An elementrary reflector is a vector that implements reflection (mathematics). It is also referred to as a triangular factor, and is a triangular matrix. Elementary reflectors are used in the Householder transformation.

The routines in LAPACK "*LARZ" apply an "elementary reflector" H to a M-by-N Matrix, from either the left or the right. The elementary reflector is expressed as

H = I − τ v· v′

where τ is a scalar and v is a vector. H is a product of k elementary reflectors.

A block reflector is formed out of k elementary reflectors by the routines "*LARZT", which forms the triangular factor T of a block reflector H of order > n.

LAPACK ROUTINES

• "*larf" applies an elementary reflector to a general rectangular matrix.
• "*larfc" applies the conjugate transpose of an "elementary reflector" to a general matrix.
• "*larfg" generates an elementary reflector Householder matrix.
• "*larzc" applies (multiplies by) the conjugate transpose of an elementary reflector as returned by "*tzrzf" to a general matrix.
• "*larz" applies an elementary reflector as returned by "*tzrzf" to a general matrix.
• "*tzrzf" reduces the upper trapezoidal matrix A to upper triangular matrix.