# 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(cl::sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, cl::sycl::buffer<T,1> &a, std::int64_t lda, cl::sycl::buffer<T,1> &d, cl::sycl::buffer<T,1> &e, 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.

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.

Throws

This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here.

oneapi::mkl::unimplemented

oneapi::mkl::unsupported_device

oneapi::mkl::lapack::invalid_argument

oneapi::mkl::lapack::computation_error

Exception is thrown in case of problems during calculations. The info code of the problem can be obtained by info() method of exception object:

If $$\text{info}=-i$$, the $$i$$-th parameter had an illegal value.

If info equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should not be less than value return by detail() method of exception object.

## sytrd (USM Version)¶

Syntax

namespace oneapi::mkl::lapack {
cl::sycl::event sytrd(cl::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<cl::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.

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

Throws

This routine shall throw the following exceptions if the associated condition is detected. An implementation may throw additional implementation-specific exception(s) in case of error conditions not covered here.

oneapi::mkl::unimplemented

oneapi::mkl::unsupported_device

oneapi::mkl::lapack::invalid_argument

oneapi::mkl::lapack::computation_error

Exception is thrown in case of problems during calculations. The info code of the problem can be obtained by info() method of exception object:

If $$\text{info}=-i$$, the $$i$$-th parameter had an illegal value.

If info equals to value passed as scratchpad size, and detail() returns non zero, then passed scratchpad is of insufficient size, and required size should not be less than value return by detail() method of exception object.

Return Values

Output event to wait on to ensure computation is complete.