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: