# 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 $$a$$.

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 $$P$$.

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.

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 $$P$$.