Enum LinearAlgebraOperation

Namespace

DotCompute.Algorithms

Assembly

DotCompute.Algorithms.dll

Linear algebra operations supported by DotCompute kernel library.

public enum LinearAlgebraOperation

Fields

CholeskyDecomposition = 7

Cholesky decomposition for symmetric positive-definite matrices.

Computes L where A = LL^T for symmetric positive-definite A. Twice as efficient as LU decomposition for this matrix class.

Requirements: Matrix must be symmetric and positive-definite

Applications: Linear least squares, optimization, Monte Carlo

Numerical Stability: Highly stable for well-conditioned matrices

EigenDecomposition = 9

Eigenvalue decomposition for diagonalizable matrices.

Computes A = VΛV^(-1) where Λ is diagonal matrix of eigenvalues, V contains corresponding eigenvectors. Critical for spectral analysis.

Algorithm: QR algorithm with Hessenberg reduction

Complexity: O(N³) with iterative refinement

Applications: Stability analysis, quantum mechanics, vibration modes

HouseholderTransform = 2

Householder transformation application to matrices.

Applies Householder reflector H = I - 2vv^T to matrix A. Critical operation in QR decomposition and eigenvalue algorithms.

HouseholderVector = 1

Householder reflection vector computation for QR decomposition.

Computes Householder vector v where H = I - 2vv^T reflects input vector x onto a coordinate axis. Used as building block for QR decomposition.

Numerical Stability: Designed to avoid catastrophic cancellation

JacobiSVD = 3

Jacobi Singular Value Decomposition (SVD).

Computes SVD A = UΣV^T using Jacobi iterations. Provides high accuracy singular values and vectors through iterative refinement.

Convergence: Quadratic convergence with proper pivoting

Accuracy: Machine precision singular values

Use Cases: Numerical analysis, principal component analysis, signal processing

LUDecomposition = 8

LU decomposition with partial pivoting.

Computes PA = LU where P is permutation matrix, L is lower triangular, U is upper triangular. General-purpose factorization for linear systems.

Pivoting: Partial pivoting for numerical stability

Use Cases: Solving linear systems, matrix inversion, determinants

MatrixMultiply = 0

Matrix-matrix multiplication (GEMM - General Matrix Multiply).

Computes C = αAB + βC where A is M×K, B is K×N, C is M×N. Optimized implementations use tiled algorithms with shared memory.

Performance: Up to 4 TFLOPS on modern GPUs

Kernels: Tiled, Strassen's algorithm for large matrices

MatrixVector = 4

Matrix-vector multiplication (GEMV).

Computes y = αAx + βy where A is M×N matrix, x is N-vector, y is M-vector. Optimized for different matrix layouts (row-major, column-major).

Performance: Memory-bandwidth bound, benefits from coalesced access

ParallelReduction = 5

Parallel reduction operations (sum, max, min, etc.).

Efficient parallel reduction using tree-based algorithms in shared memory. Used as primitive for norms, dot products, and statistical operations.

Algorithm: Binary tree reduction with warp-level primitives

Performance: O(log N) depth, minimal synchronization overhead

QRAlgorithm = 6

QR algorithm for eigenvalue computation.

Iterative QR factorization algorithm for computing eigenvalues and eigenvectors of general matrices. Uses Householder or Givens rotations.

Convergence: Linear convergence, accelerated with shifts

Preprocessing: Hessenberg reduction for efficiency

Remarks

Enumerates GPU-accelerated linear algebra operations with optimized kernels for CUDA, OpenCL, and Metal backends. Each operation has specialized implementations for different matrix types and sizes.

Used by LinearAlgebraKernelLibrary to select appropriate kernel implementations based on operation type and hardware capabilities.