QRMatrix (i.e. QR decomposition, QR factorisation or orthogonal-triangular
decomposition) decomposes a scalar/complex matrix \c A into the following
matrix product:
\verbatim
A = Q*R,
\endverbatim
where
\c Q is a unitary similarity matrix,
\c R is an upper triangular matrix.
Usage
Input types:
- \c A can be a \c SquareMatrix<Type> or \c RectangularMatrix<Type>
Output types:
- \c Q is always of the type of the matrix \c A
- \c R is always of the type of the matrix \c A
Options for the output forms of \c QRMatrix (for an (m-by-n) input matrix
\c A with k = min(m, n)):
- outputTypes::FULL_R: computes only \c R (m-by-n)
- outputTypes::FULL_QR: computes both \c R and \c Q (m-by-m)
- outputTypes::REDUCED_R: computes only reduced \c R (k-by-n)
Options where to store \c R:
- storeMethods::IN_PLACE: replaces input matrix content with \c R
- storeMethods::OUT_OF_PLACE: creates new object of \c R
Options for the computation of column pivoting:
- colPivoting::FALSE: switches off column pivoting
- colPivoting::TRUE: switches on column pivoting
Direct solution of linear systems A x = b is possible by solve() alongside
the following limitations:
- \c A = a scalar square matrix
- output type = outputTypes::FULL_QR
- store method = storeMethods::IN_PLACE
Notes
- QR decomposition is not unique if \c R is not positive diagonal \c R.
- The option combination:
- outputTypes::REDUCED_R
- storeMethods::IN_PLACE
will not modify the rows of input matrix \c A after its nth row.
- Both FULL_R and REDUCED_R QR decompositions execute the same number of
operations. Yet REDUCED_R QR decomposition returns only the first n rows
of \c R if m > n for an input m-by-n matrix \c A.
- For m <= n, FULL_R and REDUCED_R will produce the same matrices
