# gesvd¶

Computes the singular value decomposition of a general rectangular matrix.

Description

gesvd supports the following precisions.

T

float

double

std::complex<float>

std::complex<double>

## gesvd (Buffer Version)¶

Description

The routine computes the singular value decomposition (SVD) of a real/complex $$m \times n$$ matrix $$A$$, optionally computing the left and/or right singular vectors. The SVD is written as

$$A = U\Sigma V^T$$ for real routines

$$A = U\Sigma V^H$$ for complex routines

where $$\Sigma$$ is an $$m \times n$$ diagonal matrix, $$U$$ is an $$m \times m$$ orthogonal/unitary matrix, and $$V$$ is an $$n \times n$$ orthogonal/unitary matrix. The diagonal elements of $$\Sigma$$ are the singular values of $$A$$; they are real and non-negative, and are returned in descending order. The first $$\min(m, n)$$ columns of $$U$$ and $$V$$ are the left and right singular vectors of $$A$$.

Syntax

namespace oneapi::mkl::lapack {
void gesvd(cl::sycl::queue &queue, oneapi::mkl::job jobu, oneapi::mkl::job jobvt, std::int64_t m, std::int64_t n, cl::sycl::buffer<T,1> &a, std::int64_t lda, cl::sycl::buffer<realT,1> &s, cl::sycl::buffer<T,1> &u, std::int64_t ldu, cl::sycl::buffer<T,1> &vt, std::int64_t ldvt, cl::sycl::buffer<T,1> &scratchpad, std::int64_t scratchpad_size)
}


Input Parameters

queue

The queue where the routine should be executed.

jobu

Must be job::allvec, job::somevec, job::overwritevec, or job::novec. Specifies options for computing all or part of the matrix $$U$$.

If jobu = job::allvec, all $$m$$ columns of $$U$$ are returned in the buffer u;

if jobu = job::somevec, the first $$\min(m, n)$$ columns of $$U$$ (the left singular vectors) are returned in the buffer u;

if jobu = job::overwritevec, the first $$\min(m, n)$$ columns of $$U$$ (the left singular vectors) are overwritten on the buffer a;

if jobu = job::novec, no columns of $$U$$ (no left singular vectors) are computed.

jobvt

Must be job::allvec, job::somevec, job::overwritevec, or job::novec. Specifies options for computing all or part of the matrix $$V^T/V^H$$.

If jobvt = job::allvec, all $$n$$ columns of $$V^T/V^H$$ are returned in the buffer vt;

if jobvt = job::somevec, the first $$\min(m, n)$$ columns of $$V^T/V^H$$ (the left singular vectors) are returned in the buffer vt;

if jobvt = job::overwritevec, the first $$\min(m, n)$$ columns of $$V^T/V^H$$ (the left singular vectors) are overwritten on the buffer a;

if jobvt = job::novec, no columns of $$V^T/V^H$$ (no left singular vectors) are computed.

jobvt and jobu cannot both be job::overwritevec.

m

The number of rows in the matrix $$A$$ ($$0 \le m$$).

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.

ldu

The leading dimension of u.

ldvt

The leading dimension of vt.

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

Output Parameters

a

On exit,

If jobu = job::overwritevec, a is overwritten with the first $$\min(m,n)$$ columns of $$U$$ (the left singular vectors stored columnwise);

If jobvt = job::overwritevec, a is overwritten with the first $$\min(m, n)$$ rows of $$V^{T}$$/$$V^{H}$$ (the right singular vectors stored rowwise);

If jobu $$\ne$$ job::overwritevec and jobvt $$\ne$$ job::overwritevec, the contents of a are destroyed.

s

Buffer containing the singular values, size at least $$\max(1, \min(m,n))$$. Contains the singular values of $$A$$ sorted so that $$s(i) \ge s(i+1)$$.

u

Buffer containing $$U$$; the second dimension of u must be at least $$\max(1, m)$$ if jobu = job::allvec, and at least $$\max(1, \min(m, n))$$ if jobu = job::somevec.

If jobu = job::allvec, u contains the $$m \times m$$ orthogonal/unitary matrix $$U$$.

If jobu = job::somevec, u contains the first $$\min(m, n)$$ columns of $$U$$ (the left singular vectors stored column-wise).

If jobu = job::novec or job::overwritevec, u is not referenced.

vt

Buffer containing $$V^{T}$$; the second dimension of vt must be at least $$\max(1, n)$$.

If jobvt = job::allvec, vt contains the $$n \times n$$ orthogonal/unitary matrix $$V^{T}$$/$$V^{H}$$.

If jobvt = job::somevec, vt contains the first $$\min(m, n)$$ rows of $$V^{T}$$/$$V^{H}$$ (the right singular vectors stored row-wise).

If jobvt = job::novec or job::overwritevec, vt is not referenced.

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 info=-i, the $$i$$-th parameter had an illegal value.

If info=i, then if bdsqr did not converge, $$i$$ specifies how many superdiagonals of the intermediate bidiagonal form $$B$$ did not converge to zero, and scratchpad(2:min(m,n)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix $$B$$ whose diagonal is in s (not necessarily sorted). $$B$$ satisfies $$A = UBV^{T}$$, so it has the same singular values as $$A$$, and singular vectors related by $$U$$ and $$V^{T}$$.

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.

## gesvd (USM Version)¶

Description

The routine computes the singular value decomposition (SVD) of a real/complex $$m \times n$$ matrix $$A$$, optionally computing the left and/or right singular vectors. The SVD is written as

$$A = U\Sigma V^T$$ for real routines

$$A = U\Sigma V^H$$ for complex routines

where $$\Sigma$$ is an $$m \times n$$ diagonal matrix, $$U$$ is an $$m \times m$$ orthogonal/unitary matrix, and $$V$$ is an $$n \times n$$ orthogonal/unitary matrix. The diagonal elements of $$\Sigma$$ are the singular values of $$A$$; they are real and non-negative, and are returned in descending order. The first $$\min(m, n)$$ columns of $$U$$ and $$V$$ are the left and right singular vectors of $$A$$.

Syntax

namespace oneapi::mkl::lapack {
cl::sycl::event gesvd(cl::sycl::queue &queue, oneapi::mkl::job jobu, oneapi::mkl::job jobvt, std::int64_t m, std::int64_t n, T *a, std::int64_t lda, RealT *s, T *u, std::int64_t ldu, T *vt, std::int64_t ldvt, T *scratchpad, std::int64_t scratchpad_size, const cl::sycl::vector_class<cl::sycl::event> &events = {})
}


Input Parameters

queue

The queue where the routine should be executed.

jobu

Must be job::allvec, job::somevec, job::overwritevec, or job::novec. Specifies options for computing all or part of the matrix $$U$$.

If jobu = job::allvec, all $$m$$ columns of $$U$$ are returned in the array u;

if jobu = job::somevec, the first $$\min(m, n)$$ columns of $$U$$ (the left singular vectors) are returned in the array u;

if jobu = job::overwritevec, the first $$\min(m, n)$$ columns of $$U$$ (the left singular vectors) are overwritten on the array a;

if jobu = job::novec, no columns of $$U$$ (no left singular vectors) are computed.

jobvt

Must be job::allvec, job::somevec, job::overwritevec, or job::novec. Specifies options for computing all or part of the matrix $$V^T/V^H$$.

If jobvt = job::allvec, all $$n$$ columns of $$V^T/V^H$$ are returned in the array vt;

if jobvt = job::somevec, the first $$\min(m, n)$$ columns of $$V^T/V^H$$ (the left singular vectors) are returned in the array vt;

if jobvt = job::overwritevec, the first $$\min(m, n)$$ columns of $$V^T/V^H$$ (the left singular vectors) are overwritten on the array a;

if jobvt = job::novec, no columns of $$V^T/V^H$$ (no left singular vectors) are computed.

jobvt and jobu cannot both be job::overwritevec.

m

The number of rows in the matrix $$A$$ ($$0 \le m$$).

a

Pointer to array a, size (lda,*), containing the matrix $$A$$. The second dimension of a must be at least $$\max(1, m)$$.

lda

The leading dimension of a.

ldu

The leading dimension of u.

ldvt

The leading dimension of vt.

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

events

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

Output Parameters

a

On exit,

If jobu = job::overwritevec, a is overwritten with the first $$\min(m,n)$$ columns of $$U$$ (the left singular vectors stored columnwise);

If jobvt = job::overwritevec, a is overwritten with the first $$\min(m, n)$$ rows of $$V^{T}$$/$$V^{H}$$ (the right singular vectors stored rowwise);

If jobu $$\ne$$ job::overwritevec and jobvt $$\ne$$ job::overwritevec, the contents of a are destroyed.

s

Array containing the singular values, size at least $$\max(1, \min(m,n))$$. Contains the singular values of $$A$$ sorted so that $$s(i) \ge s(i+1)$$.

u

Array containing $$U$$; the second dimension of u must be at least $$\max(1, m)$$ if jobu = job::allvec, and at least $$\max(1, \min(m, n))$$ if jobu = job::somevec.

If jobu = job::allvec, u contains the $$m \times m$$ orthogonal/unitary matrix $$U$$.

If jobu = job::somevec, u contains the first $$\min(m, n)$$ columns of $$U$$ (the left singular vectors stored column-wise).

If jobu = job::novec or job::overwritevec, u is not referenced.

vt

Array containing $$V^{T}$$; the second dimension of vt must be at least $$\max(1, n)$$.

If jobvt = job::allvec, vt contains the $$n \times n$$ orthogonal/unitary matrix $$V^{T}$$/$$V^{H}$$.

If jobvt = job::somevec, vt contains the first $$\min(m, n)$$ rows of $$V^{T}$$/$$V^{H}$$ (the right singular vectors stored row-wise).

If jobvt = job::novec or job::overwritevec, vt is not referenced.

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 info=-i, the $$i$$-th parameter had an illegal value.

If info=i, then if bdsqr did not converge, $$i$$ specifies how many superdiagonals of the intermediate bidiagonal form $$B$$ did not converge to zero, and scratchpad(2:min(m,n)) contains the unconverged superdiagonal elements of an upper bidiagonal matrix $$B$$ whose diagonal is in s (not necessarily sorted). $$B$$ satisfies $$A = UBV^{T}$$, so it has the same singular values as $$A$$, and singular vectors related by $$U$$ and $$V^{T}$$.

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.