Pivoted QR Decomposition

Given the matrix X
of size $LaTex Math image.$ , the problem is to compute the QR decomposition with column pivoting XP = QR
, where

Q
is an orthogonal matrix of size $LaTex Math image.$

R
is a rectangular upper triangular matrix of size $LaTex Math image.$

P
is a permutation matrix of size $LaTex Math image.$

The library requires $LaTex Math image.$ . In this case:

$LaTex Math image.$

where the matrix $LaTex Math image.$ has the size $LaTex Math image.$ and $LaTex Math image.$ has the size $LaTex Math image.$ .

Batch Processing

Algorithm Input

Pivoted QR decomposition accepts the input described below. Pass the

Input ID

as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Input ID	Input
data	Pointer to the numeric table that represents the $LaTex Math image.$ matrix X to be factorized. The input can be an object of any class derived from NumericTable .

Algorithm Parameters

Pivoted QR decomposition has the following parameters:

Parameter	Default Value	Description
algorithmFPType	float	The floating-point type that the algorithm uses for intermediate computations. Can be float or double .
method	defaultDense	Performance-oriented computation method, the only method supported by the algorithm.
permutedColumns	Not applicable	Pointer to the numeric table with the $LaTex Math image.$ matrix with the information for the permutation: If the i -th element is zero, the i -th column of the input matrix is a free column and may be permuted with any other free column during the computation. If the i -th element is non-zero, the i -th column of the input matrix is moved to the beginning of XP before the computation and remains in its place during the computation. By default, this parameter is an object of the HomogenNumericTable class, filled by zeros. However, you can define this parameter as an object of any class derived from NumericTable except the PackedSymmetricMatrix class, CSRNumericTable class, and PackedTriangularMatrix class with the lowerPackedTriangularMatrix layout.

Algorithm Output

Pivoted QR decomposition calculates the results described below. Pass the

Result ID

as a parameter to the methods that access the results of your algorithm. For more details, see Algorithms.

Result ID	Result
matrixQ	Pointer to the numeric table with the $LaTex Math image.$ matrix $LaTex Math image.$ . By default, this result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except PackedSymmetricMatrix , PackedTriangularMatrix , and CSRNumericTable .
matrixR	Pointer to the numeric table with the $LaTex Math image.$ upper triangular matrix $LaTex Math image.$ . By default, this result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except the PackedSymmetricMatrix class, CSRNumericTable class, and PackedTriangularMatrix class with the lowerPackedTriangularMatrix layout.
permutationMatrix	Pointer to the numeric table with the $LaTex Math image.$ matrix such that $LaTex Math image.$ if the column k of the full matrix X is permuted into the position i in XP . By default, this result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except the PackedSymmetricMatrix class, CSRNumericTable class, and PackedTriangularMatrix class with the lowerPackedTriangularMatrix layout.