# sytrd¶

Reduces a real symmetric matrix to tridiagonal form.

Description

sytrd supports the following precisions.

T

float

double

The routine reduces a real symmetric matrix $$A$$ to symmetric tridiagonal form $$T$$ by an orthogonal similarity transformation: $$A = QTQ^T$$. The orthogonal matrix $$Q$$ is not formed explicitly but is represented as a product of $$n-1$$ elementary reflectors. Routines are provided for working with $$Q$$ in this representation .

## sytrd (Buffer Version)¶

Syntax

namespace oneapi::mkl::lapack {
void sytrd(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, sycl::buffer<T,1> &a, std::int64_t lda, sycl::buffer<T,1> &d, sycl::buffer<T,1> &e, sycl::buffer<T,1> &tau, sycl::buffer<T,1> &scratchpad, std::int64_t scratchpad_size)
}


Input Parameters

queue

The queue where the routine should be executed.

upper_lower

Must be uplo::upper or uplo::lower.

If upper_lower = uplo::upper, a stores the upper triangular part of $$A$$.

If upper_lower = uplo::lower, a stores the lower triangular part of $$A$$.

n

The order of the matrices $$A$$ $$(0 \le n)$$.

a

The buffer a, size (lda,*). Contains the upper or lower triangle of the symmetric matrix $$A$$, as specified by upper_lower.

The second dimension of a must be at least $$\max(1,n)$$.

lda

The leading dimension of a; at least $$\max(1,n)$$.

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

Output Parameters

a

On exit,

if upper_lower = uplo::upper, the diagonal and first superdiagonal of $$A$$ are overwritten by the corresponding elements of the tridiagonal matrix $$T$$, and the elements above the first superdiagonal, with the buffer tau, represent the orthogonal matrix $$Q$$ as a product of elementary reflectors;

if upper_lower = uplo::lower, the diagonal and first subdiagonal of $$A$$ are overwritten by the corresponding elements of the tridiagonal matrix $$T$$, and the elements below the first subdiagonal, with the buffer tau, represent the orthogonal matrix $$Q$$ as a product of elementary reflectors.

d

Buffer containing the diagonal elements of the matrix $$T$$. The dimension of d must be at least $$\max(1, n)$$.

e

Buffer containing the off diagonal elements of the matrix $$T$$. The dimension of e must be at least $$\max(1, n-1)$$.

tau

Buffer, size at least $$\max(1, n)$$. Stores $$(n-1)$$ scalars that define elementary reflectors in decomposition of the unitary matrix $$Q$$ in a product of $$n-1$$ elementary reflectors. $$\tau(n)$$ is used as workspace.

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

## sytrd (USM Version)¶

Syntax

namespace oneapi::mkl::lapack {
sycl::event sytrd(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, T *a, std::int64_t lda, T *d, T *e, T *tau, T *scratchpad, std::int64_t scratchpad_size, const std::vector<sycl::event> &events = {})
}


Input Parameters

queue

The queue where the routine should be executed.

upper_lower

Must be uplo::upper or uplo::lower.

If upper_lower = uplo::upper, a stores the upper triangular part of $$A$$.

If upper_lower = uplo::lower, a stores the lower triangular part of $$A$$.

n

The order of the matrices $$A$$ $$(0 \le n)$$.

a

The pointer to matrix $$A$$, size (lda,*). Contains the upper or lower triangle of the symmetric matrix $$A$$, as specified by upper_lower. The second dimension of a must be at least $$\max(1,n)$$.

lda

The leading dimension of a; at least $$\max(1,n)$$.

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

events

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

Output Parameters

a

On exit,

if upper_lower = uplo::upper, the diagonal and first superdiagonal of $$A$$ are overwritten by the corresponding elements of the tridiagonal matrix $$T$$, and the elements above the first superdiagonal, with the array tau, represent the orthogonal matrix $$Q$$ as a product of elementary reflectors;

if upper_lower = uplo::lower, the diagonal and first subdiagonal of $$A$$ are overwritten by the corresponding elements of the tridiagonal matrix $$T$$, and the elements below the first subdiagonal, with the array tau, represent the orthogonal matrix $$Q$$ as a product of elementary reflectors.

d

Pointer to diagonal elements of the matrix $$T$$. The dimension of d must be at least $$\max(1, n)$$.

e

Pointer to off diagonal elements of the matrix $$T$$. The dimension of e must be at least $$\max(1, n-1)$$.

tau

Pointer to array of size at least $$\max(1, n)$$. Stores $$(n-1)$$ scalars that define elementary reflectors in decomposition of the unitary matrix $$Q$$ in a product of $$n-1$$ elementary reflectors. $$\tau(n)$$ is used as workspace.