computes the symmetric matrix-vector multiplication α a.x+β y and resets y to the result.


  • To use SYMV, you first need to load the BLAS Package using Needs["LinearAlgebra`BLAS`"].
  • The following arguments must be given:
  • ulinput stringupper/lower triangular string
    αinput expressionscalar mutliple
    ainput expressionsquare symmetric matrix
    xinput expressionvector
    βinput expressionscalar multiple
    yinput/output symbolvector; the symbol value is modified in place
  • The matrix is assumed symmetric, and only the upper or lower triangular part of a is used.
  • The upper/lower triangular string ul may be specified as:
  • "U"the upper triangular part of a is to be used
    "L"the lower triangular part of a is to be used
  • Dimensions of the matrix and vector arguments must be such that the dot product and addition are well defined.


open allclose all

Basic Examples  (1)

Load the BLAS package:

Compute a.x+2y and save it in y:

Scope  (4)

Real symmetric matrix and vectors:

Complex symmetric matrix and vectors:

Arbitrary-precision symmetric matrix and vectors:

Symbolic symmetric matrix and vectors:

Properties & Relations  (3)

SYMV["U",α,a,x,β,y] is equivalent to y=α a.x+β y if a is symmetric:

For a symmetric matrix, using the upper or lower triangular part generally produces the same result:

SYMV works with a non-symmetric matrices:

However, the upper and lower parts give different results:

The effective computation of yU is the following:

Possible Issues  (2)

The last argument must be a symbol:

The last argument must be initialized to a vector: