geqrf

Computes the QR factorization of a general \(m \times n\) matrix.

Description

geqrf supports the following precisions:

T

float

double

std::complex<float>

std::complex<double>

The routine forms the QR factorization of a general \(m \times n\) matrix \(A\). No pivoting is performed.

The routine does not form the matrix \(Q\) explicitly. Instead, \(Q\) is represented as a product of \(\min(m, n)\) elementary reflectors. Routines are provided to work with \(Q\) in this representation.

geqrf (Buffer Version)

Syntax

namespace oneapi::mkl::lapack {
  void geqrf(cl::sycl::queue &queue, std::int64_t m, std::int64_t n, cl::sycl::buffer<T,1> &a, std::int64_t lda, cl::sycl::buffer<T,1> &tau, cl::sycl::buffer<T,1> &scratchpad, std::int64_t scratchpad_size)
}

Input Parameters

queue

The queue where the routine should be executed.

m

The number of rows in the matrix \(A\) (\(0 \le m\)).

n

The number of columns in \(A\) (\(0 \le n\)).

a

Buffer holding input matrix \(A\). Must have size at least \(\text{lda} \cdot n\).

lda

The leading dimension of \(A\); at least \(\max(1, m)\).

scratchpad_size

Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by geqrf_scratchpad_size function.

Output Parameters

a

Output buffer, overwritten by the factorization data as follows:

The elements on and above the diagonal of the array contain the \(\min(m,n) \times n\) upper trapezoidal matrix \(R\) (\(R\) is upper triangular if \(m \ge n\)); the elements below the diagonal, with the array tau, represent the orthogonal matrix \(Q\) as a product of \(\min(m,n)\) elementary reflectors.

tau

Output buffer, size at least \(\max(1, \min(m, n))\). Contains scalars that define elementary reflectors for the matrix \(Q\) in its decomposition in a product of elementary reflectors.

scratchpad

Buffer holding scratchpad memory to be used by routine for storing intermediate results.

geqrf (USM Version)

Syntax

namespace oneapi::mkl::lapack {
  cl::sycl::event geqrf(cl::sycl::queue &queue, std::int64_t m, std::int64_t n, T *a, std::int64_t lda, T *tau, T *scratchpad, std::int64_t scratchpad_size, const cl::sycl::vector_class<cl::sycl::event> &events = {})
}

Input Parameters

queue

The queue where the routine should be executed.

m

The number of rows in the matrix \(A\) (\(0 \le m\)).

n

The number of columns in \(A\) (\(0 \le n\)).

a

Pointer to memory holding input matrix \(A\). Must have size at least \(\text{lda} \cdot n\).

lda

The leading dimension of \(A\); at least \(\max(1, m)\).

scratchpad_size

Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by geqrf_scratchpad_size function.

events

List of events to wait for before starting computation. Defaults to empty list.

Output Parameters

a

Overwritten by the factorization data as follows:

The elements on and above the diagonal of the array contain the \(\min(m,n) \times n\) upper trapezoidal matrix \(R\) (\(R\) is upper triangular if \(m \ge n\)); the elements below the diagonal, with the array tau, represent the orthogonal matrix \(Q\) as a product of \(\min(m,n)\) elementary reflectors.

tau

Array, size at least \(\max(1, \min(m, n))\). Contains scalars that define elementary reflectors for the matrix \(Q\) in its decomposition in a product of elementary reflectors.

scratchpad

Pointer to scratchpad memory to be used by routine for storing intermediate results.

Return Values

Output event to wait on to ensure computation is complete.

Parent topic: LAPACK Linear Equation Routines