Level 3¶

The prototypes for the following routines can be found at include/elemental/blas-like/decl.hpp, while the implementations are in include/elemental/blas-like/level3/.

Gemm¶

General matrix-matrix multiplication: updates \(C := \alpha A^\# B^\sharp + \beta C\), where \(M^\#\) and \(M^\sharp\) are individually defined to be one of \(\{M,M^T,M^H\}\).

void Gemm(Orientation orientationOfA, Orientation orientationOfB, T alpha, const Matrix<T> &A, const Matrix<T> &B, T beta, Matrix<T> &C)¶

void Gemm(Orientation orientationOfA, Orientation orientationOfB, T alpha, const DistMatrix<T> &A, const DistMatrix<T> &B, T beta, DistMatrix<T> &C)¶

Hemm¶

Hermitian matrix-matrix multiplication: updates \(C := \alpha A B + \beta C\), or \(C := \alpha B A + \beta C\), depending upon whether side is set to LEFT or RIGHT, respectively. In both of these types of updates, \(A\) is implicitly Hermitian and only the triangle specified by uplo is accessed.

void Hemm(LeftOrRight side, UpperOrLower uplo, T alpha, const Matrix<T> &A, const Matrix<T> &B, T beta, Matrix<T> &C)¶

void Hemm(LeftOrRight side, UpperOrLower uplo, T alpha, const DistMatrix<T> &A, const DistMatrix<T> &B, T beta, DistMatrix<T> &C)¶

Her2k¶

Hermitian rank-2K update: updates \(C := \alpha (A B^H + B A^H) + \beta C\), or \(C := \alpha (A^H B + B^H A) + \beta C\), depending upon whether orientation is set to NORMAL or ADJOINT, respectively. Only the triangle of \(C\) specified by the uplo parameter is modified.

void Her2k(UpperOrLower uplo, Orientation orientation, T alpha, const Matrix<T> &A, const Matrix<T> &B, T beta, Matrix<T> &C)¶

void Her2k(UpperOrLower uplo, Orientation orientation, T alpha, const DistMatrix<T> &A, const DistMatrix<T> &B, T beta, DistMatrix<T> &C)¶

Please see SetLocalTrr2kBlocksize<T>() and LocalTrr2kBlocksize<T>() in the Tuning parameters section for information on tuning the distributed Her2k().

Herk¶

Hermitian rank-K update: updates \(C := \alpha A A^H + \beta C\), or \(C := \alpha A^H A + \beta C\), depending upon whether orientation is set to NORMAL or ADJOINT, respectively. Only the triangle of \(C\) specified by the uplo parameter is modified.

void Herk(UpperOrLower uplo, Orientation orientation, T alpha, const Matrix<T> &A, T beta, Matrix<T> &C)¶

void Herk(UpperOrLower uplo, Orientation orientation, T alpha, const DistMatrix<T> &A, T beta, DistMatrix<T> &C)¶

Please see SetLocalTrrkBlocksize<T>() and LocalTrrkBlocksize<T>() in the Tuning parameters section for information on tuning the distributed Herk().

Multi-shift QuasiTrsm¶

Solve for \(X\) in the linear system

\[T^\# X - X D^\# = Y\]

\[X T^\# - D^\# X = Y\]

where \(T\) is quasi-triangular, \(D\) is diagonal, and \(A^\#\) is defined to be one of \(\{A,A^T,A^H\}\). The data movement requires almost no modification from that of QuasiTrsm().

Note that the term quasi-triangular is in the context of real Schur decompositions, which produce triangular matrices with mixes of \(1 \times 1\) and \(2 \times 2\) diagonal blocks.

Note

There is no corresponding BLAS routine, but it is a natural extension of Trsm.

void MultiShiftQuasiTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, F alpha, const Matrix<F> &T, const Matrix<F> &shifts, Matrix<F> &X)¶

void MultiShiftQuasiTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, F alpha, const DistMatrix<F> &T, const DistMatrix<F, VR, STAR> &shifts, DistMatrix<F> &X)¶: Overwrite the columns of X with the solutions to the shifted linear systems.

void MultiShiftQuasiTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, Complex<Real> alpha, const Matrix<Real> &T, const Matrix<Complex<Real>> &shifts, Matrix<Real> &XReal, Matrix<Real> &XImag)¶

void MultiShiftQuasiTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, Complex<Real> alpha, const DistMatrix<Real> &T, const DistMatrix<Complex<Real>, VR, STAR> &shifts, DistMatrix<Real> &XReal, DistMatrix<Real> &XImag)¶: Overwrite the columns of the real and imaginary parts of X with the solutions to the shifted linear systems.

Multi-shift Trsm¶

Solve for \(X\) in the linear system

\[T^\# X - X D^\# = Y\]

\[X T^\# - D^\# X = Y\]

where \(T\) is triangular, \(D\) is diagonal, and \(A^\#\) is defined to be one of \(\{A,A^T,A^H\}\). The data movement requires almost no modification from that of Trsm().

Note

There is no corresponding BLAS routine, but it is a natural modification of Trsm.

void MultiShiftTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, F alpha, const Matrix<F> &T, const Matrix<F> &shifts, Matrix<F> &X)¶

void MultiShiftTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, F alpha, const DistMatrix<F> &T, const DistMatrix<F, VR, STAR> &shifts, DistMatrix<F> &X)¶: Overwrite the columns of X with the solutions to the shifted linear systems.

void MultiShiftTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, Complex<Real> alpha, const Matrix<Real> &T, const Matrix<Complex<Real>> &shifts, Matrix<Real> &XReal, Matrix<Real> &XImag)¶

void MultiShiftTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, Complex<Real> alpha, const DistMatrix<Real> &T, const DistMatrix<Complex<Real>, VR, STAR> &shifts, DistMatrix<Real> &XReal, DistMatrix<Real> &XImag)¶: Overwrite the columns of the real and imaginary parts of X with the solutions to the shifted linear systems.

QuasiTrsm¶

Solve for \(X\) in the linear system

\[T^\# X = Y\]

\[X T^\# = Y\]

where \(T\) is quasi-triangular and \(A^\#\) is defined to be one of \(\{A,A^T,A^H\}\). The algorithm is very similar to that of Trsm().

Note that the term quasi-triangular is in the context of real Schur decompositions, which produce triangular matrices with mixes of \(1 \times 1\) and \(2 \times 2\) diagonal blocks.

Note

There is no corresponding BLAS routine, but it is a natural extension of Trsm.

void QuasiTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, F alpha, const Matrix<F> &T, const Matrix<F> &shifts, Matrix<F> &X, bool checkIfSingular = false)¶

void QuasiTrsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, F alpha, const DistMatrix<F> &T, const DistMatrix<F, VR, STAR> &shifts, DistMatrix<F> &X, bool checkIfSingular = false)¶: Overwrite the columns of X with the solutions to the shifted linear systems.

Symm¶

Symmetric matrix-matrix multiplication: updates \(C := \alpha A B + \beta C\), or \(C := \alpha B A + \beta C\), depending upon whether side is set to LEFT or RIGHT, respectively. In both of these types of updates, \(A\) is implicitly symmetric and only the triangle specified by uplo is accessed.

void Symm(LeftOrRight side, UpperOrLower uplo, T alpha, const Matrix<T> &A, const Matrix<T> &B, T beta, Matrix<T> &C, bool conjugate = false)¶

void Symm(LeftOrRight side, UpperOrLower uplo, T alpha, const DistMatrix<T> &A, const DistMatrix<T> &B, T beta, DistMatrix<T> &C, bool conjugate = false)¶

Syr2k¶

Symmetric rank-2K update: updates \(C := \alpha (A B^T + B A^T) + \beta C\), or \(C := \alpha (A^T B + B^T A) + \beta C\), depending upon whether orientation is set to NORMAL or TRANSPOSE, respectively. Only the triangle of \(C\) specified by the uplo parameter is modified.

void Syr2k(UpperOrLower uplo, Orientation orientation, T alpha, const Matrix<T> &A, const Matrix<T> &B, T beta, Matrix<T> &C)¶

void Syr2k(UpperOrLower uplo, Orientation orientation, T alpha, const DistMatrix<T> &A, const DistMatrix<T> &B, T beta, DistMatrix<T> &C)¶

Please see SetLocalTrr2kBlocksize<T>() and LocalTrr2kBlocksize<T>() in the Tuning parameters section for information on tuning the distributed Syr2k().

Syrk¶

Symmetric rank-K update: updates \(C := \alpha A A^T + \beta C\), or \(C := \alpha A^T A + \beta C\), depending upon whether orientation is set to NORMAL or TRANSPOSE, respectively. Only the triangle of \(C\) specified by the uplo parameter is modified.

void Syrk(UpperOrLower uplo, Orientation orientation, T alpha, const Matrix<T> &A, T beta, Matrix<T> &C)¶

void Syrk(UpperOrLower uplo, Orientation orientation, T alpha, const DistMatrix<T> &A, T beta, DistMatrix<T> &C)¶

Please see SetLocalTrrkBlocksize<T>() and LocalTrrkBlocksize<T>() in the Tuning parameters section for information on tuning the distributed Syrk().

Trmm¶

Triangular matrix-matrix multiplication: performs \(C := \alpha A^\# B\), or \(C := \alpha B A^\#\), depending upon whether side was chosen to be LEFT or RIGHT, respectively. Whether \(A\) is treated as lower or upper triangular is determined by uplo, and \(A^\#\) is defined to be one of \(\{A,A^T,A^H\}\) (and diag determines whether \(A\) is treated as unit-diagonal or not).

void Trmm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, UnitOrNonUnit diag, T alpha, const Matrix<T> &A, Matrix<T> &B)¶

void Trmm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, UnitOrNonUnit diag, T alpha, const DistMatrix<T> &A, DistMatrix<T> &B)¶

Trr2k¶

Triangular rank-2k update: performs \(E := \alpha ( A^\# B^\sharp + C^\Diamond D^\triangle ) + \beta E\), where only the triangle of E specified by uplo is modified, and the orientation of each input matrix, e.g., \(A^\# \in \{A,A^T,A^H\}\), is determined by orientationOfX for each \(X \in \left\{A,B,C,D\right\}\).

Note

There is no corresponding BLAS routine, but it is a natural generalization of “symmetric” and “Hermitian” updates.

void Trr2k(UpperOrLower uplo, Orientation orientationOfA, Orientation orientationOfB, Orientation orientationOfC, Orientation orientationOfD, T alpha, const Matrix<T> &A, const Matrix<T> &B, const Matrix<T> &C, const Matrix<T> &D, T beta, Matrix<T> &E)¶

void Trr2k(UpperOrLower uplo, Orientation orientationOfA, Orientation orientationOfB, Orientation orientationOfC, Orientation orientationOfD, T alpha, const DistMatrix<T> &A, const DistMatrix<T> &B, const DistMatrix<T> &C, const DistMatrix<T> &D, T beta, DistMatrix<T> &E)¶

Trrk¶

Triangular rank-k update: performs \(C := \alpha A^\# B^\sharp + \beta C\), where only the triangle of C specified by uplo is modified, and the orientations \(A^\#\) and \(B^\sharp\) are determined by orientationOfA and orientationOfB, respectively.

Note

There is no corresponding BLAS routine, but this type of update is frequently encountered, even in serial. For instance, the symmetric rank-k update performed during an LDL factorization is symmetric but one of the two update matrices is scaled by D.

void Trrk(UpperOrLower uplo, Orientation orientationOfA, Orientation orientationOfB, T alpha, const Matrix<T> &A, const Matrix<T> &B, T beta, Matrix<T> &C)¶

void Trrk(UpperOrLower uplo, Orientation orientationOfA, Orientation orientationOfB, T alpha, const DistMatrix<T> &A, const DistMatrix<T> &B, T beta, DistMatrix<T> &C)¶

Trtrmm¶

Note

This routine loosely corresponds with the LAPACK routines ?lauum.

Symmetric/Hermitian triangular matrix-matrix multiply: performs \(L := L^T L\), \(L := L^H L\), \(U := U U^T\), or \(U := U U^H\), depending upon the choice of the orientation and uplo parameters.

void Trtrmm(Orientation orientation, UpperOrLower uplo, Matrix<T> &A)¶

void Trtrmm(Orientation orientation, UpperOrLower uplo, DistMatrix<T> &A)¶

Trdtrmm¶

Note

This is a modification of Trtrmm for LDL factorizations.

Symmetric/Hermitian triangular matrix-matrix multiply (with diagonal scaling): performs \(L := L^T D^{-1} L\), \(L := L^H D^{-1} L\), \(U := U D^{-1} U^T\), or \(U := U D^{-1} U^H\), depending upon the choice of the orientation and uplo parameters. Note that \(L\) and \(U\) are unit-diagonal and their diagonal is overwritten with \(D\).

void Trdtrmm(Orientation orientation, UpperOrLower uplo, Matrix<F> &A)¶

void Trdtrmm(Orientation orientation, UpperOrLower uplo, DistMatrix<F> &A)¶

Trsm¶

Triangular solve with multiple right-hand sides: performs \(C := \alpha A^{-\#} B\), or \(C := \alpha B A^{-\#}\), depending upon whether side was chosen to be LEFT or RIGHT, respectively. Whether \(A\) is treated as lower or upper triangular is determined by uplo, and \(A^{-\#}\) can be \(A^{-1}\), \(A^{-T}\), or \(A^{-H}\) (and diag determines whether \(A\) is treated as unit-diagonal or not).

void Trsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, UnitOrNonUnit diag, F alpha, const Matrix<F> &A, Matrix<F> &B)¶

void Trsm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, UnitOrNonUnit diag, F alpha, const DistMatrix<F> &A, DistMatrix<F> &B)¶

Trstrm¶

Performs a triangular solve against a triangular matrix. Only the Left Lower Normal option is currently supported.

void Trstrm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, UnitOrNonUnit diag, F alpha, const Matrix<F> &A, Matrix<F> &X, bool checkIfSingular = true)¶

void Trstrm(LeftOrRight side, UpperOrLower uplo, Orientation orientation, UnitOrNonUnit diag, F alpha, const DistMatrix<F> &A, DistMatrix<F> &X, bool checkIfSingular = true)¶

Two-sided Trmm¶

Performs a two-sided triangular multiplication with multiple right-hand sides which preserves the symmetry of the input matrix, either \(A := L^H A L\) or \(A := U A U^H\).

void TwoSidedTrmm(UpperOrLower uplo, UnitOrNonUnit diag, Matrix<T> &A, const Matrix<T> &B)¶

void TwoSidedTrmm(UpperOrLower uplo, UnitOrNonUnit diag, DistMatrix<T> &A, const DistMatrix<T> &B)¶

Two-sided Trsm¶

Performs a two-sided triangular solves with multiple right-hand sides which preserves the symmetry of the input matrix, either \(A := L^{-1} A L^{-H}\) or \(A := U^{-H} A U^{-1}\).

void TwoSidedTrsm(UpperOrLower uplo, UnitOrNonUnit diag, Matrix<F> &A, const Matrix<F> &B)¶

void TwoSidedTrsm(UpperOrLower uplo, UnitOrNonUnit diag, DistMatrix<F> &A, const DistMatrix<F> &B)¶

Level 3¶

Gemm¶

Hemm¶

Her2k¶

Herk¶

Multi-shift QuasiTrsm¶

Multi-shift Trsm¶

QuasiTrsm¶

Symm¶

Syr2k¶

Syrk¶

Trmm¶

Trr2k¶

Trrk¶

Trtrmm¶

Trdtrmm¶

Trsm¶

Trstrm¶

Two-sided Trmm¶

Two-sided Trsm¶

Usage documentation

Developer resources