# sygvd¶

Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem using a divide and conquer method.

Description

sygvd supports the following precisions.

T

float

double

The routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form

$$Ax = \lambda Bx$$, $$ABx = \lambda x$$, or $$BAx = \lambda x$$ .

Here $$A$$ and $$B$$ are assumed to be symmetric and $$B$$ is also positive definite.

It uses a divide and conquer algorithm.

## sygvd (Buffer Version)¶

Syntax

namespace oneapi::mkl::lapack {
void sygvd(sycl::queue &queue, std::int64_t itype, oneapi::mkl::job jobz, oneapi::mkl::uplo upper_lower, std::int64_t n, sycl::buffer<T,1> &a, std::int64_t lda, sycl::buffer<T,1> &b, std::int64_t ldb, sycl::buffer<T,1> &w, sycl::buffer<T,1> &scratchpad, std::int64_t scratchpad_size)
}


Input Parameters

queue

The queue where the routine should be executed.

itype

Must be 1 or 2 or 3. Specifies the problem type to be solved:

if $$\text{itype} = 1$$, the problem type is $$Ax = \lambda Bx$$;

if $$\text{itype} = 2$$, the problem type is $$ABx = \lambda x$$;

if $$\text{itype} = 3$$, the problem type is $$BAx = \lambda x$$.

jobz

Must be job::novec or job::vec.

If jobz = job::novec, then only eigenvalues are computed.

If jobz = job::vec, then eigenvalues and eigenvectors are computed.

upper_lower

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

If upper_lower = job::upper, a and b store the upper triangular part of $$A$$ and $$B$$.

If upper_lower = job::lower, a and b stores the lower triangular part of $$A$$ and $$B$$.

n

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

a

Buffer, size a(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 $$\max(1, n)$$.

b

Buffer, size b (ldb,*) contains the upper or lower triangle of the symmetric matrix $$B$$, as specified by upper_lower. The second dimension of b must be at least $$\max(1, n)$$.

ldb

The leading dimension of b; 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 sygvd_scratchpad_size function.

Output Parameters

a

On exit, if jobz = job::vec, then if $$\text{info} = 0$$, a contains the matrix $$Z$$ of eigenvectors. The eigenvectors are normalized as follows:

if $$\text{itype} = 1$$ or $$2$$ , $$Z^{T}BZ = I$$;

if $$\text{itype} = 3$$ , $$Z^{T}B^{-1}Z = I$$;

If jobz = job::novec, then on exit the upper triangle (if upper_lower = uplo::upper) or the lower triangle (if upper_lower = uplo::lower) of $$A$$, including the diagonal, is destroyed.

b

On exit, if $$\text{info} \le n$$, the part of b containing the matrix is overwritten by the triangular factor $$U$$ or $$L$$ from the Cholesky factorization $$B = U^{T}U$$ or $$B = LL^{T}$$.

w

Buffer, size at least $$n$$. If $$\text{info} = 0$$, contains the eigenvalues of the matrix $$A$$ in ascending order.

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

## sygvd (USM Version)¶

Syntax

namespace oneapi::mkl::lapack {
sycl::event sygvd(sycl::queue &queue, std::int64_t itype, oneapi::mkl::job jobz, oneapi::mkl::uplo upper_lower, std::int64_t n, T *a, std::int64_t lda, T *b, std::int64_t ldb, T *w, T *scratchpad, std::int64_t scratchpad_size, const std::vector<sycl::event> &events = {})
}


Input Parameters

queue

The queue where the routine should be executed.

itype

Must be 1 or 2 or 3. Specifies the problem type to be solved:

if $$\text{itype} = 1$$, the problem type is $$Ax = \lambda Bx$$;

if $$\text{itype} = 2$$, the problem type is $$ABx = \lambda x$$;

if $$\text{itype} = 3$$, the problem type is $$BAx = \lambda x$$.

jobz

Must be job::novec or job::vec.

If jobz = job::novec, then only eigenvalues are computed.

If jobz = job::vec, then eigenvalues and eigenvectors are computed.

upper_lower

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

If upper_lower = job::upper, a and b store the upper triangular part of $$A$$ and $$B$$.

If upper_lower = job::lower, a and b stores the lower triangular part of $$A$$ and $$B$$.

n

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

a

Pointer to array of size a(lda,*) containing 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 $$\max(1, n)$$.

b

Pointer to array of size b (ldb,*) contains the upper or lower triangle of the symmetric matrix $$B$$, as specified by upper_lower. The second dimension of b must be at least $$\max(1, n)$$.

ldb

The leading dimension of b; 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 sygvd_scratchpad_size function.

events

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

Output Parameters

a

On exit, if jobz = job::vec, then if $$\text{info} = 0$$, $$a$$ contains the matrix $$Z$$ of eigenvectors. The eigenvectors are normalized as follows:

if $$\text{itype} = 1$$ or $$2$$, $$Z^{T}BZ = I$$;

if $$\text{itype} = 3$$, $$Z^{T}B^{-1}Z = I$$;

If jobz = job::novec, then on exit the upper triangle (if upper_lower = uplo::upper) or the lower triangle (if upper_lower = uplo::lower) of $$A$$, including the diagonal, is destroyed.

b

On exit, if $$\text{info} \le n$$, the part of b containing the matrix is overwritten by the triangular factor $$U$$ or $$L$$ from the Cholesky factorization $$B$$ = $$U^{T}U$$ or $$B = LL^{T}$$.

w

Pointer to array of size at least n. If $$\text{info} = 0$$, contains the eigenvalues of the matrix $$A$$ in ascending order.