Compute a dense generalized Schur decomposition

generalized_schur() is eigencore's dense QZ surface for general matrix pencils A x = lambda B x. It computes the generalized Schur pair S, T and, when requested, left/right Schur vectors Q, Z such that A = Q S Z* and B = Q T Z* for complex inputs, with transpose replacing conjugate-transpose for real inputs. Sparse and operator inputs are not silently densified.

Usage

generalized_schur(A, B, sort = NULL, vectors = TRUE, ...)

Arguments

A

Base dense square matrix.

B

Base dense square matrix with the same dimension as A.

sort

Optional LAPACK sorting class. Use NULL or "none" for no sorting, "finite" to move finite generalized eigenvalues first, "infinite" to move beta-zero nonzero-alpha eigenvalues first. Custom predicates and undefined alpha-zero/beta-zero sorting are not part of the public contract.

vectors

Whether to compute Schur vectors Q and Z.

...

Reserved for future options.

Value

A classed generalized Schur result with fields S, T, Q, Z, alpha, beta, values, classification, sdim, method, plan, and certificate.

Examples

A <- matrix(c(0, -1, 1, 0), 2, 2)
B <- diag(2)
qz <- generalized_schur(A, B)
values(qz)
#> [1] 0+1i 0-1i

pencil <- generalized_schur(diag(c(2, 3, 0)), diag(c(1, 0, 0)),
                            sort = "infinite")
pencil$classification
#> [1] "infinite"  "finite"    "undefined"