import { ComplexType, NumLikeType } from './Complex';
import { type ElementType, MultiArray } from './MultiArray';
type Jacobi2x2Result = {
    c: number;
    s: ComplexType;
};
type LAPACKConfig = {
    /**
     * Maximum iteration factor.
     */
    maxIterationFactor: number;
};
export declare const LAPACKConfigKeyTable: (keyof LAPACKConfig)[];
/**
 * # LAPACK (Linear Algebra PACKage)
 *
 * ## References
 * - https://en.wikipedia.org/wiki/LAPACK
 * - https://www.netlib.org/lapack/
 * - https://github.com/Reference-LAPACK/lapack
 */
declare abstract class LAPACKunused {
    /**
     * `LAPACK` default settings.
     */
    static readonly defaultSettings: LAPACKConfig;
    /**
     * `LAPACK` current settings.
     */
    static readonly settings: LAPACKConfig;
    /**
     * Set configuration options for `LAPACK`.
     * @param config Configuration options.
     */
    static readonly set: (config: Partial<LAPACKConfig>) => void;
    /**
     * Apply a sequence of row interchanges to a MultiArray.
     * LAPACK-like signature: laswp(A, k1, k2, ipiv, incx, colStart = 0, colEnd = numCols-1)
     * @param A Multidimensional matrix.
     * @param k1 1-based start index of rows to process (inclusive).
     * @param k2 1-based end index of rows to process (inclusive).
     * @param ipiv Integer array of pivot indices (LAPACK convention: 1-based). The routine will auto-detect if ipiv looks 0-based and adapt.
     * @param incx Increment (typically +1 or -1). When incx > 0 loop i=k1..k2 step incx; when incx<0 loop i=k1..k2 step incx.
     * @param colStart Optional 0-based column start to which the swaps are applied (inclusive).
     * @param colEnd Optional 0-based column end to which the swaps are applied (inclusive).
     * @returns
     */
    static readonly laswp: (A: MultiArray, k1: number, k2: number, ipiv: number[], incx: number, colStart?: number, colEnd?: number) => void;
    /**
     * Swap two rows of an N-dimensional MultiArray across all pages in place.
     * It is `LAPACK.laswp` for a single pivot applied across slices). `row1`,
     * `row2` are 0-based indices within the first dimension.
     * @param M Matrix.
     * @param row1 First row index to swap.
     * @param row2 Second row index to swap.
     */
    static readonly laswp_rows: (M: MultiArray, row1: number, row2: number) => void;
    /**
     * Swap two columns of an N-dimensional MultiArray across all pages in
     * place. There is no direct equivalent to LAPACK, but it is symmetrical
     * to `laswp_rows`.
     * @param M Matrix.
     * @param col1 First column index to swap.
     * @param col2 Second column index to swap.
     */
    static readonly laswp_cols: (M: MultiArray, col1: number, col2: number) => void;
    /**
     * Returns a 2D or ND-aware identity matrix (`ComplexType[][]`), row-major.
     * Optimized for internal `BLAS`/`LAPACK` operations.
     * Supports arbitrary number of dimensions.
     *
     * Usage:
     *  `LAPACK.eye(2,3,5,4)`   -> MultiArray with dims [2,3,5,4]
     *  `LAPACK.eye([2,3,5,4])` -> same as above
     *
     * Diagonal filled with `1` (`Complex.one()`), rest zeros.
     * Each 2-D page `[m,n]` in `ND` array gets its own identity.
     * @param dims `number[] | ...number`
     */
    static readonly eye: (...dims: any[]) => ComplexType[][];
    /**
     * Returns a 2D or ND-aware null matrix (`ComplexType[][]`), row-major.
     * Optimized for internal `BLAS`/`LAPACK` operations.
     * Supports arbitrary number of dimensions.
     *
     * Usage:
     *  `LAPACK.zeros(2,3,5,4)`   -> MultiArray with dims [2,3,5,4]
     *  `LAPACK.zeros([2,3,5,4])` -> same as above
     *
     * All elements filled with `0` (`Complex.zero()`).
     * @param dims `number[] | ...number`
     * @returns
     */
    static readonly zeros: (...dims: any[]) => ComplexType[][];
    static readonly lassq: (A: MultiArray) => {
        scale: ComplexType;
        sumsq: ComplexType;
    };
    static readonly lange: (norm: "F" | "M", A: MultiArray) => ComplexType;
    /**
     * Zero the lower triangular part (i > j), keeping only the upper triangle. LAPACK-like TRIU operation.
     * @param M
     */
    static readonly triu_inplace: (M: MultiArray) => void;
    /**
     * Zero the upper triangular part (j > i), keeping only the lower triangle. LAPACK-like TRIL operation.
     * @param M
     */
    static readonly tril_inplace: (M: MultiArray) => void;
    /**
     * Solve for X in U * X = B where U is upper triangular k x k stored at
     * A[kblock]. We implement in-place update of Bblock (A12 region). Block
     * target at rows k..k+kb-1, cols k+kb..n-1
     * @param A full MultiArray; U is at rows k..k+kb-1, cols k..k+kb-1
     * @param k
     * @param kb
     */
    static readonly trsm_left_upper_block: (A: MultiArray, k: number, kb: number) => void;
    /**
     *
     * @param jpvt
     * @returns
     */
    static readonly lapmt_matrix: (jpvt: number[]) => MultiArray;
    /**
     * ## LAPACK.larfg (complex)
     * Construct a complex Householder reflector H = I - tau * v * vᴴ
     * such that:
     *
     *      H * [ alpha ] = [ beta ]
     *          [   x   ]   [  0   ]
     *
     * v is stored as:
     *      v[0] = 1
     *      v[1:] overwrites x
     *
     * This implementation is faithful to LAPACK ZLARFG.
     *
     * @param alpha Complex scalar (modified in-place to beta)
     * @param x Vector below alpha (modified in-place to v[1:])
     * @returns beta as alpha and tau both Complex scalar
     */
    static larfg_complex(alpha: ComplexType, x: ComplexType[]): {
        alpha: ComplexType;
        tau: ComplexType;
    };
    /**
     * Generate a Householder reflector for a vector x (length m).
     * Produces tau, v (with v[0] = 1) and alpha (the value to write at x[0]).
     *
     * This is the vector-version of larfg. It does NOT read or write a matrix:
     * it only uses the vector x (ComplexType[]) and returns the reflector data.
     *
     * Conventions match LAPACK ZLARFG: alpha = -phi * ||x||, tau = (alpha - x0)/alpha,
     * v[0] = 1, v[i] = x[i] / (x0 - alpha) for i>=1. If sigma == 0 then tau = 0.
     */
    static readonly larfg_vector: (x: ComplexType[]) => {
        tau: ComplexType;
        v: ComplexType[];
        phi: ComplexType;
        alpha: ComplexType;
    };
    /**
     * ## `LAPACK.larfg_left`
     * LAPACK-style LARFG - Compute Householder vector and tau for complex
     * vectors `x = A[k:m-1, k]`. It's a complex householder generator for
     * left-side application. Generates `tau`, `v`, `phi` and `alpha` for
     * `H = I - tau*v*vᴴ`.
     * ### Notes:
     * - Uses `BLAS.dotc_col(A, k, k)` to compute `sigma = sum_{i>k} |x_i|^2`.
     * - Follows the ZLARFG convention (`alpha = -phi * ||x||`, with `phi = x0/|x0|` when `x0!=0`).
     *
     * NOTE: v is returned unscaled; callers (geqr2/geqp2/geqp3) are responsible for storing tau * v if needed.
     *
     * @param A Target MultiArray (only used for shape reference).
     * @param m Number of rows
     * @param k Start index of reflector
     * @returns object { tau, v, phi, alpha } where:
     * - `v` is a ComplexType[] with `v[0] = 1` and `length = m-k`.
     * - `tau` is ComplexType.
     * - `phi` is ComplexType.
     * - `alpha` is the resulting leading value (the value that replaces `A[k,k]`)
     */
    static readonly larfg_left: (A: MultiArray, dim: number, k: number) => {
        tau: ComplexType;
        v: ComplexType[];
        phi: ComplexType;
        alpha: ComplexType;
    };
    /**
     * LAPACK-style LARFG - build a Householder reflector acting **on the right** (row-wise).
     *
     * Computes tau, v, phi, alpha for the row vector
     *    x = A[k, k:n-1]ᵀ
     *
     * Generates H = I - tau * v * vᴴ such that H * x = [alpha, 0, ...].
     *
     * Householder conventions follow LAPACK ZLARFG:
     *   - alpha = -phi * ||x||, phi = x0/|x0| if x0!=0, else 1
     *   - tau = (alpha - x0)/alpha
     *   - v[0] = 1, v[j>0] = x[j] / (x0 - alpha)
     *
     * @param A MultiArray (row-wise target)
     * @param n number of columns
     * @param k start index of reflector (column)
     * @returns { tau, v, phi, alpha }:
     *   - v: ComplexType[] with v[0]=1, length = n-k
     *   - tau: ComplexType
     *   - phi: ComplexType
     *   - alpha: ComplexType (replaces A[k][k])
     */
    static readonly larfg_right: (A: MultiArray, n: number, k: number) => {
        tau: ComplexType;
        v: ComplexType[];
        phi: ComplexType;
        alpha: ComplexType;
    };
    /**
     * ## LAPACK.larfg
     *
     * Generates a LAPACK-style Householder reflector H such that:
     *
     *   H = I - tau * v * vᴴ
     *
     * where:
     *   - For side = 'L' (left):  H * x = [alpha, 0, ..., 0]ᵀ
     *   - For side = 'R' (right): x * H = [alpha, 0, ..., 0]
     *
     * The vector x is extracted from matrix A starting at index `k`:
     *   - side = 'L': x = A[k:m-1, k] (column)
     *   - side = 'R': x = A[k, k:n-1] (row)
     *
     * The Householder vector `v` is returned as a 1D array (ComplexType[]).
     * It is the caller's responsibility to interpret it as a column or row vector:
     *   - side = 'L' → v is conceptually a column vector
     *   - side = 'R' → v is conceptually a row vector
     *
     * ### LAPACK ZLARFG conventions
     * - phi = x0 / |x0|   if x0 != 0, else 1
     * - alpha = -phi * ||x||_2
     * - tau = (alpha - x0) / alpha
     * - v[0] = 1
     *
     * Special case:
     * - If the tail of x is zero (sigma = 0), tau = 0 and H = I (identity reflector)
     *
     * @param side 'L' for left (column-wise) or 'R' for right (row-wise) application
     * @param A Target matrix (MultiArray)
     * @param dim Dimension of the subvector (m for 'L', n for 'R')
     * @param k Start index of the reflector in A
     * @returns Object containing:
     *   - `tau` (ComplexType): scalar factor of the reflector
     *   - `v` (ComplexType[]): Householder vector with v[0] = 1
     *   - `phi` (ComplexType): phase factor used to define alpha
     *   - `alpha` (ComplexType): resulting leading element after applying H
     *
     * ### Notes
     * - `v` is always normalized such that v[0] = 1; the remaining entries are scaled accordingly.
     * - The caller decides whether `v` is treated as a column or row vector when constructing H.
     * - Follows LAPACK-style handling of complex vectors.
     */
    static readonly larfg_old: (side: "L" | "R", A: MultiArray, dim: number, k: number) => {
        tau: ComplexType;
        v: ComplexType[];
        phi: ComplexType;
        alpha: ComplexType;
    };
    static readonly larfgLQ: (A: MultiArray, k: number, n: number) => {
        tau: ComplexType;
        v: ComplexType[];
        alpha: ComplexType;
    };
    static readonly gelq2_final: (A: MultiArray) => {
        L: MultiArray;
        taus: ComplexType[];
    };
    static readonly gelq2_nova: (A: MultiArray) => {
        L: MultiArray;
        taus: ComplexType[];
    };
    static readonly gelq2_nao_funciona: (A: MultiArray) => {
        L: MultiArray;
        taus: ComplexType[];
    };
    /**
     * Generate Householder vector and apply it immediately to A.
     * LAPACK-style: side-aware (left/right), complex.
     * @param side 'L' for left, 'R' for right
     * @param A Matrix to transform (modified in place)
     * @param k Index for the reflector (row/column start)
     */
    /**
     *
     * @param side
     * @param A
     * @param k
     * @returns Object { tau, v, alpha }
     */
    static readonly larfg_apply: (side: "L" | "R", A: MultiArray, k: number) => {
        tau: ComplexType;
        v: ComplexType[];
        alpha: ComplexType;
    };
    /**
     *
     * @param A
     * @param k
     * @returns Object { tau, v, alpha }
     */
    static readonly larfg_left_nd: (A: MultiArray, k: number) => {
        tau: ComplexType;
        v: ComplexType[];
        alpha: ComplexType;
    };
    /**
     *
     * @param A
     * @param k
     * @returns Object { tau, v, alpha }
     */
    static readonly larfg_right_nd: (A: MultiArray, k: number) => {
        tau: ComplexType;
        v: ComplexType[];
        alpha: ComplexType;
    };
    /**
     *
     * @param side
     * @param A
     * @param k
     * @returns
     */
    static readonly larfg_nd: (side: "L" | "R", A: MultiArray, k: number) => {
        tau: ComplexType;
        v: ComplexType[];
        alpha: ComplexType;
    };
    /**
     * Apply an elementary reflector H = I - tau * v * vᴴ to A from the left,
     * but restricted to columns colStart .. colEnd-1.
     *
     * A: MultiArray (row-major)
     * v: ComplexType[] (v[0] == 1, length = m - rowStart)
     * tau: ComplexType
     * rowStart: starting row index of the reflector (k)
     * colStart: first column to update (usually k+1)
     * colEnd: (optional) one-past-last column to update (default = A.dimension[1])
     */
    static readonly larf_left_block_anterior: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number, colEnd?: number) => void;
    /**
     * Apply Householder reflector H = I - tau * v * vᴴ to A from the left,
     * in a blocked fashion:
     *
     *   A[rowStart..m-1, colStart..n-1] := H * A[rowStart..m-1, colStart..n-1]
     *
     * v is ComplexType[] of length = m - rowStart (v[0] == 1).
     *
     * Uses temporary block buffers and BLAS.gemm_block to compute
     * S = vᴴ * A_block and then A_block -= tau * v * S
     */
    static readonly larf_left_block: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number, blockSize?: number) => void;
    /**
     * Apply a Householder reflector from the RIGHT:Vamos mapear ZGEMV/ZGERC do LAPACK e alinhar com nosso código.
     *
     *  A := A * (I - tau * v * vᴴ)
     *
     * Where:
     *  - A is the target matrix
     *  - v is the Householder vector (length = block size)
     *  - tau is the scalar
     *  - rowStart is the first row of the block
     *  - colStart is the first column of the block
     *
     * This exactly matches LAPACK xLARF(side='R') behavior.
     */
    static readonly larf_right: (C: MultiArray, v: ComplexType[], tau: ComplexType, i0: number, j0: number) => void;
    /**
     * Blocked version of applying Hᴴ = I - conj(tau) * v * vᴴ from the RIGHT:
     *
     *   A[rowStart..m-1, colStart..colStart+vlen-1] := A[rowStart..m-1, colStart..colStart+vlen-1] * Hᴴ
     *
     * v length = number of columns in the block (vlen).
     *
     * Uses temporary buffers and BLAS.gemm_block where appropriate.
     */
    static readonly larf_right_conj_block: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number, blockSize?: number) => void;
    /**
     * Apply the conjugate-right Householder reflector Hᴴ = I - conj(tau) * v * vᴴ to A from the right.
     * Equivalent to A := A * Hᴴ for rows i = rowStart..m-1 and columns j = colStart..colStart+v.length-1.
     *
     * This is the companion routine for larf_left when you need to perform
     * A := H * A * Hᴴ (first call larf_left(A, v, tau, rowStart, colStartLeft)
     * then call larf_right_conj(A, v, tau, rowStart, colStartRight)).
     *
     * @param A         MultiArray to modify in-place.
     * @param v         Householder vector (ComplexType[]), v[0] == 1, length = block width.
     * @param tau       Householder scalar (ComplexType).
     * @param rowStart  first row of the block (usually 0 or k).
     * @param colStart  first column of the block (where v aligns horizontally).
     */
    static readonly larf_right_conj: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number) => void;
    /**
     * Same operation as larf_right_conj — but returns new matrix instead of modifying A.
     *
     * Compute:
     *      A_new = A * Hᴴ
     * where Hᴴ = I - conj(tau) * v * vᴴ
     *
     * Input:
     *   A         original matrix (not modified)
     *   v         Householder vector, v[0] == 1
     *   tau       Householder scalar
     *   rowStart  first affected row
     *   colStart  first affected column
     *
     * Output:
     *   MultiArray A_new = A * Hᴴ
     */
    static readonly larf_right_conj_return: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number) => MultiArray;
    /**
     *
     * @param A
     * @param v
     * @param tau
     * @param rowStart
     * @param colStart
     * @returns
     */
    static readonly larf_left_nd: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number) => void;
    /**
     *
     * @param A
     * @param v
     * @param tau
     * @param rowStart
     * @param colStart
     * @returns
     */
    static readonly larf_right_nd: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number) => void;
    static readonly larf_nd: (side: "L" | "R", A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number) => void;
    /**
     * Unblocked LU factorization (panel) - modifies A in place.
     * A is m x n stored as MultiArray (row-major physical layout).
     * Performs LU on A[k..m-1, k..n-1] and writes pivots into piv starting at offset k.
     * Returns number of pivots performed (panel width) or info.
     *
     * This is analogous to LAPACK's xGETF2 applied to the submatrix.
     *
     * This routine will update A in-place (compact LU) and fill piv[k..k+panelWidth-1].
     *
     * @param A MultiArray (m x n)
     * @param k starting column/row index for panel
     * @param panelWidth number of columns to factor (<= min(m-k, n-k))
     * @param piv global pivot array (zero-based), length >= min(m,n)
     */
    static readonly getf2: (A: MultiArray, k: number, panelWidth: number, piv: number[]) => void;
    /**
     * Blocked GETRF for complex matrices (LU with partial pivoting).
     * This routine will modify A in-place, writing LU (L below diag, U on and above diag).
     * @param A
     * @returns { LU: MultiArray (same reference as input A, modified), piv: number[], swaps: number }
     */
    static readonly getrf: (A: MultiArray, blockSize?: number) => {
        LU: MultiArray;
        piv: number[];
        swaps: number;
    };
    /**
     * Compute A22 := A22 - A21 * A12  (standard update in GETRF).
     * All indices are absolute within A.array.
     * A21: rows (k+kb .. m-1) x cols (k .. k+kb-1)
     * A12: rows (k .. k+kb-1) x cols (k+kb .. n-1)
     * A22: rows (k+kb .. m-1) x cols (k+kb .. n-1)
     * @param A
     * @param k
     * @param kb
     * @param blockSizeInner
     */
    static readonly gemm_blocked: (A: MultiArray, k: number, kb: number, blockSizeInner?: number) => void;
    /**
     * Blocked LU factorization that calls getf2 for panel factorization,
     * then solves and updates the trailing submatrix.
     * @param A
     * @returns Object { LU: A, piv, info, swaps }
     */
    static readonly getrf_blocked: (A: MultiArray, blockSize?: number) => {
        LU: MultiArray;
        piv: number[];
        info: number;
        swaps: number;
    };
    /**
     * Solve systems A X = B given LU factorization in-place and pivots.
     * This follows LAPACK GETRS semantics (no transpose).
     * @param LU MultiArray containing LU as returned by getrf
     * @param piv pivot vector (zero-based) length = min(m,n)
     * @param B MultiArray of size m x nrhs (modified in place)
     * @returns new MultiArray (`B`) with solution
     */
    static readonly getrs: (LU: MultiArray, piv: number[], B: MultiArray) => MultiArray;
    /**
     * Reduce a Hermitian matrix A (n × n) to real tridiagonal form T using
     * Householder reflectors.
     *
     * This routine performs a similarity transformation:
     *
     *     A = Q * T * Qᴴ
     *
     * where:
     *   - A is the original Hermitian matrix,
     *   - T is a real symmetric tridiagonal matrix,
     *   - Q is a unitary matrix formed as a product of Householder reflectors.
     *
     * The reduction is carried out implicitly as:
     *
     *     A ← H₀ᴴ H₁ᴴ ... Hₙ₋₂ᴴ · A · Hₙ₋₂ ... H₁ H₀
     *
     * with:
     *
     *     Q = H₀ H₁ ... Hₙ₋₂
     *
     * Each Householder reflector has the form:
     *
     *     H_k = I - tau_k · v_k · v_kᴴ
     *
     * where v_k is a Householder vector with the convention v_k[0] = 1.
     *
     * ---
     * Storage conventions (LAPACK-compatible):
     *
     * This routine overwrites the input matrix A in-place. After completion:
     *
     *   - diag[k] contains the diagonal entry T[k, k]
     *   - offdiag[k] contains the sub/super-diagonal entry T[k+1, k]
     *   - taus[k] contains the scalar factor tau_k for reflector H_k
     *
     *   - The Householder vector v_k is stored in:
     *
     *         A[k+1:n-1, k]
     *
     *     with:
     *
     *         v_k[0] = 1
     *         v_k[i] = A[k+1+i, k],  i = 1 .. n-k-2
     *
     * The strictly upper triangle of A is not referenced after the reduction.
     *
     * ---
     * Algorithmic structure:
     *
     * For each k = 0 .. n-2:
     *   1) A Householder reflector H_k is generated from column k
     *   2) The trailing (n-k-1) × (n-k-1) submatrix is updated via a
     *      Hermitian rank-2 update
     *
     * The concrete implementation of the reflector application is injected
     * via the `larf` parameter, allowing different backends:
     *
     *   - real symmetric reduction   → larf_left
     *   - complex Hermitian reduction → her2_zhtrd_update
     *
     * This mirrors the separation of concerns used in LAPACK (e.g. `zhetrd`)
     * and simplifies validation and reuse.
     *
     * ---
     * Dependencies:
     *
     *   - LAPACK.larfg_vector(x): generates Householder vector v and scalar tau
     *   - larf(A, v, tau, rowStart, colStart): applies reflector to trailing block
     *
     * All numeric arrays are represented as ComplexType[] to conform to the
     * MathJSLab Complex facade, even though the resulting tridiagonal matrix T
     * is real-valued.
     *
     * @param A
     *   Hermitian square MultiArray (n × n). Modified in-place.
     *
     * @param larf
     *   Function responsible for applying the Householder reflector to the
     *   trailing submatrix. Its exact behavior depends on whether the reduction
     *   is real symmetric or complex Hermitian.
     *
     * @returns
     *   An object containing:
     *     - diag    : diagonal entries of T
     *     - offdiag : sub/super-diagonal entries of T
     *     - taus    : scalar Householder factors tau_k
     */
    /**
     * Reduce a Hermitian matrix A to real symmetric tridiagonal form T:
     *
     *     A = Q · T · Qᴴ
     *
     * using Householder reflectors, following exactly the LAPACK ZHETRD
     * unblocked algorithm (lower triangle variant).
     *
     * On exit:
     *   - diag[k]     = T[k,k]
     *   - offdiag[k]  = T[k+1,k]  (real in exact arithmetic)
     *   - taus[k]     = Householder scalar τₖ
     *
     * Storage convention (LAPACK-compatible):
     *   - The Householder vector vₖ is stored in A[k+1:n-1, k]
     *   - vₖ[0] = 1 is implicit (not stored)
     *
     * @param A    Hermitian matrix (n×n), modified in-place
     * @param hetrd_update Function that applies a reflector from the left (Hermitian-aware)
     */
    static readonly sytrd: (A: MultiArray, hetrd_update: (A: MultiArray, v: ComplexType[], tau: ComplexType, rowStart: number, colStart: number) => void) => {
        diag: ComplexType[];
        offdiag: ComplexType[];
        taus: ComplexType[];
    };
    /**
     * Generate the unitary matrix Q from a Hermitian tridiagonal reduction.
     *
     * This routine reconstructs the unitary matrix Q from the Householder
     * reflectors produced by `sytrd` (Hermitian reduction to tridiagonal form).
     * It is the complex/Hermitian analogue of LAPACK's `ungtr`.
     *
     * The reduction performed by `sytrd` satisfies:
     *
     *     A = Q * T * Qᴴ
     *
     * where:
     *   - A is the original Hermitian matrix,
     *   - T is the real tridiagonal matrix returned by `sytrd`,
     *   - Q is a unitary matrix defined as the product of Householder reflectors.
     *
     * After a call to `sytrd(A, larf)`, the input matrix `A` contains the data
     * needed to reconstruct Q:
     *
     *   - For each k = 0 .. n-2, the Householder vector v_k is stored in
     *     A[k+1:n-1, k], with the convention:
     *
     *         v_k[0] = 1
     *         v_k[i] = A[k+1+i, k],  i = 1 .. n-k-2
     *
     *   - The scalar factor tau_k associated with each reflector H_k is stored
     *     in TAU[k].
     *
     * Each Householder reflector is defined as:
     *
     *     H_k = I - tau_k * v_k * v_kᴴ
     *
     * and the unitary matrix Q is given by:
     *
     *     Q = H_0 * H_1 * ... * H_{n-2}
     *
     * This routine builds Q explicitly by applying the reflectors in reverse
     * order:
     *
     *     Q = H_{n-2} * ... * H_1 * H_0 * I
     *
     * Notes:
     *   - The input matrix `A` is NOT modified by this routine.
     *   - This function assumes that `sytrd` stored the Householder vectors
     *     exactly according to the LAPACK convention (v[0] = 1).
     *   - No attempt is made to symmetrize or normalize the reflectors here;
     *     correctness depends on a consistent `sytrd` implementation.
     *
     * @param A
     *   Square MultiArray (n × n) that was previously reduced by `sytrd`.
     *   The strictly lower triangular part of A contains the Householder
     *   vectors v_k.
     *
     * @param TAU
     *   Array of length n containing the scalar factors tau_k returned by
     *   `sytrd`. Only entries TAU[0 .. n-2] are used.
     *
     * @returns
     *   A new MultiArray (n × n) containing the unitary matrix Q such that
     *   A_original = Q * T * Qᴴ.
     */
    static readonly ungtr0: (A: MultiArray, TAU: ComplexType[]) => MultiArray;
    /**
     * Generate the unitary matrix Q from a Hermitian tridiagonal reduction.
     *
     * This routine reconstructs
     *
     *     Q = H₀ H₁ ... Hₙ₋₂
     *
     * where each Householder reflector is
     *
     *     Hₖ = I − τₖ vₖ vₖᴴ
     *
     * The reflectors {vₖ, τₖ} are assumed to be stored exactly as produced by
     * sytrd / zhetrd:
     *
     *   - vₖ is stored in A[k+1:n-1, k]
     *   - vₖ[0] = 1 is implicit (not stored)
     *   - τₖ is stored in TAU[k]
     *
     * This implementation is semantically equivalent to LAPACK ZUNGTR
     * (internally using the same logic as ZUNMTR applied to the identity).
     *
     * On exit, Q satisfies:
     *
     *     A ≈ Q · T · Qᴴ
     *
     * provided sytrd was correct.
     *
     * @param A   Matrix containing Householder vectors (modified sytrd output)
     * @param TAU Householder scalar factors
     * @returns   Unitary matrix Q
     */
    static readonly ungtr: (A: MultiArray, TAU: ComplexType[]) => MultiArray;
    /**
     * Versão estendida de depuração da sytrd()
     * Reconstrói Q, T e imprime o processo completo de tridiagonalização.
     */
    static readonly sytrd_debug: (A: MultiArray) => {
        diag: import("./ComplexInterface").ComplexInterface<import("./Complex").RealType, number, unknown>[];
        offdiag: import("./ComplexInterface").ComplexInterface<import("./Complex").RealType, number, unknown>[];
        taus: import("./ComplexInterface").ComplexInterface<import("./Complex").RealType, number, unknown>[];
        Q: MultiArray<import("./ComplexInterface").ComplexInterface<import("./Complex").RealType, number, unknown> | import("./CharString").CharString | import("./Structure").Structure | import("./FunctionHandle").FunctionHandle>;
        T: MultiArray<import("./ComplexInterface").ComplexInterface<import("./Complex").RealType, number, unknown> | import("./CharString").CharString | import("./Structure").Structure | import("./FunctionHandle").FunctionHandle>;
        QtA: import("./MathOperation").MathObject;
    };
    /**
     * Blocked reduction of a Hermitian matrix A to tridiagonal form.
     *
     * This is a blocked variant of sytrd that uses larf_left_block and
     * larf_right_conj_block to apply reflectors in blocks and BLAS.gemm_block
     * for inter-block multiplications. It preserves the storage convention:
     * - Householder vectors v are stored in A[k+1:n-1, k] with v[0] at A[k+1,k]
     * - taus[k] stores the scalar for the k-th reflector
     *
     * @param A    Hermitian MultiArray (n x n). Modified in-place.
     * @param blockSize   block size (in columns) used for panel processing; default 32
     * @returns    { diag, offdiag, taus } as ComplexType[] arrays
     */
    static readonly sytrd_blocked: (A: MultiArray, blockSize?: number) => {
        diag: ComplexType[];
        offdiag: ComplexType[];
        taus: ComplexType[];
    };
    /**
     * Blocked reduction of a Hermitian matrix A to tridiagonal form
     * using LAPACK-style panel accumulation with matrix W.
     *
     * Householder vectors are stored in the lower triangle of A (v[0] at A[k+1,k])
     * taus[k] stores the scalar for the k-th reflector.
     *
     * @param A    Hermitian MultiArray (n x n), modified in-place
     * @param blockSize   block size (panel width)
     * @returns    { diag, offdiag, taus }
     */
    static readonly sytrd_blocked_w: (A: MultiArray, blockSize?: number) => {
        diag: ComplexType[];
        offdiag: ComplexType[];
        taus: ComplexType[];
    };
    /**
     * Compute eigenvalues of a symmetric tridiagonal matrix T given by
     *    diag[] and offdiag[].
     *
     * This is the QR algorithm with Wilkinson shifts - simplified version
     * that returns only eigenvalues. Next step will include eigenvectors.
     *
     * @param diag      ComplexType[] - diagonal elements of T
     * @param offdiag   ComplexType[] - sub/super diagonal (e1...e[n-1])
     * @param maxIter   optional - default 1000*n
     */
    /**
     * Compute eigenvalues only (returns MultiArray column [n x 1]).
     * This wrapper currently reuses steqr_vectors internally for correctness.
     * Signature:
     *   steqr_values(diag, offdiag, maxIter?) -> MultiArray [n x 1]
     */
    static readonly steqr_values: (diag: ComplexType[], offdiag: ComplexType[], maxIter?: number) => ComplexType[];
    /**
     * Compute eigenvalues AND eigenvectors of a symmetric tridiagonal matrix T
     * using the implicit QR algorithm with Wilkinson shifts.
     *
     * The matrix is represented by:
     *   diag[i]   = diagonal entries  (length n)
     *   offdiag[i]= sub/super diagonal (length n-1)
     *
     * Returns:
     *   { D, V }
     *     D : eigenvalues  (ComplexType[n])
     *     V : eigenvectors (ComplexType[n][n])  -- V[:,i] is eigenvector of D[i]
     */
    /**
     * Compute eigenvalues AND eigenvectors of a symmetric tridiagonal matrix T
     * using the implicit QR algorithm with Wilkinson shifts.
     *
     * Input:
     *  - diag:  ComplexType[] length n  (diagonal of T)
     *  - offdiag: ComplexType[] length n-1 (sub/super diagonal)
     *
     * Returns:
     *  { D: MultiArray, V: MultiArray }
     *   - D : MultiArray [n x 1]  (eigenvalues)
     *   - V : MultiArray [n x n]  (eigenvectors as columns)
     */
    static readonly steqr_vectors_1: (diag: ComplexType[], offdiag: ComplexType[], maxIter?: number) => {
        D: MultiArray;
        V: MultiArray;
    };
    /**
     * Stable steqr_vectors: compute eigenvalues and eigenvectors of symmetric tridiagonal T
     * Inputs:
     *  - diag: ComplexType[]  (length n)
     *  - offdiag: ComplexType[] (length n-1, last may be unused)
     * Returns:
     *  { D: MultiArray (n x 1), V: MultiArray (n x n) }
     *
     * Numeric improvements:
     * - compute r using Math.hypot of magnitudes to avoid overflow
     * - deflation test for tiny subdiagonals
     * - reasonable max iterations and early exit
     */
    /**
     * steqr_vectors (numeric-real kernel)
     * - diag: ComplexType[]  (length n)  -- assumed real values
     * - offdiag: ComplexType[] (length n-1)
     * Returns { D: MultiArray (n x 1), V: MultiArray (n x n) }
     *
     * Numeric kernel works in JS numbers for D/E and uses Complex only to update V.
     */
    static readonly steqr_vectors_0: (diag: ComplexType[], offdiag: ComplexType[], maxIteration?: number) => {
        D: MultiArray;
        V: MultiArray;
    };
    static readonly steqr_vectors_2: (diag: ComplexType[], offdiag: ComplexType[], maxIteration?: number) => {
        D: MultiArray;
        V: MultiArray;
    };
    static readonly numeric_jacobi_symmetric_dense: (Dnum_in: number[], Enum_in: number[], maxItsLocal: number) => {
        Dnum: number[];
        Vnum: ComplexType[][];
    };
    /**
     * Build dense Hermitian matrix from tridiagonal representation
     *
     * @param diag     diagonal entries
     * @param offdiag  sub/superdiagonal entries (length n-1)
     * @returns dense Hermitian matrix
     */
    static readonly tridiag_her_to_dense: (diag: ComplexType[], offdiag: ComplexType[]) => ComplexType[][];
    /**
     * Compute Frobenius norm of residual || T * V - V * diag(evals) ||_F using real-numeric aggregation.
     * @param T ComplexType[][] dense Hermitian
     * @param V MultiArray (ComplexType entries)
     * @param evalsNum number[] (real eigenvalues)
     * @returns numeric Frobenius norm (number)
     */
    static readonly compute_residual_numeric: (T: ComplexType[][], V: MultiArray, evalsNum: number[]) => number;
    /**
     * Compute Frobenius norm of residual R = A * V - V * diag(evals),
     * @param A
     * @param V
     * @param evals `evals` is `ComplexType[]` or `number[]`.
     * @returns A `number` (real Frobenius norm).
     */
    static readonly compute_residual_complex: (T: ComplexType[][], V: MultiArray, evals: ComplexType[] | number[]) => number;
    /**
     * Find the element with the highest magnitude in the column (pivot).
     * @param array Row-major order bidimensional MultiArray structure.
     * @param column Column to find the pivot.
     * @param rowStart Row start (to deal with multidimensional operations and functions). Default is `0`.
     * @param rowEnd Row end (to deal with multidimensional operations and functions). Default is `array.length`.
     * @returns An object containing:
     * - pivotRow - Row index of pivot element (relative to rowStart).
     * - pivotAbs - Absolute value of pivot.
     * - pivotIsZero - `true` if pivot element is zero (improbable null column.).
     */
    static readonly column_pivot: (array: ComplexType[][], column: number, rowStart?: number, rowEnd?: number) => {
        pivotRow: number;
        pivotAbs: ComplexType;
        pivotIsZero: boolean;
    };
    /**
     * Normalize eigenvector phases so that pivot element in each column becomes real positive.
     * Modifies V in-place (ComplexType[][]).
     * @param V
     * @param pivotPositive
     */
    static readonly normalize_eigenvector_phases: (V: ComplexType[][], pivotPositive?: boolean) => void;
    /**
     *
     * @param V
     * @returns
     */
    static readonly test_orthonormality: (V: MultiArray) => {
        maxOffDiag: number;
        maxDiagDeviation: number;
    };
    /**
     *
     * @param A
     * @param V
     * @param Dcol
     * @returns
     */
    static readonly test_residual: (A: ComplexType[][], V: MultiArray, Dcol: MultiArray) => number;
    /**
     * Diagnóstico por autovetor:
     * Para cada coluna j:
     *  - lambda = Dcol[j,0]
     *  - rq = (vᴴ * A * v) / (vᴴ * v)   (Rayleigh quotient)
     *  - res_j = || A*v - lambda*v ||_2   (norma euclidiana do residual)
     *
     * Imprime uma tabela (j, lambda, Re(rq), Im(rq), |lambda - rq|, res_j )
     */
    /**
     *
     * @param A
     * @param V
     * @param Dcol
     * @returns
     */
    static readonly diag_eigenpairs: (A: ComplexType[][], V: MultiArray, Dcol: MultiArray) => any[];
    /**
     * Compara listas: ordena autovalores fornecidos e compara com Rayleighs - procura permutações.
     * Retorna um mapeamento aproximado index->index por menor diferença absoluta.
     *
     * Inputs:
     *  - evals: ComplexType[] (autovalores retornados)
     *  - rqs: ComplexType[] (rayleighs calculados para cada coluna v_j)
     *
     * Imprime o pareamento escolhido e as diferenças.
     */
    /**
     *
     * @param evals
     * @param rqs
     * @returns
     */
    static readonly match_evals_to_rayleighs: (evals: ComplexType[], rqs: ComplexType[]) => {
        evalIndex: number;
        rqIndex: number;
        diff: number;
    }[];
    static readonly steqr_vectors_original: (diag: ComplexType[], offdiag: ComplexType[], maxIter?: number) => {
        D: MultiArray;
        V: MultiArray;
        complex: boolean;
    };
    static readonly steqr_vectors: (diag: ComplexType[], offdiag: ComplexType[], maxIter?: number) => {
        D: ComplexType[];
        V: ComplexType[][];
        complex: boolean;
    };
    /**
     * Reconstruct Q from the output of sytrd_blocked_w (Hermitian to tridiagonal reduction)
     * using LAPACK-style blocked panel accumulation with matrix W.
     *
     * @param A     MultiArray containing Householder vectors in lower triangle
     * @param taus  ComplexType[] scalars for each reflector (from sytrd_blocked_w)
     * @param blockSize    block size (panel width, same as sytrd_blocked_w)
     * @returns     Q unitary MultiArray (n x n)
     */
    static readonly orgtr_blocked_w: (A: MultiArray, taus: ComplexType[], blockSize?: number) => MultiArray;
    static readonly eig_symmetric: (A: MultiArray, computeVectors?: boolean) => {
        D: MultiArray;
        V?: MultiArray;
    };
    static readonly eig_symmetric0: (A: MultiArray, computeVectors?: boolean, maxIter?: number) => {
        D: MultiArray;
        V?: MultiArray;
    };
    static readonly eig_symmetric1: (A: MultiArray, computeVectors?: boolean, maxIter?: number) => {
        D: number[];
        V: MultiArray;
    };
    static readonly eig_symmetric2: (A: MultiArray, computeVectors?: boolean, maxIter?: number) => {
        D: MultiArray;
        V?: MultiArray;
    };
    /**
     * eig_symmetric - eigenvalues + eigenvectors for Hermitian/symmetric matrix A.
     *
     * @param A     MultiArray (n x n)
     * @param order 'none' | 'asc' | 'desc'  (default = 'asc')
     *
     * @returns { V, D, evals } where
     *   V     = MultiArray n x n (eigenvectors column-wise)
     *   D     = MultiArray n x n diagonal eigenvalue matrix
     *   evals = ComplexType[] eigenvalues (sorted according to `order`)
     */
    static readonly eig_symmetric3: (A: MultiArray, order?: "none" | "asc" | "desc") => {
        V: MultiArray;
        D: MultiArray;
        evals: ComplexType[];
    };
    static readonly eig_symmetric4: (A: MultiArray, computeVectors?: boolean, maxIter?: number) => {
        D: MultiArray;
        V?: MultiArray;
    };
    static readonly eig_symmetric5: (A: MultiArray, computeVectors?: boolean, maxIter?: number) => {
        D: MultiArray;
        V?: MultiArray;
    };
    /**
     * eig_symmetric_blocked - blocked eigen-decomposition for Hermitian/symmetric A
     *
     * @param A      MultiArray (n x n) Hermitian / real symmetric (not modified)
     * @param order  'asc' | 'desc' | 'none'  (default = 'asc')
     * @param blockSize     block size (default 32)
     *
     * @returns { D: ComplexType[], V: MultiArray }  (D = eigenvalues, V columns = eigenvectors)
     */
    static readonly eig_symmetric_blocked: (A: MultiArray, order?: "asc" | "desc" | "none", blockSize?: number) => {
        D: ComplexType[];
        V: MultiArray;
    };
    /**
     * ===================================================================================
     * =================================== 2th round =====================================
     * ===================================================================================
     */
    static readonly jacobi_complex_hermitian_dense_1: (T: ComplexType[][], maxSweeps?: NumLikeType, tol?: NumLikeType) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    /**
     * Jacobi rotation for a 2×2 Hermitian complex block:
     *
     *      [ a   b ]
     *      [ b*  d ]
     *
     * Returns c (real) and s (complex) such that:
     *
     *      Jᴴ A J is diagonal
     *
     * with:
     *
     *      J = [  c        s ]
     *          [ -conj(s)  c ]
     */
    static readonly jacobi_hermitian_2x2: (a: ComplexType, b: ComplexType, d: ComplexType) => {
        c: ComplexType;
        s: ComplexType;
    };
    static readonly jacobi_complex_hermitian_dense: (A: ComplexType[][], maxSweeps?: NumLikeType, tol?: NumLikeType) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    /**
     * Jacobi cyclic for Hermitian matrices. `T` is mutated in-place during the algorithm.
     * @param T
     * @param maxSweeps
     * @param tol
     * @returns
     */
    static readonly jacobi_symmetric_hermitian: (T: ComplexType[][], maxSweeps?: number, tol?: number) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    static readonly jacobi_real_symmetric_structured_2n: (T: ComplexType[][], maxSweeps?: number, tol?: number) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    static readonly jacobi_hermitian_real2n: (T: ComplexType[][], maxSweeps?: number, tol?: number) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    static readonly jacobi_hermitian_real2n_direct: (diag: ComplexType[], offdiag: ComplexType[], maxSweeps?: number, tol?: number) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    static readonly numeric_jacobi_hermitian_via_real2n_final: (T: MultiArray, maxSweeps?: number, tol?: number) => {
        D: ComplexType[];
        Z: MultiArray;
    };
    static readonly jacobi_hermitian_dense: (T: MultiArray, tol: number, maxSweeps: number) => {
        A: MultiArray;
        Z: MultiArray;
    };
    static readonly numeric_jacobi_hermitian_direct: (D: ComplexType[], E: ComplexType[], maxSweeps?: number, tol?: number) => {
        D_work: ComplexType[];
        Z: MultiArray;
    };
    static readonly jacobi_hermitian_full: (T: MultiArray, maxSweeps?: number, tol?: number) => MultiArray;
    static readonly numeric_jacobi_hermitian_direct_02: (D: ComplexType[], E: ComplexType[], maxSweeps?: number, tol?: number) => {
        D_work: ComplexType[];
        Z: MultiArray;
    };
    static readonly numeric_jacobi_hermitian_direct_03: (D: ComplexType[], E: ComplexType[], maxSweeps?: number, tol?: number) => {
        D_work: ComplexType[];
        Z: MultiArray;
    };
    /**
     * Rotação de Jacobi para um bloco hermitiano 2×2
     *
     * [ a   b ]
     * [ b*  d ]
     */
    static readonly jacobiHermitian2x2: (a: number, d: number, b: ComplexType, tol?: number) => Jacobi2x2Result;
    static readonly jacobi_hermitian_rotation: (T: MultiArray, Z: MultiArray, p: number, q: number) => void;
    static readonly jacobi_rotate_hermitian: (A: MultiArray, Z: MultiArray, p: number, q: number) => void;
    /**
     * Atualiza a matriz acumuladora Z no estilo LAPACK:
     *
     *      Z ← Z · G
     *
     * onde G é a rotação de Givens complexa no plano (k, k+1):
     *
     *      [  c      s  ]
     *      [ -s*     c  ]
     *
     * c é real, s é complexo.
     */
    static readonly apply_givens_right_Z: (Z: ComplexType[][], k: number, c: ComplexType, s: ComplexType) => void;
    static readonly apply_givens_left_tridiagonal: (D: ComplexType[], E: ComplexType[], k: number, c: ComplexType, s: ComplexType) => void;
    static readonly apply_givens_tridiagonal_hermitian: (D: ComplexType[], E: ComplexType[], k: number, c: ComplexType, s: ComplexType) => void;
    static readonly apply_givens_right_tridiagonal: (D: ComplexType[], E: ComplexType[], k: number, c: ComplexType, s: ComplexType) => void;
    static readonly compute_complex_givens: (x: ComplexType, z: ComplexType) => {
        c: ComplexType;
        s: ComplexType;
        r: ComplexType;
    };
    static readonly complexGivens: (x: ComplexType, y: ComplexType) => {
        c: ComplexType;
        s: ComplexType;
        r: ComplexType;
    };
    static readonly applyGivensToZ: (Z: MultiArray, k: number, c: ComplexType, s: ComplexType) => void;
    static readonly applyGivensTridiagonal: (D: ComplexType[], E: ComplexType[], k: number, c: ComplexType, s: ComplexType) => void;
    static readonly complex_gram_schmidt: (Z: MultiArray) => void;
    static readonly implicitQRStepTridiagonalHermitian: (D: ComplexType[], E: ComplexType[], l: number, m: number, Z?: ComplexType[][]) => void;
    /**
     * One implicit-shift QR sweep on a Hermitian tridiagonal block [l..m+1].
     * Literal extraction from steqr_vectors_tridiagonal (no changes).
     */
    static readonly numeric_qr_sweep_tridiagonal_hermitian: (D: ComplexType[], E: ComplexType[], l: number, m: number, Z?: ComplexType[][]) => void;
    static readonly numeric_qr_bulge_chasing_hermitian_1: (D: ComplexType[], E: ComplexType[], maxIts?: number) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    static readonly numeric_qr_bulge_chasing_hermitian: (D: ComplexType[], E: ComplexType[], maxSweeps?: number) => ComplexType[][];
    static readonly numeric_steqr_tridiagonal_bulge: (D: ComplexType[], E: ComplexType[], maxIts?: number) => {
        D: ComplexType[];
        V: ComplexType[][];
    };
    static readonly numeric_qr_bulge_chasing_hermitian_refined: (D: ComplexType[], E: ComplexType[], maxSweeps?: number, tol?: number) => ComplexType[][];
    static readonly numeric_qr_hermitian_tridiagonal: (D: ComplexType[], E: ComplexType[], maxSweeps?: number, tol?: number) => ComplexType[][];
    static readonly qr_step_tridiagonal_hermitian: (D: ComplexType[], E: ComplexType[], Z: ComplexType[][]) => void;
    static readonly qr_hermitian_tridiagonal: (D: ComplexType[], E: ComplexType[], tol: number, maxSweeps: number) => ComplexType[][];
    static readonly numeric_tridiagonal_hermitian_bulge_chasing: (D: ComplexType[], // diagonal da matriz
    E: ComplexType[], // subdiagonal (size n-1)
    maxSweeps?: number) => {
        D: ComplexType[];
        E: ComplexType[];
        Z: MultiArray;
    };
    static readonly zsteqr: (D: ComplexType[], E: ComplexType[], Z?: ComplexType[][], tol?: NumLikeType, maxIts?: NumLikeType) => {
        D: ComplexType[];
        Z?: ComplexType[][];
    };
}
export type { ElementType };
export { LAPACKunused };
declare const _default: {
    LAPACKunused: typeof LAPACKunused;
};
export default _default;