gesvda_batch (USM Strided Version)

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ID 772045

Date 3/22/2024

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gesvda_batch (USM Strided Version)

Computes the truncated SVD factorizations for a batch of general matrices. This routine belongs to the oneapi::mkl::lapack namespace.

Description
API

Description

The routine computes the truncated SVD decomposition (or a reduced rank approximation) for a batch of general matrices A_i, as:

A_i = U_i * S_i * V_i^T

where U_i and V_i are orthogonal matrices, S_i is a diagonal matrix with singular values on the diagonal. Singular values are nonnegative and listed in decreasing order. A truncated SVD of a given matrix will produce matrices with the specified number of columns, where the number of columns is defined by the user or determined at runtime with the help of the user-defined tolerance threshold.

An approximation of each matrix can be also obtained as a product of two low-rank matrices (e.g. low rank product):

A_i = P_i * Q_i

where

P_i = U_i * S_i, Q_i = V_i^T

if the number of rows is greater or equal to the number of columns,

P_i = U_i, Q_i = S_i` *V_i^T

otherwise.

The routine provides 3 possible ways for computing truncated SVD:

Compute truncated SVD with the help of the input array rank where rank[i] specifies the number of singular values and vectors to be computed in parameter U_i, V_i and S_i for each matrix A_i.
Compute truncated SVD using a tolerance threshold. While computing SVD, singular values that are less than the user-defined tolerance are treated as zero, and they are not computed but set to zero.
Compute truncated SVD using the effective rank. The effective rank is determined by treating as zero those singular values that are less than the user-defined tolerance threshold times the largest singular value.

The routine can be also used for computing singular values only.

API

Syntax

namespace oneapi::mkl::lapack {
  void gesvda_batch(sycl::queue &queue,
  std::int64_t *iparm,
  std::int64_t *irank,
  std::int64_t m,
  std::int64_t n,
  T *a,
  std::int64_t lda,
  std::int64_t stride_a,
  RealT *s,
  std::int64_t stride_s,
  T * u,
  std::int64_t ldu,
  std::int64_t stride_u,
  T * vt,
  std::int64_t ldvt,
  std::int64_t stride_vt,
  RealT tolerance,
  RealT *residual,
  std::int64_t batch_size,
  T *scratchpad,
  std::int64_t scratchpad_size,
  const std::vector<sycl::event> &events = {})
 }

gesvda_batch supports the following precisions and devices:

T	Devices supported
`float`	CPU, GPU*
`double`	CPU, GPU*
`std::complex<float>`	CPU, GPU*
`std::complex<double>`	CPU, GPU*

*Interface support only; all computations are performed on the CPU.

Input Parameters

queue: Device queue where calculations will be performed.
iparm: Array of dimension 16 specifying options to compute truncated SVD. It also specifies the type of returned SVD decomposition form. The table below describes all individual components of the iparm parameter. Default values are denoted with an asterisk (*). NOTE: iparm[4]-iparm[15] are reserved for future use.

Component description	Values
`iparm[0]` specifies a criterion for treating singular values as zeros	If `iparm[0]=-1`, default iparm values `(iparm[0-2]=0, iparm[3]=1)` are used If `iparm[0]=0`^*, the truncated SVD is computed with the help of the input array `irank` If `iparm[0]=1`, the truncated SVD is computed using the parameter `tolerance` If `iparm[0]=2`, the truncated SVD is computed using the effective rank. The effective rank is determined by treating as zero those singular values that are less than the user-defined tolerance times the largest singular value
`iparm[1]` specifies option for computing singular vectors	If `iparm[1]=0`^*, both singular values and singular vectors are computed If `iparm[1]=1`, only singular values are computed
`iparm[2]` specifies type of returned SVD decomposition	If `iparm[2]=0`^, the truncated SVD as a product of three matrices `A`_i = `U`_i `S`_i * `V`_i^T is computed. If `iparm[2]=1`, the truncated as as a low rank product `A`_i = `P`_i * `Q`_i is computed.
`iparm[3]` specifies option for computing the residual vector	If `iparm[2]=0`, the residual vector is not computed If `iparm[2]=1`^*, the residual vector is computed

irank: If iparm[0]=0 or iparm[0]=-1, element irank[i] specifies the number of singular values and/or singular vectors to be computed in U_i, V_i^Tand S_i for each matrix A_i.
m: The number of rows in the matrices A_i.
n: The number of columns in the matrices A_i.
a: Array holding input matrices A_i.
lda: The leading dimension of a. Must be at least max(1, m).
stride_a: The stride between the beginnings of matrices A_i inside the batch array a. Must be at least max(1, lda * n).
stride_s: The stride between the beginnings of arrays S_i inside the array s. Must be at least min(m,n).
ldu: The leading dimension of U_i. Must be at least max(1, m).
stride_u: The stride between the beginnings of matrices U_i inside the batch array u. Must be at least max(1, ldu * m).
ldvt: The leading dimension of V_i^T . Must be at least max(1, n).
stride_vt: The stride between the beginnings of matrices V_i^Tinside the batch array vt. Must be at least max(1, ldvt * n).
tolerance: Specifies the tolerance threshold. It is only used for computing truncated SVD in the cases of iparm[0]=1 and iparm[0]=2. Not used otherwise.
batch_size: The number of problems in a batch.
scratchpad: Scratchpad memory to be used by routine for storing intermediate results.
scratchpad_size: Size of scratchpad memory as a number of floating point elements of type T. Size should not be less than the value returned by gesvda_batch_scratchpad_size (Strided Version) function.

Output Parameters

irank

If iparm[0]=-1 or iparm[0]=0, element irank[i] is the number of computed singular values and/or singular vectors for matrix A_i.

a

Unchanged on exit, if the residual vector is not required. Otherwise contains the residual matrix

A_i - U_i * S_i * V_i^Tif iparm[2]=0,

A_i - P_i * Q_i if iparm[2]=1.

s

Array of size min(m,n)*batch_size to store batch of singular values S_i.

u

u is array of size at least stride_u*batch_size to store batch of U_i if iparm[2]=0, or to store batch of P_i if iparm[2]=1.

vt

vt is array of size at least stride_vt*batch_size to store batch of V_i^Tif iparm[2]=0, or to store batch of Q_i if iparm[2]=1.

residual

Array of dimension batch_size. If iparm[3]=1, residual[i] is the Frobenius norm of the matrix:

|| A_i - U_i * S_i * V_i^T ||

if iparm[2]=0, and

|| A_i - P_i * Q_i ||

if iparm[2]=1.

Exceptions

Exception	Description
`mkl::lapack::batch_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 = -n`, the `n`-th parameter had an illegal value. If `info = 1`, an internal memory allocation failed. If `info = 2`, an input parameter contains an invalid value. If `info = 3`, an error in algorithm while computing singular values . If `info = 4`, the routine encountered an empty structure or matrix array.

Exception

Description

mkl::lapack::batch_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 = -n, the n-th parameter had an illegal value.

If info = 1, an internal memory allocation failed.

If info = 2, an input parameter contains an invalid value.

If info = 3, an error in algorithm while computing singular values .

If info = 4, the routine encountered an empty structure or matrix array.

Return Values

Output event to wait on to ensure computation is complete.

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oneMKL - Data Parallel C++ Developer Reference

gesvda_batch (USM Strided Version)

Description

API