMatrix
StrictlyUpperMatrix< CompressedMatrix< StaticMatrix<int,3UL,3UL> > > A( 5 );

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

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


Performance Considerations

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


Positive Impact: Matrix/Matrix Multiplication

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

StrictlyUpperMatrix< DynamicMatrix<double,rowMajor> > A;
StrictlyUpperMatrix< 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 and medium-sized matrices. Therefore is it highly recommended to use the StrictlyUpperMatrix adaptor when a matrix is known to be strictly upper triangular. 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 strictly upper triangular matrix in a matrix/vector multiplication:

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

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

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

StrictlyUpperMatrix< DynamicMatrix<double> > A, B, C;
C = A + B; // Results in a strictly upper matrix; no runtime overhead
C = A - B; // Results in a strictly upper matrix; no runtime overhead
C = A * B; // Results in a strictly upper matrix; no runtime overhead

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