---
title: Get started with eigencore
output:
  rmarkdown::html_vignette:
    toc: yes
    toc_depth: 2.0
    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{Get started with eigencore}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup-opts, 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
)
```

```{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
))
```

```{r helpers, include = FALSE}
# Shared palette + two small base-graphics helpers used throughout the
# vignette. Kept out of the reader's view: the reader sees the API call and
# the picture, never the plotting boilerplate.
ec_blue <- "#2166ac"; ec_red <- "#b2182b"; ec_tol <- "#ef8a62"; ec_grey <- "grey78"

# Per-pair backward error as a stem/dot plot, against the tolerance line.
plot_backward_error <- function(cert, main = NULL, tol = cert$tolerance) {
  be <- pmax(cert$backward_error, .Machine$double.eps)
  k  <- length(be)
  col_pt <- ifelse(cert$converged, ec_blue, ec_red)
  rng  <- range(c(be, tol))
  ylim <- c(rng[1] / 6, rng[2] * 6)
  op <- par(mar = c(4, 4.6, if (is.null(main)) 1 else 2.4, 1)); on.exit(par(op))
  plot(seq_len(k), be, log = "y", type = "n", xlim = c(0.5, k + 0.5),
       ylim = ylim, xaxt = "n", bty = "n", main = main,
       xlab = "returned pair", ylab = "backward error (log scale)")
  axis(1, at = seq_len(k))
  segments(seq_len(k), ylim[1], seq_len(k), be, col = col_pt, lwd = 2)
  points(seq_len(k), be, pch = 19, cex = 1.3, col = col_pt)
  abline(h = tol, col = ec_tol, lty = 2, lwd = 2)
  text(k + 0.45, tol, sprintf("tol = %.0e", tol), col = ec_tol,
       pos = 3, cex = 0.85, xpd = NA)
}

# A computed slice (blue) drawn on top of the full dense spectrum (grey).
plot_spectrum <- function(all_vals, 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(seq_len(k), 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)
}
```

eigencore turns spectral computation into a five-step workflow:

1. **Build an operator** describing the action of `A`.
2. **Describe the problem** (eigen or SVD; metric `B`; spectral target).
3. **Plan a solver** and inspect what it will run.
4. **Solve.**
5. **Inspect the certificate** — the numerical evidence that the result is trustworthy.

This vignette walks through the workflow on three problem classes: a standard
Hermitian eigenproblem, a generalized SPD eigenproblem with a metric `B`, and
a partial SVD. It also shows the RSpectra-compatible shim for users migrating
from existing code.

For the V2 CRAN release, planner labels are part of the contract: promoted
paths are native and benchmark-backed in their documented regimes, while
reference, prototype, oracle, and diagnostic paths remain clearly labelled.

```{r setup}
library(eigencore)
```

## 1. Standard Hermitian eigenproblem

Build a small symmetric matrix and ask for the five largest eigenvalues.

```{r make-A}
set.seed(1)
n <- 200
A <- crossprod(matrix(rnorm(n * n), n, n)) / n + diag(n)
```

eigencore wraps the matrix in a *block-native operator* the moment you hand it
to a problem constructor. You can do this explicitly:

```{r as-operator}
Aop <- as_operator(A)
Aop
```

The operator carries dimensions, structure tags, and a flag indicating whether
the underlying storage has a native kernel (in this case dense double — yes).

### Build the problem and inspect the plan

```{r plan}
P    <- eigen_problem(A, structure = hermitian(), target = largest())
plan <- plan_solver(P, k = 5)
plan
```

The plan tells you *exactly* which kernel will be invoked. There is no
silent dispatch.

### Solve and read the certificate

```{r solve}
fit <- solve(P, k = 5)
fit
```

```{r certificate}
fit$certificate
```

A passing certificate means: every requested pair has

- residual `||A v - lambda v|| / s` below `tol`, where `s` is the labeled norm
  bound (here `frobenius_exact`),
- backward error below `tol`,
- inter-vector orthogonality loss below the orthogonality tolerance.

The certificate is not one number — it is one verdict *per returned pair*. Plot
the per-pair backward error and the tolerance becomes a line you can see every
pair clearing.

```{r cert-bars, echo = FALSE, fig.cap = "Per-pair backward error for the five returned eigenpairs. Every pair sits well below the dashed tolerance line, so the overall certificate passes.", fig.alt = "Stem plot of backward error for five eigenpairs on a log scale; all five points fall far below the dashed tolerance line at 1e-8."}
plot_backward_error(fit$certificate)
```

If any pair were above the line, the certificate's `failed_indices` slot would
name it. Here the worst pair clears the tolerance by orders of magnitude.

```{r values}
fit$values
```

These five values are a *certified slice* of a 200-dimensional spectrum.
Plotting them against the full dense spectrum shows exactly which part of the
problem you paid to compute.

```{r spectrum, echo = FALSE, fig.cap = "The five largest eigenvalues (blue) located within the full spectrum of A (grey). eigencore computes only the requested slice, then certifies it.", fig.alt = "Scatter plot of all 200 eigenvalues sorted from largest to smallest in grey, with the five largest highlighted in blue at the top-left."}
all_vals <- eigen(A, symmetric = TRUE, only.values = TRUE)$values
plot_spectrum(all_vals, fit$values, ylab = "eigenvalue")
```

With `n = 200` we asked for 5 of 200 pairs. In a production problem with
`n = 1e6`, computing the full spectrum is impossible — the *partial* result is
the only result, which is exactly why a certificate matters.

## 2. Generalized SPD eigenproblem (`A v = lambda B v`)

Pass a metric `B` to `eigen_problem()` (or to `eig_partial()` via the `B`
argument).

```{r generalized}
B <- diag(seq(1, 5, length.out = n))
fit_gen <- eig_partial(A, k = 5, target = largest(), B = B,
                       method = lobpcg(maxit = 200))
fit_gen
```

The certificate's residual is `||A v - lambda B v|| / (||A|| + |lambda| ||B||)`,
and orthogonality is measured in the `B`-inner product where appropriate. The
`norm_bound_type` now reports a bound for both `A` and `B`.

## 3. Partial SVD

For rectangular problems use `svd_partial()`:

```{r svd}
M <- matrix(rnorm(400 * 50), 400, 50)
svd_fit <- svd_partial(M, rank = 5, target = largest())
svd_fit
```

The same mental model applies to singular values: you compute the leading few
and leave the tail untouched.

```{r svd-scree, echo = FALSE, fig.cap = "The five leading singular values (blue) computed by eigencore, shown against the full singular-value spectrum of M (grey).", fig.alt = "Scatter plot of all 50 singular values of M sorted descending in grey, with the top five highlighted in blue."}
all_sv <- svd(M, nu = 0, nv = 0)$d
plot_spectrum(all_sv, svd_fit$d, ylab = "singular value")
```

The reported `method` identifies the path — for very small or near-square
problems eigencore may use a dense LAPACK SVD fallback rather than running its
iterative Golub–Kahan kernel. Either way the certificate covers both
`||A v - sigma u||` and `||A^T u - sigma v||`.

## 4. RSpectra-compatible workflow

If your existing code uses `RSpectra::eigs_sym()`, you can call eigencore in
the same shape — the return list extends RSpectra's by adding `certificate`
and `diagnostics`:

```{r rspectra}
res <- eigs_sym(A, k = 5, which = "LA")
str(res, max.level = 1)
```

```{r rspectra-cert}
res$certificate
```

`eigs()`, `eigs_sym()`, and `svds()` accept the same `which` codes as
`RSpectra` — `"LM"`, `"SM"`, `"LA"`, `"SA"`, `"LR"`, `"SR"`, `"LI"`, `"SI"`,
and `"BE"`.

## Where to go next

- `vignette("certificates")` is the deep dive on reading the numerical
  evidence — what each field means and what to do when a check fails.
- Run `help(package = "eigencore")` to browse the installed help index.
- `?certificate` documents the certificate fields in detail.
- `?plan_solver` explains how operator structure, target, and method
  combine to choose a kernel.
- `linear_operator()` lets you wrap a matrix-free `apply` callback as a
  first-class operator; eigencore's planner will warn when an R-level
  callback is being driven from a block-native solver hot loop, so you
  can decide whether to invest in a native operator implementation.
