/**
 * Iterative symmetric eigensolver — Lanczos tridiagonalization + Rayleigh-Ritz.
 *
 * For large symmetric problems where the dense `eigs` (full O(n^3)
 * eigendecomposition of the whole matrix) is prohibitive, `eigsh` extracts
 * just the `k` largest or smallest eigenpairs. It builds an orthonormal
 * Krylov basis `V` via the Lanczos iteration (with full reorthogonalization
 * against every prior Lanczos vector, for numerical stability at the small
 * sizes this is exercised at), forming the small tridiagonal projection
 * `T = Vᵀ A V`. `T`'s eigenproblem is solved directly (cyclic Jacobi
 * rotations — `T` is dense-symmetric-small by construction) and lifted back
 * through `V` (Rayleigh-Ritz) to approximate `A`'s eigenpairs. Accepts either
 * a dense matrix or a matvec callback (a linear operator, matching the
 * `krylov.ts` convention) — the matvec form never forms `A` and requires
 * `opts.n` since the dimension can't otherwise be inferred.
 *
 * @packageDocumentation
 */
/** A symmetric linear operator: either a dense matrix or a matvec callback `x -> A x`. */
export type EigshOperatorInput = number[][] | ((x: number[]) => number[]);
/** Options accepted by {@link eigsh}. */
export interface EigshOptions {
    /** Which end of the spectrum to return: `'LM'` (largest, default) or `'SM'` (smallest). */
    which?: 'LM' | 'SM';
    /** Dimension of `A` — required when `A` is a matvec callback. */
    n?: number;
    /** Convergence tolerance for diagonalizing the small tridiagonal projection (default 1e-10). */
    tol?: number;
    /** Maximum Lanczos steps (default `min(max(2k + 20, 20), n)`). */
    maxIter?: number;
}
/**
 * Result of {@link eigsh}.
 *
 * `eigenvectors` is an `n x k` matrix with eigenvectors stored as **columns**:
 * `eigenvectors[i][j]` is the `i`-th component of the `j`-th eigenvector,
 * which corresponds to `eigenvalues[j]`.
 */
export interface EigshResult {
    /** The `k` selected eigenvalues, ordered by `which` (largest-first for `'LM'`, smallest-first for `'SM'`). */
    eigenvalues: number[];
    /** The `k` corresponding eigenvectors, as columns of an `n x k` matrix. */
    eigenvectors: number[][];
}
/**
 * `eigsh` — the `k` largest or smallest eigenpairs of a **symmetric** matrix
 * via the Lanczos iteration, for problems too large for the dense `eigs`.
 *
 * `A` may be a dense `number[][]` or a matvec callback `x => A x` (in which
 * case `opts.n` is required — the dimension can't be inferred from a
 * function). Eigenvectors are returned as **columns** of an `n x k` matrix:
 * `result.eigenvectors[i][j]` is the `i`-th component of the eigenvector for
 * `result.eigenvalues[j]`.
 *
 * @example
 * eigsh([[2, 1, 0], [1, 2, 1], [0, 1, 2]], 1, { which: 'LM' })
 * // => { eigenvalues: [2 + Math.SQRT2], eigenvectors: [[...]] }
 */
export declare function eigsh(a: EigshOperatorInput, k?: number, opts?: EigshOptions): EigshResult;
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