quadform package
Quadratic forms are polynomials with all terms of degree 2. Given a
column vector \({\mathbf
x}=(x_1,\ldots,x_n)^\top\) and an \(n\times n\) matrix \(M\) then the function \(f\colon\mathbb{R}^n\longrightarrow\mathbb{R}\)
given by \(f({\mathbf x})=x^TMx\) is a
quadratic form; we extend to complex vectors by mapping \({\mathbf z}=(z_1,\ldots, z_n)^\top\) to
\({\mathbf z}^*M{\mathbf z}\), where
\(z^*\) means the complex conjugate of
\(z^T\). These are implemented in the
package with quad.form(M,x) which is essentially
quad.form <- function(M,x){crossprod(crossprod(M, Conj(x)), x)}.
This is preferable to t(x) %*% M %*% x on several
grounds. Firstly, it streamlines and simplifies code; secondly, it is
more efficient; and thirdly it handles the complex case consistently.
The package includes similar functionality for other related
expressions.
The main motivation for the package is nicer code. For example, the
emulator package has to manipulate the following
expression:
\[ \left[H_x-H^\top A^{-1}U\right]^\top \left[H^\top\left(H^\top A^{-1}H\right)^{-1}H\right] \left[H_x-H^\top A^{-1}U\right]. \]
Direct R idiom would be:
t(Hx - t(H) %*% solve(A) %*% U) %*% t(H) %*% solve(t(H) %*% solve(A) %*% H) %*% H %*% (Hx - t(H) %*% solve(A) %*% U)But quadform idiom is:
quad.form(quad.form.inv(quad.form.inv(A,H),H), Hx - quad3.form.inv(A,H,U))and in terse form becomes:
qf(qfi(qfi(A,H),H), Hx - q3fi(A,H,U))which is certainly shorter, arguably more elegant, and possibly faster.
The package is maintained on github.