gemv

gemv, short for general matrix–vector multiplication, is a routine from the BLAS (Basic Linear Algebra Subprograms) library. It computes a linear combination of a matrix–vector product and a scaled existing vector, delivering y = alpha * op(A) * x + beta * y, where A is a general matrix, x and y are vectors, and alpha and beta are scalars. The routine supports real and complex data and can apply a transpose or conjugate transpose to A.

The operation uses an input matrix A of size m by n and two vectors x and y. The parameter TRANS determines how A is used: 'N' uses A as is (op(A) = A), 'T' uses A transposed (op(A) = A^T), and 'C' uses the conjugate transpose (op(A) = A^H) for complex data. When TRANS = 'N', y has length m and x has length n; when TRANS = 'T' or 'C', y has length n and x has length m. The routine also requires the leading dimension LDA of A (LDA >= max(1, m) for TRANS = 'N'), and increments INCX and INCY for the elements of x and y.

In practice, gemv is used for matrix–vector products within iterative solvers, projections, and various linear algebra operations. Implementations exist in many libraries (NetLIB BLAS, OpenBLAS, Intel MKL) and exploit cache-friendly and vectorized code paths. The routine is typically accessed through language bindings that adapt the Fortran BLAS interface to C, C++, or other environments, with column-major storage assumed by default in standard BLAS conventions.