# sygvd¶

Computes all eigenvalues and, optionally, eigenvectors of a real generalized symmetric definite eigenproblem using a divide and conquer method. This routine belongs to the `oneapi::mkl::lapack`namespace.

Syntax

void `sygvd`(cl::sycl::queue &queue, std::int64_t itype, mkl::job jobz, mkl::uplo uplo, std::int64_t n, cl::sycl::buffer<T> &a, std::int64_t lda, cl::sycl::buffer<T> &b, std::int64_t ldb, cl::sycl::buffer<T> &w, cl::sycl::buffer<T> &scratchpad, std::int64_t scratchpad_size)

`sygvd` supports the following precisions and devices:

T

Devices supported

`float`

Host, CPU, GPU

`double`

Host, CPU, GPU

Description

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

`A*x = λ*B*x`, `A*B*x = λ*x`, or `B*A*x = λ*x` .

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

It uses a divide and conquer algorithm.

Input Parameters

queue

Device queue where calculations will be performed.

itype

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

if itype`= 1`, the problem type is `A*x = lambda*B*x;`

if itype`= 2`, the problem type is `A*B*x = lambda*x;`

if itype`= 3`, the problem type is `B*A*x = 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.

uplo

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

If `uplo = uplo::upper`, a and b store the upper triangular part of `A` and `B`.

If `uplo = uplo::lower`, a and b stores the lower triangular part of `A` and `B`.

n

The order of the matrices `A` and `B` (`0≤n`).

a

Buffer holding the array of size `a(lda,*)` containing the upper or lower triangle of the symmetric matrix `A`, as specified by uplo.

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 holding the array of size `b(ldb,*)` containing the upper or lower triangle of the symmetric matrix `B`, as specified by uplo.

The second dimension of b must be at least `max(1, n)`.

ldb

The leading dimension of b; at least `max(1,n)`.

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

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

Output Parameters

a

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

if itype`= 1`or `2` , `Z`T`*B*Z = I`;

if itype`= 3` , `Z`T`*inv(B)*Z = I`;

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

b

On exit, if `info≤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 = L*L`T.

w

Buffer, size at least `n`. If `info = 0`, contains the eigenvalues of the matrix `A` in ascending order. See also `info`.

Exceptions

 mkl::lapack::exception This exception is thrown when problems occur during calculations. You can obtain the info code of the problem using the info() method of the exception object: If `info = -i`, the `i`-th parameter had an illegal value. For `info ≤ n`: If `info = i`, and `jobz = job::novec`, then the algorithm failed to converge; `i` indicates the number of off-diagonal elements of an intermediate tridiagonal form which did not converge to zero. If `info = i`, and `jobz = job:vec`, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and columns `info/(n+1)` through `mod(info,n+1)`. For `info > n`: If `info = n + i`, for `1 ≤ i ≤ n`, then the leading minor of order `i` of `B` is not positive-definite. The factorization of `B` could not be completed and no eigenvalues or eigenvectors were computed. If `info` is equal to the value passed as scratchpad size, and detail() returns non zero, then the passed scratchpad has an insufficient size, and the required size should not be less than the value returned by the detail() method of the exception object.