LinearAlgebra`BLAS`
LinearAlgebra`BLAS`

# HERK

HERK[ul,ts,α,a,β,b]

computes the Hermitian rank-k update α opts[a].ConjugateTranspose[opts[a]]+β b and resets the appropriate part of b to the result.

# Details and Options

• To use HERK, you first need to load the BLAS Package using Needs["LinearAlgebra`BLAS`"].
• The following arguments must be given:
•  ul input string upper/lower triangular string ts input string transposition string α input expression scalar mutliple a input expression rectangular matrix β input expression scalar multiple b input/output symbol square matrix; the symbol value is modified in place
• The upper/lower triangular string ul may be specified as:
•  "U" update the upper triangular part of b "L" update the lower triangular part of b
• The transposition strings describe the operators opts and may be specified as:
•  "N" no transposition "T" transpose "C" conjugate transpose
• The main diagonal elements of b are assumed to be real-valued.
• Dimensions of the matrix arguments must be such that the dot product and addition are well defined.

# Examples

open allclose all

## Basic Examples(1)

Apply Hermitian rank-2 update to the upper triangular part of a matrix:

## Scope(4)

Real matrices:

Complex matrices:

Arbitrary-precision matrices:

Integer-symbolic matrices:

## Properties & Relations(1)

HERK["U","N",α,a,β,b] is equivalent to b=α a.ConjugateTranspose[a]+β b applied to the upper triangular part of b:

The strictly lower triangular part of b is unchanged:

## Possible Issues(2)

The last argument must be a symbol:

The last argument must be initialized to a matrix, otherwise an error message is issued:

Wolfram Research (2017), HERK, Wolfram Language function, https://reference.wolfram.com/language/LowLevelLinearAlgebra/ref/HERK.html.

#### Text

Wolfram Research (2017), HERK, Wolfram Language function, https://reference.wolfram.com/language/LowLevelLinearAlgebra/ref/HERK.html.

#### CMS

Wolfram Language. 2017. "HERK." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/LowLevelLinearAlgebra/ref/HERK.html.

#### APA

Wolfram Language. (2017). HERK. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/LowLevelLinearAlgebra/ref/HERK.html

#### BibTeX

@misc{reference.wolfram_2022_herk, author="Wolfram Research", title="{HERK}", year="2017", howpublished="\url{https://reference.wolfram.com/language/LowLevelLinearAlgebra/ref/HERK.html}", note=[Accessed: 10-June-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_herk, organization={Wolfram Research}, title={HERK}, year={2017}, url={https://reference.wolfram.com/language/LowLevelLinearAlgebra/ref/HERK.html}, note=[Accessed: 10-June-2023 ]}