---
title: Generalized eigenvalue problems with eigencore
output:
  rmarkdown::html_vignette:
    toc: yes
    toc_depth: 2
    css: albers.css
    includes:
      in_header: albers-header.html
params:
  family: lapis
  preset: homage
resource_files:
- albers.css
- albers.js
- albers-header.html
- fonts

vignette: |
  %\VignetteIndexEntry{Generalized eigenvalue problems with eigencore}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  message = FALSE,
  warning = FALSE,
  fig.width = 7,
  fig.height = 4.1,
  fig.align = "center",
  out.width = "92%",
  dpi = 100
)
library(eigencore)
library(Matrix)

# Shared plotting helper. Kept out of the reader's view: the reader sees
# the API call and the picture, never the plotting boilerplate.
ec_blue <- "#2166ac"; ec_grey <- "grey78"

# A computed slice (blue) drawn on top of the full dense spectrum (grey).
plot_spectrum <- function(all_vals, computed, computed_ranks = seq_along(computed),
                          ylab = "value", main = NULL) {
  s <- sort(all_vals, decreasing = TRUE)
  k <- length(computed)
  op <- par(mar = c(4, 4.6, if (is.null(main)) 1 else 2.4, 1)); on.exit(par(op))
  plot(seq_along(s), s, pch = 19, cex = 0.4, col = ec_grey, bty = "n",
       xlab = "rank (largest to smallest)", ylab = ylab, main = main)
  points(computed_ranks, sort(computed, decreasing = TRUE),
         pch = 19, cex = 1.2, col = ec_blue)
  legend("topright", c("full spectrum (dense reference)", "computed by eigencore"),
         pch = 19, pt.cex = c(0.7, 1.2), col = c(ec_grey, ec_blue),
         bty = "n", cex = 0.9)
}
```

```{r albers-classes, echo=FALSE, results='asis'}
cat(sprintf(
  paste0(
    '<script>document.addEventListener("DOMContentLoaded",function(){',
    'document.body.classList.remove("palette-red","palette-lapis","palette-ochre","palette-teal","palette-green","palette-violet","preset-homage","preset-interaction","preset-study","preset-structural","preset-adobe","preset-midnight");',
    'document.body.classList.add("palette-%s","preset-%s");',
    '});</script>'
  ),
  params$family,
  params$preset
))
```

A generalized eigenvalue problem asks for nonzero vectors $x$ and scalars
$\lambda$ satisfying

$$
A x = \lambda B x.
$$

This form shows up whenever a problem needs a metric other than the
standard inner product: whitened PCA and canonical correlation analysis
weight by a covariance matrix, linear discriminant analysis weights by a
within-class scatter matrix, and normalized graph Laplacians weight by a
degree matrix. In each case `B` is not incidental — solving the plain
eigenproblem `A x = lambda x` and ignoring `B` gives you the wrong answer.

The second matrix $B$ may define a positive-definite metric, or it may be
part of a more general matrix pencil. eigencore supports both cases, along
with partial solves when you need only a few eigenpairs and QZ
decompositions when you need the full structure of a dense pencil.

## Start with a positive-definite metric

When `A` is symmetric (or Hermitian) and `B` is symmetric positive definite,
pass `B` directly to `eig_full()`. The returned vectors are normalized in the
$B$ metric.

```{r dense-spd}
A <- diag(c(2, 8, 18))
B <- diag(c(1, 2, 3))

fit <- eig_full(A, B = B)
values(fit)
certificate(fit)$passed
```

Here the generalized eigenvalues are 2, 4, and 6: each diagonal entry of `A`
is divided by the corresponding entry of `B`.

```{r validate-dense-spd, include=FALSE}
stopifnot(
  isTRUE(certificate(fit)$passed),
  isTRUE(all.equal(sort(Re(values(fit))), c(2, 4, 6)))
)
```

## Compute only part of the spectrum

Use `eig_partial()` when you need only a few eigenpairs. A target such as
`smallest()` states which part of the spectrum to compute. Scale the same
idea up to a diagonal metric problem with 150 pairs, and ask for the 5
smallest:

```{r partial-spd}
n <- 150
A_wide <- diag(seq(1, 150, length.out = n))
B_wide <- diag(seq(1, 2, length.out = n))

part <- eig_partial(A_wide, B = B_wide, k = 5, target = smallest())
sort(Re(values(part)))
part$method
certificate(part)$passed
```

```{r validate-partial-spd, include=FALSE}
truth_wide <- sort(diag(A_wide) / diag(B_wide))
stopifnot(
  isTRUE(certificate(part)$passed),
  isTRUE(all.equal(sort(Re(values(part))), head(truth_wide, 5), tolerance = 1e-7))
)
```

```{r partial-spd-spectrum, echo = FALSE, fig.cap = "The five smallest generalized eigenvalues (blue) against the full 150-pair spectrum of the pencil (A, B) (grey).", fig.alt = "Scatter plot of 150 generalized eigenvalues sorted descending in grey, with the five smallest highlighted in blue at the bottom-right."}
plot_spectrum(
  truth_wide,
  values(part),
  computed_ranks = tail(seq_along(truth_wide), length(values(part))),
  ylab = "generalized eigenvalue"
)
```

With `n = 150` this computed 5 of 150 pairs. A whitened-PCA or
normalized-Laplacian problem built from real data routinely has `n` in the
tens or hundreds of thousands, where a full generalized eigendecomposition
is often not practical and a certified partial slice is the useful result you
can afford.

The main choice is the structure of the problem and how much of its spectrum
you need:

| Problem | Function |
| --- | --- |
| Symmetric/Hermitian `A` with positive-definite `B`, full spectrum | `eig_full(A, B = B)` |
| Symmetric/Hermitian `A` with positive-definite `B`, partial spectrum | `eig_partial(A, B = B, k = ...)` |
| Dense general pencil | `eig_full(A, B = B, structure = general())` |
| Dense generalized Schur decomposition | `generalized_schur(A, B)` |

## Handle a dense general pencil

Use `structure = general()` when `B` is indefinite, singular, nonsymmetric, or
when the pair does not satisfy the symmetric/Hermitian positive-definite
contract. This path represents eigenvalues by homogeneous coordinates
$(\alpha, \beta)$, where a finite eigenvalue is $\alpha / \beta$.

```{r dense-general}
A_general <- matrix(c(1, 4, 2, 3), 2, 2)
B_general <- matrix(c(2, 1, 0, -1), 2, 2)

pencil <- eig_full(A_general, B = B_general, structure = general())
values(pencil)
alpha_beta(pencil)$classification
certificate(pencil)$passed
```

```{r validate-dense-general, include=FALSE}
stopifnot(
  isTRUE(certificate(pencil)$passed),
  all(alpha_beta(pencil)$classification == "finite")
)
```

Homogeneous coordinates are especially useful when `B` is singular. eigencore
classifies finite, infinite, and undefined eigenvalues instead of forcing every
pair into an ordinary numeric ratio.

```{r singular-pencil}
singular <- eig_full(
  diag(c(2, 3, 0)),
  B = diag(c(1, 0, 0)),
  structure = general()
)

alpha_beta(singular)$classification
certificate(singular)$failed_indices
```

```{r validate-singular-pencil, include=FALSE}
stopifnot(identical(
  sort(alpha_beta(singular)$classification),
  sort(c("finite", "infinite", "undefined"))
))
```

## Use QZ when you need the Schur form

`generalized_schur()` computes a dense generalized Schur, or QZ,
decomposition. Use `values()` for the finite ratios and `alpha_beta()` when you
need the homogeneous coordinates or finite/infinite classification.

```{r qz}
qz <- generalized_schur(A_general, B_general)
values(qz)
alpha_beta(qz)$classification
qz$method
```

```{r validate-qz, include=FALSE}
stopifnot(
  length(values(qz)) == 2L,
  all(alpha_beta(qz)$classification == "finite")
)
```

For pencils with singular `B`, `sort = "finite"` or `sort = "infinite"` moves
the requested class to the leading part of the decomposition.

```{r qz-sort}
qz_singular <- generalized_schur(
  diag(c(2, 3, 0)),
  diag(c(1, 0, 0)),
  sort = "infinite"
)
alpha_beta(qz_singular)$classification
```

## Keep sparse partial problems sparse

Sparse symmetric/Hermitian problems with a positive-definite metric use
`eig_partial()`. Set `allow_dense_fallback = "never"` when preserving sparsity
is a hard requirement.

```{r sparse-partial}
A_sparse <- Diagonal(x = c(1, 4, 9, 16, 25, 36))
B_sparse <- Diagonal(x = c(1, 2, 3, 4, 5, 6))

sparse_fit <- eig_partial(
  A_sparse,
  B = B_sparse,
  k = 3,
  target = smallest(),
  method = lanczos(max_subspace = 6),
  allow_dense_fallback = "never"
)

values(sparse_fit)
sparse_fit$method
certificate(sparse_fit)$passed
```

```{r validate-sparse-partial, include=FALSE}
stopifnot(
  isTRUE(certificate(sparse_fit)$passed),
  isTRUE(all.equal(
    sort(Re(values(sparse_fit))), c(1, 2, 3), tolerance = 1e-7
  ))
)
```

Full-spectrum functions accept base dense matrices and reject sparse or
operator inputs rather than silently converting them to dense storage.

## A note for older code

The retired `geigen` package exposed related operations, but eigencore is not a
namespace-compatible replacement. When maintaining older code, translate the
mathematical operation: use `eig_full()` for a full generalized eigensolve,
`generalized_schur()` for QZ, and `alpha_beta()` to inspect homogeneous
eigenvalue coordinates.

## Where to go next

- `vignette("sparse-pca")` covers operators and non-densifying centering for
  the standard (non-generalized) eigenproblem and SVD.
- For more on result validation, see `vignette("certificates")`.