Pivoted QR Decomposition¶

Given the matrix \(X\) of size \(n \times p\), the problem is to compute the QR decomposition with column pivoting \(XP = QR\), where

\(Q\) is an orthogonal matrix of size \(n \times n\)
\(R\) is a rectangular upper triangular matrix of size \(n \times p\)
\(P\) is a permutation matrix of size \(n \times n\)

The library requires \(n > p\). In this case:

\[\begin{split}XP = QR = [Q_1, Q_2] \cdot \begin{bmatrix} R_1 \\ 0 \end{bmatrix} = Q_1 R_1\end{split}\]

where the matrix \(Q_1\) has the size \(n \times p\) and \(R_1\) has the size \(p \times p\).

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.

Algorithm Input for Pivoted QR Decomposition (Batch Processing)¶
Input ID	Input
`data`	Pointer to the numeric table that represents the \(n \times p\) 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:

Algorithm Parameters for Pivoted QR Decomposition (Batch Processing)¶
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 \(1 \times p\) 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. Note 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.

Algorithm Output for Pivoted QR Decomposition (Batch Processing)¶
Result ID	Result
`matrixQ`	Pointer to the numeric table with the \(n \times p\) matrix \(Q_1\). Note 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 \(p \times p\) upper triangular matrix \(R_1\). Note 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 \(1 \times p\) matrix such that \(\text{permutationMatrix}(i) = k\) if the column \(k\) of the full matrix \(X\) is permuted into the position \(i\) in \(XP\). Note 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.

Examples¶

Batch Processing:

pivoted_qr_dense_batch.cpp

Distributed Processing

Outlier Detection