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

Matrix adapter for strictly lower triangular $ N \times N $ matrices. More...

#include <BaseTemplate.h>

Detailed Description

template<typename MT, bool SO = IsColumnMajorMatrix<MT>::value, bool DF = IsDenseMatrix<MT>::value>
class blaze::StrictlyLowerMatrix< MT, SO, DF >

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

General

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

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

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

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.


Special Properties of Strictly Lower Triangular Matrices

A strictly lower 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 lower triangular matrix constraint:

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


Strictly Lower 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 lower dynamic matrix
StrictlyLowerMatrix< 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 lower static matrix
StrictlyLowerMatrix< StaticMatrix<int,3UL,3UL,columnMajor> > A;
// Compilation error: the provided matrix type is not a square matrix type
StrictlyLowerMatrix< StaticMatrix<int,3UL,4UL,columnMajor> > B;


The Strictly Lower Triangular Matrix Property is Always Enforced!

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

typedef StrictlyLowerMatrix< CompressedMatrix<double,rowMajor> > CompressedStrictlyLower;
// Default constructed, row-major 3x3 strictly lower compressed matrix
CompressedStrictlyLower A( 3 );
// Initializing elements via the function call operator
A(0,0) = 9.0; // Throws an exception; invalid modification of diagonal element
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 elements via the insert() function
A.insert( 1, 0, 3.0 ); // Inserting the lower element (1,0)
A.insert( 1, 1, 9.0 ); // Throws an exception; invalid insertion of diagonal element
A.insert( 0, 2, 9.0 ); // Throws an exception; invalid insertion of upper element
// Appending an element via the append() function
A.reserve( 2, 2 ); // Reserving enough capacity in row 2
A.append( 1, 1, 9.0 ); // Throws an exception; appending a diagonal element
A.append( 2, 1, 4.0 ); // Appending the lower element (2,1)
// Access via a non-const iterator
*it = 7.0; // Modifies the lower element (2,0)
++it;
*it = 8.0; // Modifies the lower element (2,1)
// 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 strictly lower dense matrix
StaticMatrix<double,3UL,3UL> B( 0.0, 0.0, 0.0,
8.0, 0.0, 0.0,
-2.0, -1.0, 0.0 );
StrictlyLowerMatrix< DynamicMatrix<double,rowMajor> > C( B ); // OK
// Assignment of a general 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; strictly lower triangular matrix invariant would be violated!

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

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

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

// Setup of the strictly lower matrix
//
// ( 0 0 0 0 )
// A = ( 2 0 0 0 )
// ( 0 3 0 0 )
// ( 4 0 5 0 )
//
StrictlyLowerMatrix< DynamicMatrix<int> > A( 4 );
A(1,0) = 2;
A(2,1) = 3;
A(3,0) = 4;
A(3,2) = 5;
// Setting the lower elements in the 2nd row to 9 results in the matrix
//
// ( 0 0 0 0 )
// A = ( 2 0 0 0 )
// ( 9 9 0 0 )
// ( 4 0 5 0 )
//
row( A, 2 ) = 9;
// Setting the lower elements in the 1st and 2nd column to 7 results in
//
// ( 0 0 0 0 )
// A = ( 1 0 0 0 )
// ( 9 7 0 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 strictly lower matrices. Since only lower elements may be modified the matrix to be assigned must be structured such that the strictly lower triangular matrix invariant of the strictly lower matrix is preserved. Otherwise a std::invalid_argument exception is thrown:

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


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

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

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


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

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


Arithmetic Operations

A StrictlyLowerMatrix 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 StrictlyLowerMatrix within arithmetic operations:

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


Block-Structured Strictly Lower Matrices

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

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

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

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


Performance Considerations

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


Positive Impact: Matrix/Matrix Multiplication

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

StrictlyLowerMatrix< DynamicMatrix<double,rowMajor> > A;
StrictlyLowerMatrix< 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 StrictlyLowerMatrix adaptor when a matrix is known to be strictly lower 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 lower triangular matrix in a matrix/vector multiplication:

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

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

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

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

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