# acb_mat.h – matrices over the complex numbers

An `acb_mat_t` represents a dense matrix over the complex numbers, implemented as an array of entries of type `acb_struct`. The dimension (number of rows and columns) of a matrix is fixed at initialization, and the user must ensure that inputs and outputs to an operation have compatible dimensions. The number of rows or columns in a matrix can be zero.

Note

Methods prefixed with acb_mat_approx treat all input entries as floating-point numbers (ignoring the radii of the balls) and compute floating-point output (balls with zero radius) representing approximate solutions without error bounds. All other methods compute rigorous error bounds. The approx methods are typically useful for computing initial values or preconditioners for rigorous solvers. Some users may also find approx methods useful for doing ordinary numerical linear algebra in applications where error bounds are not needed.

## Types, macros and constants

type acb_mat_struct

type acb_mat_t

  Contains a pointer to a flat array of the entries (entries), an array of pointers to the start of each row (rows), and the number of rows (r) and columns (c).

  An acb_mat_t is defined as an array of length one of type acb_mat_struct, permitting an acb_mat_t to be passed by reference.

acb_mat_entry(mat, i, j)

  Macro giving a pointer to the entry at row i and column j.

acb_mat_nrows(mat)

  Returns the number of rows of the matrix.

acb_mat_ncols(mat)

  Returns the number of columns of the matrix.

## Memory management

void acb_mat_init(acb_mat_t mat, slong r, slong c)

  Initializes the matrix, setting it to the zero matrix with r rows and c columns.

void acb_mat_clear(acb_mat_t mat)

  Clears the matrix, deallocating all entries.

slong acb_mat_allocated_bytes(const acb_mat_t x)

  Returns the total number of bytes heap-allocated internally by this object. The count excludes the size of the structure itself. Add `sizeof(acb_mat_struct)` to get the size of the object as a whole.

void acb_mat_window_init(acb_mat_t window, const acb_mat_t mat, slong r1, slong c1, slong r2, slong c2)

  Initializes window to a window matrix into the submatrix of mat starting at the corner at row r1 and column c1 (inclusive) and ending at row r2 and column c2 (exclusive).

void acb_mat_window_clear(acb_mat_t window)

  Frees the window matrix.

## Conversions

void acb_mat_set(acb_mat_t dest, const acb_mat_t src)

void acb_mat_set_fmpz_mat(acb_mat_t dest, const fmpz_mat_t src)

void acb_mat_set_round_fmpz_mat(acb_mat_t dest, const fmpz_mat_t src, slong prec)

void acb_mat_set_fmpq_mat(acb_mat_t dest, const fmpq_mat_t src, slong prec)

void acb_mat_set_arb_mat(acb_mat_t dest, const arb_mat_t src)

void acb_mat_set_round_arb_mat(acb_mat_t dest, const arb_mat_t src, slong prec)

  Sets dest to src. The operands must have identical dimensions.

void acb_mat_get_real(arb_mat_t re, const acb_mat_t mat)

void acb_mat_get_imag(arb_mat_t im, const acb_mat_t mat)

  Sets re or im to the real or imaginary part of mat, respectively. The operands must have identical dimensions.

void acb_mat_set_real_imag(acb_mat_t mat, const arb_mat_t re, const arb_mat_t im)

  Sets mat to the complex matrix with real and imaginary parts re, im. The operands must have identical dimensions.

## Random generation

void acb_mat_randtest(acb_mat_t mat, flint_rand_t state, slong prec, slong mag_bits)

  Sets mat to a random matrix with up to prec bits of precision and with exponents of width up to mag_bits.

void acb_mat_randtest_eig(acb_mat_t mat, flint_rand_t state, acb_srcptr E, slong prec)

  Sets mat to a random matrix with the prescribed eigenvalues supplied as the vector E. The output matrix is required to be square. We generate a random unitary matrix via a matrix exponential, and then evaluate an inverse Schur decomposition.

## Input and output

void acb_mat_printd(const acb_mat_t mat, slong digits)

  Prints each entry in the matrix with the specified number of decimal digits.

void acb_mat_fprintd(FILE *file, const acb_mat_t mat, slong digits)

  Prints each entry in the matrix with the specified number of decimal digits to the stream file.

## Comparisons

Predicate methods return 1 if the property certainly holds and 0 otherwise.

int acb_mat_equal(const acb_mat_t mat1, const acb_mat_t mat2)

  Returns whether the matrices have the same dimensions and identical intervals as entries.

int acb_mat_overlaps(const acb_mat_t mat1, const acb_mat_t mat2)

  Returns whether the matrices have the same dimensions and each entry in mat1 overlaps with the corresponding entry in mat2.

int acb_mat_contains(const acb_mat_t mat1, const acb_mat_t mat2)

int acb_mat_contains_fmpz_mat(const acb_mat_t mat1, const fmpz_mat_t mat2)

int acb_mat_contains_fmpq_mat(const acb_mat_t mat1, const fmpq_mat_t mat2)

  Returns whether the matrices have the same dimensions and each entry in mat2 is contained in the corresponding entry in mat1.

int acb_mat_eq(const acb_mat_t mat1, const acb_mat_t mat2)

  Returns whether mat1 and mat2 certainly represent the same matrix.

int acb_mat_ne(const acb_mat_t mat1, const acb_mat_t mat2)

  Returns whether mat1 and mat2 certainly do not represent the same matrix.

int acb_mat_is_real(const acb_mat_t mat)

  Returns whether all entries in mat have zero imaginary part.

int acb_mat_is_empty(const acb_mat_t mat)

  Returns whether the number of rows or the number of columns in mat is zero.

int acb_mat_is_square(const acb_mat_t mat)

  Returns whether the number of rows is equal to the number of columns in mat.

int acb_mat_is_exact(const acb_mat_t mat)

  Returns whether all entries in mat have zero radius.

int acb_mat_is_zero(const acb_mat_t mat)

  Returns whether all entries in mat are exactly zero.

int acb_mat_is_finite(const acb_mat_t mat)

  Returns whether all entries in mat are finite.

int acb_mat_is_triu(const acb_mat_t mat)

  Returns whether mat is upper triangular; that is, all entries below the main diagonal are exactly zero.

int acb_mat_is_tril(const acb_mat_t mat)

  Returns whether mat is lower triangular; that is, all entries above the main diagonal are exactly zero.

int acb_mat_is_diag(const acb_mat_t mat)

  Returns whether mat is a diagonal matrix; that is, all entries off the main diagonal are exactly zero.

## Special matrices

void acb_mat_zero(acb_mat_t mat)

  Sets all entries in mat to zero.

void acb_mat_one(acb_mat_t mat)

  Sets the entries on the main diagonal to ones, and all other entries to zero.

void acb_mat_ones(acb_mat_t mat)

  Sets all entries in the matrix to ones.

void acb_mat_onei(acb_mat_t mat)

  Sets the entries of the main diagonal to \(i = \sqrt{-1}\) and all other entries to zero.

void acb_mat_indeterminate(acb_mat_t mat)

  Sets all entries in the matrix to indeterminate (NaN).

void acb_mat_dft(acb_mat_t mat, int type, slong prec)

  Sets mat to the DFT (discrete Fourier transform) matrix of order n where n is the smallest dimension of mat (if mat is not square, the matrix is extended periodically along the larger dimension). Here, we use the normalized DFT matrix

  \[A_{j,k} = \frac{\omega^{jk}}{\sqrt{n}}, \quad \omega = e^{-2\pi i/n}.\]

  The type parameter is currently ignored and should be set to 0. In the future, it might be used to select a different convention.

## Transpose

void acb_mat_transpose(acb_mat_t dest, const acb_mat_t src)

  Sets dest to the exact transpose src. The operands must have compatible dimensions. Aliasing is allowed for square matrices.

void acb_mat_conjugate_transpose(acb_mat_t dest, const acb_mat_t src)

  Sets dest to the conjugate transpose of src. The operands must have compatible dimensions. Aliasing is allowed for square matrices.

void acb_mat_conjugate(acb_mat_t dest, const acb_mat_t src)

  Sets dest to the elementwise complex conjugate of src.

## Norms

void acb_mat_bound_inf_norm(mag_t b, const acb_mat_t A)

  Sets b to an upper bound for the infinity norm (i.e. the largest absolute value row sum) of A.

void acb_mat_frobenius_norm(arb_t res, const acb_mat_t A, slong prec)

  Sets res to the Frobenius norm (i.e. the square root of the sum of squares of entries) of A.

void acb_mat_bound_frobenius_norm(mag_t res, const acb_mat_t A)

  Sets res to an upper bound for the Frobenius norm of A.

## Arithmetic

void acb_mat_neg(acb_mat_t dest, const acb_mat_t src)

  Sets dest to the exact negation of src. The operands must have the same dimensions.

void acb_mat_add(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

  Sets res to the sum of mat1 and mat2. The operands must have the same dimensions.

void acb_mat_sub(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

  Sets res to the difference of mat1 and mat2. The operands must have the same dimensions.

void acb_mat_mul_classical(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

void acb_mat_mul_threaded(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

void acb_mat_mul_reorder(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

void acb_mat_mul(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

  Sets res to the matrix product of mat1 and mat2. The operands must have compatible dimensions for matrix multiplication.

  The classical version performs matrix multiplication in the trivial way.

  The threaded version performs classical multiplication but splits the computation over the number of threads returned by flint_get_num_threads().

  The reorder version reorders the data and performs one to four real matrix multiplications via `arb_mat_mul()`.

  The default version chooses an algorithm automatically.

void acb_mat_mul_entrywise(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

  Sets res to the entrywise product of mat1 and mat2. The operands must have the same dimensions.

void acb_mat_sqr_classical(acb_mat_t res, const acb_mat_t mat, slong prec)

void acb_mat_sqr(acb_mat_t res, const acb_mat_t mat, slong prec)

  Sets res to the matrix square of mat. The operands must both be square with the same dimensions.

void acb_mat_pow_ui(acb_mat_t res, const acb_mat_t mat, ulong exp, slong prec)

  Sets res to mat raised to the power exp. Requires that mat is a square matrix.

void acb_mat_approx_mul(acb_mat_t res, const acb_mat_t mat1, const acb_mat_t mat2, slong prec)

  Approximate matrix multiplication. The input radii are ignored and the output matrix is set to an approximate floating-point result. For performance reasons, the radii in the output matrix will not necessarily be written (zeroed), but will remain zero if they are already zeroed in res before calling this function.

## Scalar arithmetic

void acb_mat_scalar_mul_2exp_si(acb_mat_t B, const acb_mat_t A, slong c)

  Sets B to A multiplied by \(2^c\).

void acb_mat_scalar_addmul_si(acb_mat_t B, const acb_mat_t A, slong c, slong prec)

void acb_mat_scalar_addmul_fmpz(acb_mat_t B, const acb_mat_t A, const fmpz_t c, slong prec)

void acb_mat_scalar_addmul_arb(acb_mat_t B, const acb_mat_t A, const arb_t c, slong prec)

void acb_mat_scalar_addmul_acb(acb_mat_t B, const acb_mat_t A, const acb_t c, slong prec)

  Sets B to \(B + A \times c\).

void acb_mat_scalar_mul_si(acb_mat_t B, const acb_mat_t A, slong c, slong prec)

void acb_mat_scalar_mul_fmpz(acb_mat_t B, const acb_mat_t A, const fmpz_t c, slong prec)

void acb_mat_scalar_mul_arb(acb_mat_t B, const acb_mat_t A, const arb_t c, slong prec)

void acb_mat_scalar_mul_acb(acb_mat_t B, const acb_mat_t A, const acb_t c, slong prec)

  Sets B to \(A \times c\).

void acb_mat_scalar_div_si(acb_mat_t B, const acb_mat_t A, slong c, slong prec)

void acb_mat_scalar_div_fmpz(acb_mat_t B, const acb_mat_t A, const fmpz_t c, slong prec)

void acb_mat_scalar_div_arb(acb_mat_t B, const acb_mat_t A, const arb_t c, slong prec)

void acb_mat_scalar_div_acb(acb_mat_t B, const acb_mat_t A, const acb_t c, slong prec)

  Sets B to \(A / c\).

## Vector arithmetic

void _acb_mat_vector_mul_row(acb_ptr res, acb_srcptr v, const acb_mat_t A, slong prec)

void _acb_mat_vector_mul_col(acb_ptr res, const acb_mat_t A, acb_srcptr v, slong prec)

void acb_mat_vector_mul_row(acb_ptr res, acb_srcptr v, const acb_mat_t A, slong prec)

void acb_mat_vector_mul_col(acb_ptr res, const acb_mat_t A, acb_srcptr v, slong prec)

  Sets res to the product \(vA\), (resp. \(Av\)), where res and v are seen as row (resp. column) vectors. The lengths of the vectors must match the dimensions of A.

  The underscore methods do not allow aliasing between res and v.

## Gaussian elimination and solving

int acb_mat_lu_classical(slong *perm, acb_mat_t LU, const acb_mat_t A, slong prec)

int acb_mat_lu_recursive(slong *perm, acb_mat_t LU, const acb_mat_t A, slong prec)

int acb_mat_lu(slong *perm, acb_mat_t LU, const acb_mat_t A, slong prec)

  Given an \(n \times n\) matrix \(A\), computes an LU decomposition \(PLU = A\) using Gaussian elimination with partial pivoting. The input and output matrices can be the same, performing the decomposition in-place.

  Entry \(i\) in the permutation vector perm is set to the row index in the input matrix corresponding to row \(i\) in the output matrix.

  The algorithm succeeds and returns nonzero if it can find \(n\) invertible (i.e. not containing zero) pivot entries. This guarantees that the matrix is invertible.

  The algorithm fails and returns zero, leaving the entries in \(P\) and \(LU\) undefined, if it cannot find \(n\) invertible pivot elements. In this case, either the matrix is singular, the input matrix was computed to insufficient precision, or the LU decomposition was attempted at insufficient precision.

  The classical version uses Gaussian elimination directly while the recursive version performs the computation in a block recursive way to benefit from fast matrix multiplication. The default version chooses an algorithm automatically.

void acb_mat_solve_tril_classical(acb_mat_t X, const acb_mat_t L, const acb_mat_t B, int unit, slong prec)

void acb_mat_solve_tril_recursive(acb_mat_t X, const acb_mat_t L, const acb_mat_t B, int unit, slong prec)

void acb_mat_solve_tril(acb_mat_t X, const acb_mat_t L, const acb_mat_t B, int unit, slong prec)

void acb_mat_solve_triu_classical(acb_mat_t X, const acb_mat_t U, const acb_mat_t B, int unit, slong prec)

void acb_mat_solve_triu_recursive(acb_mat_t X, const acb_mat_t U, const acb_mat_t B, int unit, slong prec)

void acb_mat_solve_triu(acb_mat_t X, const acb_mat_t U, const acb_mat_t B, int unit, slong prec)

  Solves the lower triangular system \(LX = B\) or the upper triangular system \(UX = B\), respectively. If unit is set, the main diagonal of L or U is taken to consist of all ones, and in that case the actual entries on the diagonal are not read at all and can contain other data.

  The classical versions perform the computations iteratively while the recursive versions perform the computations in a block recursive way to benefit from fast matrix multiplication. The default versions choose an algorithm automatically.

void acb_mat_solve_lu_precomp(acb_mat_t X, const slong *perm, const acb_mat_t LU, const acb_mat_t B, slong prec)

  Solves \(AX = B\) given the precomputed nonsingular LU decomposition \(A = PLU\). The matrices \(X\) and \(B\) are allowed to be aliased with each other, but \(X\) is not allowed to be aliased with \(LU\).

int acb_mat_solve(acb_mat_t X, const acb_mat_t A, const acb_mat_t B, slong prec)

int acb_mat_solve_lu(acb_mat_t X, const acb_mat_t A, const acb_mat_t B, slong prec)

int acb_mat_solve_precond(acb_mat_t X, const acb_mat_t A, const acb_mat_t B, slong prec)

  Solves \(AX = B\) where \(A\) is a nonsingular \(n \times n\) matrix and \(X\) and \(B\) are \(n \times m\) matrices.

  If \(m > 0\) and \(A\) cannot be inverted numerically (indicating either that \(A\) is singular or that the precision is insufficient), the values in the output matrix are left undefined and zero is returned. A nonzero return value guarantees that \(A\) is invertible and that the exact solution matrix is contained in the output.

  Three algorithms are provided:

  - The lu version performs LU decomposition directly in ball arithmetic. This is fast, but the bounds typically blow up exponentially with n, even if the system is well-conditioned. This algorithm is usually the best choice at very high precision.

  - The precond version computes an approximate inverse to precondition the system. This is usually several times slower than direct LU decomposition, but the bounds do not blow up with n if the system is well-conditioned. This algorithm is usually the best choice for large systems at low to moderate precision.

  - The default version selects between lu and precomp automatically.

  The automatic choice should be reasonable most of the time, but users may benefit from trying either lu or precond in specific applications. For example, the lu solver often performs better for ill-conditioned systems where use of very high precision is unavoidable.

int acb_mat_inv(acb_mat_t X, const acb_mat_t A, slong prec)

  Sets \(X = A^{-1}\) where \(A\) is a square matrix, computed by solving the system \(AX = I\).

  If \(A\) cannot be inverted numerically (indicating either that \(A\) is singular or that the precision is insufficient), the values in the output matrix are left undefined and zero is returned. A nonzero return value guarantees that the matrix is invertible and that the exact inverse is contained in the output.

void acb_mat_det_lu(acb_t det, const acb_mat_t A, slong prec)

void acb_mat_det_precond(acb_t det, const acb_mat_t A, slong prec)

void acb_mat_det(acb_t det, const acb_mat_t A, slong prec)

  Sets det to the determinant of the matrix A.

  The lu version uses Gaussian elimination with partial pivoting. If at some point an invertible pivot element cannot be found, the elimination is stopped and the magnitude of the determinant of the remaining submatrix is bounded using Hadamard’s inequality.

  The precond version computes an approximate LU factorization of A and multiplies by the inverse L and U martices as preconditioners to obtain a matrix close to the identity matrix [Rum2010]. An enclosure for this determinant is computed using Gershgorin circles. This is about four times slower than direct Gaussian elimination, but much more numerically stable.

  The default version automatically selects between the lu and precond versions and additionally handles small or triangular matrices by direct formulas.

void acb_mat_approx_solve_triu(acb_mat_t X, const acb_mat_t U, const acb_mat_t B, int unit, slong prec)

void acb_mat_approx_solve_tril(acb_mat_t X, const acb_mat_t L, const acb_mat_t B, int unit, slong prec)

int acb_mat_approx_lu(slong *P, acb_mat_t LU, const acb_mat_t A, slong prec)

void acb_mat_approx_solve_lu_precomp(acb_mat_t X, const slong *perm, const acb_mat_t A, const acb_mat_t B, slong prec)

int acb_mat_approx_solve(acb_mat_t X, const acb_mat_t A, const acb_mat_t B, slong prec)

int acb_mat_approx_inv(acb_mat_t X, const acb_mat_t A, slong prec)

  These methods perform approximate solving without any error control. The radii in the input matrices are ignored, the computations are done numerically with floating-point arithmetic (using ordinary Gaussian elimination and triangular solving, accelerated through the use of block recursive strategies for large matrices), and the output matrices are set to the approximate floating-point results with zeroed error bounds.

## Characteristic polynomial and companion matrix

void _acb_mat_charpoly(acb_ptr poly, const acb_mat_t mat, slong prec)

void acb_mat_charpoly(acb_poly_t poly, const acb_mat_t mat, slong prec)

  Sets poly to the characteristic polynomial of mat which must be a square matrix. If the matrix has n rows, the underscore method requires space for \(n + 1\) output coefficients. Employs a division-free algorithm using \(O(n^4)\) operations.

void _acb_mat_companion(acb_mat_t mat, acb_srcptr poly, slong prec)

void acb_mat_companion(acb_mat_t mat, const acb_poly_t poly, slong prec)

  Sets the n by n matrix mat to the companion matrix of the polynomial poly which must have degree n. The underscore method reads \(n + 1\) input coefficients.

## Special functions

void acb_mat_exp_taylor_sum(acb_mat_t S, const acb_mat_t A, slong N, slong prec)

  Sets S to the truncated exponential Taylor series \(S = \sum_{k=0}^{N-1} A^k / k!\). See `arb_mat_exp_taylor_sum()` for implementation notes.

void acb_mat_exp(acb_mat_t B, const acb_mat_t A, slong prec)

  Sets B to the exponential of the matrix A, defined by the Taylor series

  \[\exp(A) = \sum_{k=0}^{\infty} \frac{A^k}{k!}.\]

  The function is evaluated as \(\exp(A/2^r)^{2^r}\), where \(r\) is chosen to give rapid convergence of the Taylor series. Error bounds are computed as for `arb_mat_exp()`.

void acb_mat_trace(acb_t trace, const acb_mat_t mat, slong prec)

  Sets trace to the trace of the matrix, i.e. the sum of entries on the main diagonal of mat. The matrix is required to be square.

void _acb_mat_diag_prod(acb_t res, const acb_mat_t mat, slong a, slong b, slong prec)

void acb_mat_diag_prod(acb_t res, const acb_mat_t mat, slong prec)

  Sets res to the product of the entries on the main diagonal of mat. The underscore method computes the product of the entries between index a inclusive and b exclusive (the indices must be in range).

## Component and error operations

void acb_mat_get_mid(acb_mat_t B, const acb_mat_t A)

  Sets the entries of B to the exact midpoints of the entries of A.

void acb_mat_add_error_mag(acb_mat_t mat, const mag_t err)

  Adds err in-place to the radii of the entries of mat.

## Eigenvalues and eigenvectors

The functions in this section are experimental. There are classes of matrices where the algorithms fail to converge even as prec is increased, or for which the error bounds are much worse than necessary. In some cases, it can help to manually precondition the matrix A by applying a similarity transformation \(T^{-1} A T\).

- If A is badly scaled, take \(T\) to be a matrix such that the entries of \(T^{-1} A T\) are more uniform (this is known as balancing).

- Simply taking \(T\) to be a random invertible matrix can help if an algorithm fails to converge despite \(A\) being well-scaled. (This can be the case when dealing with multiple eigenvalues.)

int acb_mat_approx_eig_qr(acb_ptr E, acb_mat_t L, acb_mat_t R, const acb_mat_t A, const mag_t tol, slong maxiter, slong prec)

  Computes floating-point approximations of all the n eigenvalues (and optionally eigenvectors) of the given n by n matrix A. The approximations of the eigenvalues are written to the vector E, in no particular order. If L is not NULL, approximations of the corresponding left eigenvectors are written to the rows of L. If R is not NULL, approximations of the corresponding right eigenvectors are written to the columns of R.

  The parameters tol and maxiter can be used to control the target numerical error and the maximum number of iterations allowed before giving up. Passing NULL and 0 respectively results in default values being used.

  Uses the implicitly shifted QR algorithm with reduction to Hessenberg form. No guarantees are made about the accuracy of the output. A nonzero return value indicates that the QR iteration converged numerically, but this is only a heuristic termination test and does not imply any statement whatsoever about error bounds. The output may also be accurate even if this function returns zero.

void acb_mat_eig_global_enclosure(mag_t eps, const acb_mat_t A, acb_srcptr E, const acb_mat_t R, slong prec)

  Given an n by n matrix A, a length-n vector E containing approximations of the eigenvalues of A, and an n by n matrix R containing approximations of the corresponding right eigenvectors, computes a rigorous bound \(\varepsilon\) such that every eigenvalue \(\lambda\) of A satisfies \(|\lambda - \hat \lambda_k| \le \varepsilon\) for some \(\hat \lambda_k\) in E. In other words, the union of the balls \(B_k = \{z : |z - \hat \lambda_k| \le \varepsilon\}\) is guaranteed to be an enclosure of all eigenvalues of A.

  Note that there is no guarantee that each ball \(B_k\) can be identified with a single eigenvalue: it is possible that some balls contain several eigenvalues while other balls contain no eigenvalues. In other words, this method is not powerful enough to compute isolating balls for the individual eigenvalues (or even for clusters of eigenvalues other than the whole spectrum). Nevertheless, in practice the balls \(B_k\) will represent eigenvalues one-to-one with high probability if the given approximations are good.

  The output can be used to certify that all eigenvalues of A lie in some region of the complex plane (such as a specific half-plane, strip, disk, or annulus) without the need to certify the individual eigenvalues. The output is easily converted into lower or upper bounds for the absolute values or real or imaginary parts of the spectrum, and with high probability these bounds will be tight. Using `acb_add_error_mag()` and `acb_union()`, the output can also be converted to a single `acb_t` enclosing the whole spectrum of A in a rectangle, but note that to test whether a condition holds for all eigenvalues of A, it is typically better to iterate over the individual balls \(B_k\).

  This function implements the fast algorithm in Theorem 1 in [Miy2010] which extends the Bauer-Fike theorem. Approximations E and R can, for instance, be computed using `acb_mat_approx_eig_qr()`. No assumptions are made about the structure of A or the quality of the given approximations.

void acb_mat_eig_enclosure_rump(acb_t lambda, acb_mat_t J, acb_mat_t R, const acb_mat_t A, const acb_t lambda_approx, const acb_mat_t R_approx, slong prec)

  Given an n by n matrix A and an approximate eigenvalue-eigenvector pair lambda_approx and R_approx (where R_approx is an n by 1 matrix), computes an enclosure lambda guaranteed to contain at least one of the eigenvalues of A, along with an enclosure R for a corresponding right eigenvector.

  More generally, this function can handle clustered (or repeated) eigenvalues. If R_approx is an n by k matrix containing approximate eigenvectors for a presumed cluster of k eigenvalues near lambda_approx, this function computes an enclosure lambda guaranteed to contain at least k eigenvalues of A along with a matrix R guaranteed to contain a basis for the k-dimensional invariant subspace associated with these eigenvalues. Note that for multiple eigenvalues, determining the individual eigenvectors is an ill-posed problem; describing an enclosure of the invariant subspace is the best we can hope for.

  For \(k = 1\), it is guaranteed that \(AR - R \lambda\) contains the zero matrix. For \(k > 2\), this cannot generally be guaranteed (in particular, A might not diagonalizable). In this case, we can still compute an approximately diagonal k by k interval matrix \(J \approx \lambda I\) such that \(AR - RJ\) is guaranteed to contain the zero matrix. This matrix has the property that the Jordan canonical form of (any exact matrix contained in) A has a k by k submatrix equal to the Jordan canonical form of (some exact matrix contained in) J. The output J is optional (the user can pass NULL to omit it).

  The algorithm follows section 13.4 in [Rum2010], corresponding to the `verifyeig()` routine in INTLAB. The initial approximations can, for instance, be computed using `acb_mat_approx_eig_qr()`. No assumptions are made about the structure of A or the quality of the given approximations.

int acb_mat_eig_simple_rump(acb_ptr E, acb_mat_t L, acb_mat_t R, const acb_mat_t A, acb_srcptr E_approx, const acb_mat_t R_approx, slong prec)

int acb_mat_eig_simple_vdhoeven_mourrain(acb_ptr E, acb_mat_t L, acb_mat_t R, const acb_mat_t A, acb_srcptr E_approx, const acb_mat_t R_approx, slong prec)

int acb_mat_eig_simple(acb_ptr E, acb_mat_t L, acb_mat_t R, const acb_mat_t A, acb_srcptr E_approx, const acb_mat_t R_approx, slong prec)

  Computes all the eigenvalues (and optionally corresponding eigenvectors) of the given n by n matrix A.

  Attempts to prove that A has n simple (isolated) eigenvalues, returning 1 if successful and 0 otherwise. On success, isolating complex intervals for the eigenvalues are written to the vector E, in no particular order. If L is not NULL, enclosures of the corresponding left eigenvectors are written to the rows of L. If R is not NULL, enclosures of the corresponding right eigenvectors are written to the columns of R.

  The left eigenvectors are normalized so that \(L = R^{-1}\). This produces a diagonalization \(LAR = D\) where D is the diagonal matrix with the entries in E on the diagonal.

  The user supplies approximations E_approx and R_approx of the eigenvalues and the right eigenvectors. The initial approximations can, for instance, be computed using `acb_mat_approx_eig_qr()`. No assumptions are made about the structure of A or the quality of the given approximations.

  Two algorithms are implemented:

  - The rump version calls `acb_mat_eig_enclosure_rump()` repeatedly to certify eigenvalue-eigenvector pairs one by one. The iteration is stopped to return non-success if a new eigenvalue overlaps with previously computed one. Finally, L is computed by a matrix inversion. This has complexity \(O(n^4)\).

  - The vdhoeven_mourrain version uses the algorithm in [HM2017] to certify all eigenvalues and eigenvectors in one step. This has complexity \(O(n^3)\).

  The default version currently uses vdhoeven_mourrain.

  By design, these functions terminate instead of attempting to compute eigenvalue clusters if some eigenvalues cannot be isolated. To compute all eigenvalues of a matrix allowing for overlap, `acb_mat_eig_multiple_rump()` may be used as a fallback, or `acb_mat_eig_multiple()` may be used in the first place.

int acb_mat_eig_multiple_rump(acb_ptr E, const acb_mat_t A, acb_srcptr E_approx, const acb_mat_t R_approx, slong prec)

int acb_mat_eig_multiple(acb_ptr E, const acb_mat_t A, acb_srcptr E_approx, const acb_mat_t R_approx, slong prec)

  Computes all the eigenvalues of the given n by n matrix A. On success, the output vector E contains n complex intervals, each representing one eigenvalue of A with the correct multiplicities in case of overlap. The output intervals are either disjoint or identical, and identical intervals are guaranteed to be grouped consecutively. Each complete run of k identical intervals thus represents a cluster of exactly k eigenvalues which could not be separated from each other at the current precision, but which could be isolated from the other \(n - k\) eigenvalues of the matrix.

  The user supplies approximations E_approx and R_approx of the eigenvalues and the right eigenvectors. The initial approximations can, for instance, be computed using `acb_mat_approx_eig_qr()`. No assumptions are made about the structure of A or the quality of the given approximations.

  The rump algorithm groups approximate eigenvalues that are close and calls `acb_mat_eig_enclosure_rump()` repeatedly to validate each cluster. The complexity is \(O(m n^3)\) for m clusters.

  The default version, as currently implemented, first attempts to call `acb_mat_eig_simple_vdhoeven_mourrain()` hoping that the eigenvalues are actually simple. It then uses the rump algorithm as a fallback.