quadform
packageQuadratic 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.
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