.. SPDX-FileCopyrightText: 2019-2020 Intel Corporation .. .. SPDX-License-Identifier: CC-BY-4.0 .. _onemkl_lapack_potrs: potrs ===== Solves a system of linear equations with a Cholesky-factored symmetric (Hermitian) positive-definite coefficient matrix. .. container:: section .. rubric:: Description potrs supports the following precisions. .. list-table:: :header-rows: 1 * - T * - float * - double * - std::complex * - std::complex The routine solves for :math:X the system of linear equations :math:AX = B with a symmetric positive-definite or, for complex data, Hermitian positive-definite matrix :math:A, given the Cholesky factorization of :math:A: .. list-table:: :header-rows: 1 * - :math:A = U^TU for real data, :math:A = U^HU for complex data - if upper_lower=oneapi::mkl::uplo::upper * - :math:A = LL^T for real data, :math:A = LL^H for complex data - if upper_lower=oneapi::mkl::uplo::lower where :math:L is a lower triangular matrix and :math:U is upper triangular. The system is solved with multiple right-hand sides stored in the columns of the matrix :math:B. Before calling this routine, you must call :ref:onemkl_lapack_potrf to compute the Cholesky factorization of :math:A. potrs (Buffer Version) ---------------------- .. container:: section .. rubric:: Syntax .. code-block:: cpp namespace oneapi::mkl::lapack { void potrs(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, std::int64_t nrhs, sycl::buffer &a, std::int64_t lda, sycl::buffer &b, std::int64_t ldb, sycl::buffer &scratchpad, std::int64_t scratchpad_size) } .. container:: section .. rubric:: Input Parameters queue The queue where the routine should be executed. upper_lower Indicates how the input matrix has been factored: If upper_lower = oneapi::mkl::uplo::upper, the upper triangle :math:U of :math:A is stored, where :math:A = :math:U^{T}U for real data, :math:A = :math:U^{H}U for complex data. If upper_lower = oneapi::mkl::uplo::lower, the lower triangle :math:L of :math:A is stored, where :math:A = :math:LL^{T} for real data, :math:A = :math:LL^{H} for complex data. n The order of matrix :math:A (:math:0 \le n).\ nrhs The number of right-hand sides (:math:0 \le \text{nrhs}). a Buffer containing the factorization of the matrix A, as returned by :ref:onemkl_lapack_potrf. The second dimension of a must be at least :math:\max(1, n). lda The leading dimension of a. b The array b contains the matrix :math:B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least :math:\max(1,\text{nrhs}). ldb The leading dimension of b. 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 :ref:onemkl_lapack_potrs_scratchpad_size function. .. container:: section .. rubric:: Output Parameters b Overwritten by the solution matrix :math:X. scratchpad Buffer holding scratchpad memory to be used by routine for storing intermediate results. potrs (USM Version) ---------------------- .. container:: section .. rubric:: Syntax .. code-block:: cpp namespace oneapi::mkl::lapack { sycl::event potrs(sycl::queue &queue, oneapi::mkl::uplo upper_lower, std::int64_t n, std::int64_t nrhs, T *a, std::int64_t lda, T *b, std::int64_t ldb, T *scratchpad, std::int64_t scratchpad_size, const std::vector &events = {}) } .. container:: section .. rubric:: Input Parameters queue The queue where the routine should be executed. upper_lower Indicates how the input matrix has been factored: If upper_lower = oneapi::mkl::uplo::upper, the upper triangle :math:U of :math:A is stored, where :math:A = :math:U^{T}U for real data, :math:A = :math:U^{H}U for complex data. If upper_lower = oneapi::mkl::uplo::lower, the lower triangle :math:L of :math:A is stored, where :math:A = :math:LL^{T} for real data, :math:A = :math:LL^{H} for complex data. n The order of matrix :math:A (:math:0 \le n).\ nrhs The number of right-hand sides (:math:0 \le \text{nrhs}). a Pointer to array containing the factorization of the matrix :math:A, as returned by :ref:onemkl_lapack_potrf. The second dimension of a must be at least :math:\max(1, n). lda The leading dimension of a. b The array b contains the matrix :math:B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least :math:\max(1,\text{nrhs}). ldb The leading dimension of b. 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 :ref:onemkl_lapack_potrs_scratchpad_size function. events List of events to wait for before starting computation. Defaults to empty list. .. container:: section .. rubric:: Output Parameters b Overwritten by the solution matrix :math:X. scratchpad Pointer to scratchpad memory to be used by routine for storing intermediate results. .. container:: section .. rubric:: Return Values Output event to wait on to ensure computation is complete. **Parent topic:** :ref:onemkl_lapack-linear-equation-routines`