geqrf.h File Reference

Routines to compute a Householder QR decomposition. More...

#include <myramath/MYRAMATH_EXPORT.h>
#include <myramath/utility/Number.h>
#include <myramath/dense/Vector.h>

Include dependency graph for geqrf.h:

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Classes
class	myra::CMatrixRange< Number >
	Represents a const MatrixRange. More...
class	myra::MatrixRange< Number >
	Represents a mutable MatrixRange. More...

Functions
Vector< NumberS >	myra::geqrf_inplace (const MatrixRange< NumberS > &A)
	Factors A=QR, overwriting triu(A) with R and tril(A) with Householder reflectors. Returns tau[].
Vector< NumberD >	myra::geqrf_inplace (const MatrixRange< NumberD > &A)
	Factors A=QR, overwriting triu(A) with R and tril(A) with Householder reflectors. Returns tau[].
Vector< NumberC >	myra::geqrf_inplace (const MatrixRange< NumberC > &A)
	Factors A=QR, overwriting triu(A) with R and tril(A) with Householder reflectors. Returns tau[].
Vector< NumberZ >	myra::geqrf_inplace (const MatrixRange< NumberZ > &A)
	Factors A=QR, overwriting triu(A) with R and tril(A) with Householder reflectors. Returns tau[].

Detailed Description

Routines to compute a Householder QR decomposition.

The orthogonal matrix Q is not tabulated explicitly, rather it 
is stored in factored form (a rank-K perturbation of the identity,
a product of rank-1 Householder reflectors). This factored form 
overwrites the strictly lower triangle portion of the input A. 
The triangular (or trapezoidal) matrix R is tabulated explicity 
and overwrites the upper triangle of A. There are three shapes 
to consider:

Case 1 (A tall/skinny), I > J:
    
                       [\ R ]
    A   =   Q  *   R   [ \  ]
   IxJ     IxI    IxJ  [  \ ]
         (rank J)      [   \]
                       [ H  ]
                       [    ]
                       [    ]

Case 2 (A short/fat), I < J:
   
   A    =   Q  *   R   [\          ]
  IxJ      IxI    IxJ  [ \   R     ]
         (rank I)      [H \        ]

Case 3 (A square), I = J = K:

                       [\  R]
    A   =   Q  *   R   [ \  ]
   KxK     KxK    KxK  [  \ ]
         (rank K)      [H  \]

In all cases, if K = min(I,J), then Q is a rank-K pertubation of the IxI identity, and R is IxJ.