gebrd

Reduces a general matrix to bidiagonal form.

Description

gebrd supports the following precisions.

T
float
double
std::complex<float>
std::complex<double>

The routine reduces a general \(m \times n\) matrix \(A\) to a bidiagonal matrix \(B\) by an orthogonal (unitary) transformation.

If \(m \ge n\), the reduction is given by \(A=QBP^H=\begin{pmatrix}B_1\\0\end{pmatrix}P^H=Q_1B_1P_H\)

where \(B_{1}\) is an \(n \times n\) upper diagonal matrix, \(Q\) and \(P\) are orthogonal or, for a complex \(A\), unitary matrices; \(Q_{1}\) consists of the first \(n\) columns of \(Q\).

If \(m < n\), the reduction is given by

\(A = QBP^H = Q\begin{pmatrix}B_1\\0\end{pmatrix}P^H = Q_1B_1P_1^H\),

where \(B_{1}\) is an \(m \times m\) lower diagonal matrix, \(Q\) and \(P\) are orthogonal or, for a complex \(A\), unitary matrices; \(P_{1}\) consists of the first \(m\) columns of \(P\).

The routine does not form the matrices \(Q\) and \(P\) explicitly, but represents them as products of elementary reflectors. Routines are provided to work with the matrices \(Q\) and \(P\) in this representation:

If the matrix \(A\) is real,

• to compute \(Q\) and \(P\) explicitly, call orgbr.

If the matrix \(A\) is complex,

• to compute \(Q\) and \(P\) explicitly, call ungbr

gebrd (Buffer Version)

Syntax

namespace oneapi::mkl::lapack {
  void gebrd(sycl::queue &queue, std::int64_t m, std::int64_t n, sycl::buffer<T,1> &a, std::int64_t lda, sycl::buffer<realT,1> &d, sycl::buffer<realT,1> &e, sycl::buffer<T,1> &tauq, sycl::buffer<T,1> &taup, 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 the matrix \(A\) (\(0 \le n\)).

a

The buffer \(a\), size (lda,*). The buffer a contains the matrix \(A\). The second dimension of a must be at least \(\max(1, m)\).

lda

The leading dimension of \(a\).

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 gebrd_scratchpad_size function.

Output Parameters

a

If \(m \ge n\), the diagonal and first super-diagonal of a are overwritten by the upper bidiagonal matrix \(B\). The elements below the diagonal, with the buffer tauq, represent the orthogonal matrix \(Q\) as a product of elementary reflectors, and the elements above the first superdiagonal, with the buffer taup, represent the orthogonal matrix \(P\) as a product of elementary reflectors.

If \(m<n\), the diagonal and first sub-diagonal of a are overwritten by the lower bidiagonal matrix \(B\). The elements below the first subdiagonal, with the buffer tauq, represent the orthogonal matrix \(Q\) as a product of elementary reflectors, and the elements above the diagonal, with the buffer taup, represent the orthogonal matrix \(P\) as a product of elementary reflectors.

d

Buffer, size at least \(\max(1, \min(m,n))\). Contains the diagonal elements of \(B\).

e

Buffer, size at least \(\max(1, \min(m,n) - 1)\). Contains the off-diagonal elements of \(B\).

tauq

Buffer, size at least \(\max(1, \min(m, n))\). The scalar factors of the elementary reflectors which represent the orthogonal or unitary matrix \(Q\).

taup

Buffer, size at least \(\max(1, \min(m, n))\). The scalar factors of the elementary reflectors which represent the orthogonal or unitary matrix \(P\).

scratchpad

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

gebrd (USM Version)

Syntax

namespace oneapi::mkl::lapack {
  sycl::event gebrd(sycl::queue &queue, std::int64_t m, std::int64_t n, T *a, std::int64_t lda, RealT *d, RealT *e, T *tauq, T *taup, T *scratchpad, std::int64_t scratchpad_size, const std::vector<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 the matrix \(A\) (\(0 \le n\)).

a

Pointer to matrix \(A\). The second dimension of a must be at least \(\max(1, m)\).

lda

The leading dimension of a.

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 gebrd_scratchpad_size function.

events

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

Output Parameters

a

If \(m \ge n\), the diagonal and first super-diagonal of a are overwritten by the upper bidiagonal matrix \(B\). The elements below the diagonal, with the array tauq, represent the orthogonal matrix \(Q\) as a product of elementary reflectors, and the elements above the first superdiagonal, with the array taup, represent the orthogonal matrix \(P\) as a product of elementary reflectors.

If \(m<n\), the diagonal and first sub-diagonal of a are overwritten by the lower bidiagonal matrix \(B\). The elements below the first subdiagonal, with the array tauq, represent the orthogonal matrix \(Q\) as a product of elementary reflectors, and the elements above the diagonal, with the array taup, represent the orthogonal matrix \(P\) as a product of elementary reflectors.

d

Pointer to memory of size at least \(\max(1, \min(m,n))\). Contains the diagonal elements of \(B\).

e

Pointer to memory of size at least \(\max(1, \min(m,n) - 1)\). Contains the off-diagonal elements of \(B\).

tauq

Pointer to memory of size at least \(\max(1, \min(m, n))\). The scalar factors of the elementary reflectors which represent the orthogonal or unitary matrix \(Q\).

taup

Pointer to memory of size at least \(\max(1, \min(m, n))\). The scalar factors of the elementary reflectors which represent the orthogonal or unitary matrix \(P\).

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 Singular Value and Eigenvalue Problem Routines