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# UTRRealMatrix Class Reference

`#include <Matrix.h>`

Inheritance diagram for UTRRealMatrix:

List of all members.

## Detailed Description

Real upper triangular matrix. Index base for rows and columns must be equal. To make the subscripting operator as fast as possible it is limited to access elements on or above the diagonal (where the marix is known to be zero). Out of bounds error below the diagonal

Subscripting speed is crucial. For this reason upper triangular matrices often stand in for symmetric matrices which have a slower subscripting operator. The upper triangular matrix then represents the symmetric matrix of which it is the upper half. This expalains why we implement operations for upper triangular matrics which make sense for symmetric matrices (eigen value analysis, pseudo square roots,...).

Definition at line 1335 of file Matrix.h.

## Public Member Functions

UTRRealMatrix (int d, int b=0)
template<int n>  UTRRealMatrix (Real A[n][n], int b=0)
RealMatrixsymmetricCompletion () const
UTRRealMatrix & inverse () const
LTRRealMatrixltrRoot () const
UTRRealMatrix & utrRoot () const
RealMatrixrankReducedRoot (int r) const
UTRRealMatrix & exp () const
RealMatrixmatrixFunction (Real(*f)(Real))
void analyseFactors (int r) const
void testFactorization (int r, string message="") const

## Constructor & Destructor Documentation

 UTRRealMatrix::UTRRealMatrix ( int d, int b = 0 ) ` [inline]`

Constructor, initializes all components with zeroes.

Parameters:
 d dimension of square matrix b index base: b<=i,j

Definition at line 1343 of file Matrix.h.

References Matrix< Real, UpperTriangular< Real > >::b, and Real.

 template UTRRealMatrix::UTRRealMatrix ( Real A[n][n], int b = 0 ) ` [inline]`

Construct from data array, only upper triangular half is used.

Parameters:
 A data array b index base (default 0).

Definition at line 1351 of file Matrix.h.

References Matrix< Real, UpperTriangular< Real > >::b, and Real.

## Member Function Documentation

 RealMatrix& UTRRealMatrix::symmetricCompletion ( ) const
 The symmetric matrix of which this is the upper half. Return by value is deliberate.

 UTRRealMatrix& UTRRealMatrix::inverse ( ) const
 The matrix inverse. This is upper triangular of the same size.

 LTRRealMatrix& UTRRealMatrix::ltrRoot ( ) const
 Cholesky root L (lower triangular, LL'=A where A is the symmetric matrix with upper half `this`). Terminates with error message if this is not positive definite.

 UTRRealMatrix& UTRRealMatrix::utrRoot ( ) const
 Upper triangular root U of A, where A is the symmetric matrix of which `this`) is the upper half. Satisfies UU'=A. Differs from the Cholesky root as it is upper triangular instead of lower triangular. Terminates with error message if this is not positive definite.

 RealMatrix& UTRRealMatrix::rankReducedRoot ( int r ) const
 Let C be the symmetric matrix with upper half `this`. This function computes the matrix R which best approximates D as a product RR' and has rank r. The matrix R is computed by diagonalizing D, setting all but the r largest eigenvalues equal to zero and taking the square root of the remaining eigenvalues. The row index base remains the same, columns of the root are indexed from zero.

 UTRRealMatrix& UTRRealMatrix::exp ( ) const
 Matrix exponential. Reimplemented from Matrix< Real, UpperTriangular< Real > >.

 RealMatrix& UTRRealMatrix::matrixFunction ( Real(* f)(Real) )
 Computes the function f(A) of the symmetric matrix A of which `this` is the upper half. The matrix funtion f(A) is computed as follows: diagonalize A as is the diagonal matrix with the eigenvalues of A on the diagonal the columns of U are associated eigenvectors. Set (f is applied to each eigenvalue). Then

 void UTRRealMatrix::analyseFactors ( int r ) const
 Matrix is interpreted as the upper half of a multinormal covariance matrix. This method prints how much variability is captured by the r largest eigenvalues of C.

 void UTRRealMatrix::testFactorization ( int r, string message = "" ) const

Let C be the symmetric matrix with upper half `this`. C must be positive semidefinite. Computes the best approximate factorization , where R has rank r and returns the relative error in the trace norm

Parameters:
 message put test in context.

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