Namespaces
namespace	backends

namespace	cuda

namespace	detail

namespace	markov

namespace	mpi

namespace	pde

namespace	quantum

namespace	spectral

Classes
class	BandedMatrix
	Banded matrix with efficient storage for direct solvers. More...

struct	BandedSolverResult
	Result from banded solver. More...

class	BasicVector
	Dense vector with optional GPU storage, templated over scalar type T. More...

class	CellList2D

class	CellList3D

class	Circuit
	Chainable quantum circuit builder. More...

struct	ClusterResult

struct	ComplexTriDiag
	Constant-coefficient complex tridiagonal solver (precomputed LU). More...

struct	CrankNicolsonADI

struct	EigenResult
	Full eigendecomposition result: A = V * diag(values) * V^T. More...

class	FieldSolver

struct	GateView
	Compact gate description used by visualisation code. More...

struct	GivensRotation
	Givens plane rotation: G = [c, s; -s, c]. More...

class	Grid3D

struct	Histogram

struct	IntRange
	Lightweight read-only range over a contiguous int array (C++17-safe span). More...

struct	LanczosResult
	Result of a Lanczos eigensolver. More...

struct	LUResult
	Result of an LU factorization with partial pivoting (PA = LU) More...

class	MagneticSolver

class	Matrix
	Dense row-major matrix with optional GPU storage. More...

struct	ODEResult
	Result returned by general ODE integrators. More...

struct	PBCLattice2D
	4-neighbor periodic-boundary index arrays for an N×N lattice. More...

struct	PowerResult
	Result of a single-eigenvalue iteration. More...

struct	QRResult
	Result of a QR factorization: A = Q * R. More...

struct	Result
	Measurement outcome from Circuit::run() More...

struct	Rng
	Seeded pseudo-random number generator (Mersenne Twister). Pass a pointer to rng_* functions to draw samples. More...

struct	RootResult

struct	RunningStats

class	ScalarField3D

struct	SmallMatrix
	Constexpr fixed-size row-major matrix (stack-allocated). More...

struct	SmallVec
	Constexpr fixed-size dense vector (stack-allocated). More...

struct	SolverResult

class	SparseMatrix
	Sparse matrix in Compressed Sparse Row (CSR) format. More...

struct	SPHKernel
	Dimension-generic SPH smoothing kernels. Dim = 2 or 3. More...

struct	SVDResult
	Result of a Singular Value Decomposition: A = U * diag(S) * Vᵀ More...

struct	SymplecticResult
	Result returned by symplectic integrators. More...

struct	VectorField3D

class	VerletList2D

Typedefs
using	ScalarFn = std::function< real(real)>
	Real-valued scalar function f(x)

using	VectorFn = std::function< void(real, real , real )>
	Real-valued multivariate function f(x) where x is a scalar parameter and the function returns a vector of residuals – used in nonlinear systems.

using	real = double

using	idx = std::size_t

using	cplx = std::complex< real >

using	Vector = BasicVector< real >
	Real-valued dense vector with full backend dispatch (CPU + GPU)

using	CVector = BasicVector< cplx >
	Complex-valued dense vector (sequential; no GPU)

using	MatVecFn = std::function< void(const Vector &, Vector &)>
	Callable type for matrix-free matvec: computes y = A*x.

using	ODERhsFn = std::function< void(real t, const Vector &y, Vector &dydt)>
	Right-hand side callable: fills dydt = f(t, y) in-place.

using	StepCallback = std::function< void(real t, const Vector &y)>

using	AccelFn = std::function< void(const Vector &q, Vector &acc)>

using	SymplecticCallback = std::function< void(real t, const Vector &q, const Vector &v)>

Enumerations
enum class	Backend { seq , blocked , simd , blas , omp , gpu }
	Selects which backend handles a linalg operation. More...

Functions
real	trapz (ScalarFn f, real a, real b, idx n=100, Backend backend=Backend::seq)
	Trapezoidal rule with n panels.

real	simpson (ScalarFn f, real a, real b, idx n=100, Backend backend=Backend::seq)
	Simpson's 1/3 rule with n panels (n must be even)

real	gauss_legendre (ScalarFn f, real a, real b, idx p=5)
	Gauss-Legendre quadrature (exact for polynomials up to degree 2p-1)

real	adaptive_simpson (ScalarFn f, real a, real b, real tol=1e-8, idx max_depth=50)
	Adaptive Simpson quadrature.

real	romberg (ScalarFn f, real a, real b, real tol=1e-10, idx max_levels=12)
	Romberg integration (Richardson extrapolation on trapezoidal rule)

RootResult	bisection (ScalarFn f, real a, real b, real tol=1e-10, idx max_iter=1000)
	Bisection method.

RootResult	newton (ScalarFn f, ScalarFn df, real x0, real tol=1e-10, idx max_iter=1000)
	Newton-Raphson method.

RootResult	secant (ScalarFn f, real x0, real x1, real tol=1e-10, idx max_iter=1000)
	Secant method (Newton without derivative)

RootResult	brent (ScalarFn f, real a, real b, real tol=1e-10, idx max_iter=1000)
	Brent's method (bisection + secant + inverse quadratic interpolation)

void	matvec (const Matrix &A, const Vector &x, Vector &y, Backend b=default_backend)
	y = A * x

void	matmul (const Matrix &A, const Matrix &B, Matrix &C, Backend b=default_backend)
	C = A * B.

void	matadd (real alpha, const Matrix &A, real beta, const Matrix &B, Matrix &C, Backend b=default_backend)
	C = alphaA + betaB.

void	matmul_blocked (const Matrix &A, const Matrix &B, Matrix &C, idx block_size=64)
	C = A * B (cache-blocked)

void	matmul_register_blocked (const Matrix &A, const Matrix &B, Matrix &C, idx block_size=64, idx reg_size=4)
	C = A * B (register-blocked)

void	matmul_simd (const Matrix &A, const Matrix &B, Matrix &C, idx block_size=64)
	C = A * B (SIMD-accelerated)

void	matvec_simd (const Matrix &A, const Vector &x, Vector &y)
	y = A * x (SIMD-accelerated)

template<idx N>
constexpr SmallVec< N >	operator+ (SmallVec< N > a, const SmallVec< N > &b) noexcept

template<idx N>
constexpr SmallVec< N >	operator* (real s, SmallVec< N > v) noexcept

template<int N, typename T >
constexpr T	ipow (T x) noexcept
	Compute x^N at compile time via repeated squaring.

real	bessel_j (real nu, real x)
	J_nu(x) – Bessel function of the first kind.

real	bessel_y (real nu, real x)
	Y_nu(x) – Bessel function of the second kind (Neumann function)

real	bessel_i (real nu, real x)
	I_nu(x) – modified Bessel function of the first kind.

real	bessel_k (real nu, real x)
	K_nu(x) – modified Bessel function of the second kind.

real	sph_bessel_j (unsigned int n, real x)
	j_n(x) – spherical Bessel function of the first kind

real	sph_bessel_y (unsigned int n, real x)
	y_n(x) – spherical Neumann function (spherical Bessel of the second kind)

real	legendre (unsigned int n, real x)
	P_n(x) – Legendre polynomial of degree n.

real	assoc_legendre (unsigned int n, unsigned int m, real x)
	P_n^m(x) – associated Legendre polynomial.

real	sph_legendre (unsigned int l, unsigned int m, real theta)
	Y_l^m(theta) – spherical harmonic (real part, theta in radians)

real	hermite (unsigned int n, real x)
	H_n(x) – (physicists') Hermite polynomial.

real	laguerre (unsigned int n, real x)
	L_n(x) – Laguerre polynomial.

real	assoc_laguerre (unsigned int n, unsigned int m, real x)
	L_n^m(x) – associated Laguerre polynomial.

real	ellint_K (real k)
	K(k) – complete elliptic integral of the first kind.

real	ellint_E (real k)
	E(k) – complete elliptic integral of the second kind.

real	ellint_Pi (real n, real k)
	Pi(n, k) – complete elliptic integral of the third kind.

real	ellint_F (real k, real phi)
	F(k, phi) – incomplete elliptic integral of the first kind.

real	ellint_Ei (real k, real phi)
	E(k, phi) – incomplete elliptic integral of the second kind.

real	ellint_Pi_inc (real n, real k, real phi)
	Pi(n, k, phi) – incomplete elliptic integral of the third kind.

real	expint (real x)
	Ei(x) – exponential integral.

real	zeta (real x)
	zeta(x) – Riemann zeta function

real	beta (real a, real b)
	B(a, b) – beta function.

std::vector< real >	linspace (real start, real stop, idx n)
	Evenly spaced values from start to stop, inclusive. MATLAB/NumPy linspace.

std::vector< int >	int_range (int start, int n)
	Integer sequence [start, start+1, ..., start+n-1]. Wraps std::iota.

real	rng_uniform (Rng *r, real lo, real hi)
	Uniform real in [lo, hi).

real	rng_normal (Rng *r, real mean, real stddev)
	Normal (Gaussian) sample with given mean and standard deviation.

int	rng_int (Rng *r, int lo, int hi)
	Uniform integer in [lo, hi] (inclusive on both ends).

void	scale (Vector &v, real alpha, Backend b=default_backend)
	v *= alpha

void	add (const Vector &x, const Vector &y, Vector &z, Backend b=default_backend)
	z = x + y

void	axpy (real alpha, const Vector &x, Vector &y, Backend b=default_backend)
	y += alpha * x

real	dot (const Vector &x, const Vector &y, Backend b=default_backend)
	dot product

real	norm (const Vector &x, Backend b=default_backend)
	Euclidean norm.

void	scale (CVector &v, cplx alpha)
	v *= alpha

void	axpy (cplx alpha, const CVector &x, CVector &y)
	y += alpha * x

cplx	dot (const CVector &x, const CVector &y)
	Conjugate inner product <x, y> = Sigma conj(x_i) * y_i.

real	norm (const CVector &x)
	Euclidean norm sqrt(Sigma \|v_i\|^2)

BandedSolverResult	banded_lu (BandedMatrix &A, idx *ipiv)
	LU factorization of banded matrix with partial pivoting.

void	banded_lu_solve (const BandedMatrix &A, const idx *ipiv, Vector &b)
	Solve banded system using precomputed LU factorization.

void	banded_lu_solve_multi (const BandedMatrix &A, const idx ipiv, real B, idx nrhs)
	Solve multiple right-hand sides using LU factorization.

BandedSolverResult	banded_solve (const BandedMatrix &A, const Vector &b, Vector &x)
	Solve banded system Ax = b (convenience function)

void	banded_matvec (const BandedMatrix &A, const Vector &x, Vector &y, Backend backend=default_backend)
	Banded matrix-vector product y = A*x.

void	banded_gemv (real alpha, const BandedMatrix &A, const Vector &x, real beta, Vector &y, Backend backend=default_backend)
	Banded matrix-vector product y = alphaAx + beta*y.

real	banded_rcond (const BandedMatrix &A, const idx *ipiv, real anorm)
	Estimate reciprocal condition number of banded matrix.

real	banded_norm1 (const BandedMatrix &A)
	Compute 1-norm of banded matrix.

EigenResult	eig_sym (const Matrix &A, real tol=1e-12, idx max_sweeps=100, Backend backend=Backend::seq)
	Full eigendecomposition of a real symmetric matrix.

LanczosResult	lanczos (MatVecFn matvec, idx n, idx k, real tol=1e-10, idx max_steps=0, Backend backend=Backend::seq)
	Lanczos eigensolver for large sparse symmetric matrices.

PowerResult	power_iteration (const Matrix &A, real tol=1e-10, idx max_iter=1000, Backend backend=default_backend)
	Power iteration – finds the eigenvalue largest in absolute value.

PowerResult	inverse_iteration (const Matrix &A, real sigma, real tol=1e-10, idx max_iter=1000, Backend backend=default_backend)
	Inverse iteration – finds the eigenvalue closest to a shift sigma.

PowerResult	rayleigh_iteration (const Matrix &A, const Vector &x0, real tol=1e-10, idx max_iter=50, Backend backend=default_backend)
	Rayleigh quotient iteration – cubically convergent.

Vector	expv (real t, const MatVecFn &matvec, idx n, const Vector &v, int m_max=30, real tol=1e-8)
	Compute exp(tA)v via Krylov-Padé approximation (matrix-free)

Vector	expv (real t, const SparseMatrix &A, const Vector &v, int m_max=30, real tol=1e-8)
	Compute exp(tA)v via Krylov-Padé approximation (sparse matrix)

LUResult	lu (const Matrix &A)
	LU factorization of a square matrix A with partial pivoting.

void	lu_solve (const LUResult &f, const Vector &b, Vector &x)
	Solve A*x = b using a precomputed LU factorization.

void	lu_solve (const LUResult &f, const Matrix &B, Matrix &X)
	Solve A*X = B for multiple right-hand sides.

real	lu_det (const LUResult &f)
	Determinant of A from its LU factorization.

Matrix	lu_inv (const LUResult &f)
	Inverse of A from its LU factorization.

QRResult	qr (const Matrix &A)
	QR factorization of an mxn matrix A (m >= n) via Householder reflections.

void	qr_solve (const QRResult &f, const Vector &b, Vector &x)
	Solve the least-squares problem min \|\|A*x - b\|\|_2.

void	thomas (const Vector &a, const Vector &b, const Vector &c, const Vector &d, Vector &x, Backend backend=Backend::seq)
	Thomas algorithm (LU for tridiagonal systems), O(n).

SolverResult	cg (const Matrix &A, const Vector &b, Vector &x, real tol=1e-10, idx max_iter=1000, Backend backend=default_backend)
	Conjugate gradient solver for Ax = b.

SolverResult	cg_matfree (MatVecFn matvec, const Vector &b, Vector &x, real tol=1e-6, idx max_iter=1000)
	Matrix-free conjugate gradient for Ax = b where A is SPD.

SolverResult	gauss_seidel (const Matrix &A, const Vector &b, Vector &x, real tol=1e-10, idx max_iter=1000, Backend backend=default_backend)
	Gauss-Seidel iterative solver for Ax = b.

SolverResult	jacobi (const Matrix &A, const Vector &b, Vector &x, real tol=1e-10, idx max_iter=1000, Backend backend=default_backend)
	Jacobi iterative solver for Ax = b.

SolverResult	gmres (MatVecFn matvec, idx n, const Vector &b, Vector &x, real tol=1e-6, idx max_iter=1000, idx restart=30)
	Restarted GMRES(restart) – matrix-free interface.

SolverResult	gmres (const SparseMatrix &A, const Vector &b, Vector &x, real tol=1e-6, idx max_iter=1000, idx restart=30)
	Restarted GMRES with a sparse (CSR) matrix.

SolverResult	gmres (const Matrix &A, const Vector &b, Vector &x, real tol=1e-6, idx max_iter=1000, idx restart=30, Backend backend=default_backend)
	Restarted GMRES with a dense matrix.

void	sparse_matvec (const SparseMatrix &A, const Vector &x, Vector &y)
	y = A * x

SVDResult	svd (const Matrix &A, Backend backend=Backend::seq, real tol=1e-12, idx max_sweeps=100)
	Full SVD of an mxn matrix via one-sided Jacobi.

SVDResult	svd_truncated (const Matrix &A, idx k, Backend backend=default_backend, idx oversampling=10, Rng *rng=nullptr)
	Randomized truncated SVD – top-k singular triplets.

ODEResult	ode_euler (ODERhsFn f, Vector y0, real t0, real t1, real h, StepCallback on_step=nullptr)

ODEResult	ode_rk4 (ODERhsFn f, Vector y0, real t0, real t1, real h, StepCallback on_step=nullptr)
	Classic 4th-order Runge-Kutta (fixed step).

ODEResult	ode_rk45 (ODERhsFn f, Vector y0, real t0, real t1, real rtol=1e-6, real atol=1e-9, real h0=1e-3, idx max_steps=1000000, StepCallback on_step=nullptr)

SymplecticResult	ode_verlet (AccelFn accel, Vector q0, Vector v0, real t0, real t1, real h, SymplecticCallback on_step=nullptr)

SymplecticResult	ode_yoshida4 (AccelFn accel, Vector q0, Vector v0, real t0, real t1, real h, SymplecticCallback on_step=nullptr)

template<typename T >
void	laplacian_stencil_2d (const BasicVector< T > &x, BasicVector< T > &y, int N)

template<typename T >
void	laplacian_stencil_2d_periodic (const BasicVector< T > &x, BasicVector< T > &y, int N)

template<typename T , typename F >
void	col_fiber_sweep (BasicVector< T > &data, int N, F &&f)

template<typename T , typename F >
void	row_fiber_sweep (BasicVector< T > &data, int N, F &&f)

void	neg_laplacian_3d (const Vector &x, Vector &y, int nx, int ny, int nz, double inv_dx2)

void	gradient_3d (const Grid3D &phi, Grid3D &gx, Grid3D &gy, Grid3D &gz)

void	divergence_3d (const Grid3D &fx, const Grid3D &fy, const Grid3D &fz, Grid3D &out)

void	curl_3d (const Grid3D &ax, const Grid3D &ay, const Grid3D &az, Grid3D &bx, Grid3D &by, Grid3D &bz)

template<typename InCluster , typename Neighbors >
ClusterResult	connected_components (int n_sites, InCluster &&in_cluster, Neighbors &&neighbors)

real	autocorr_time (const real *data, idx n, real c=6.0)

EigenResult	eig_sym (const Matrix &A_in, real tol, idx max_sweeps)

Circuit	bell_pair (int q0=0, int q1=1)
	Bell state \|Phi+> = (\|00> + \|11>)/sqrt2 on qubits q0, q1.

Circuit	ghz_state (int n_qubits)
	n-qubit GHZ state (\|00...0> + \|11...1>)/sqrt2

Circuit	qft_circuit (int n_qubits)
	n-qubit Quantum Fourier Transform circuit Applied to \|0...0> unless you prepend state-preparation gates.

void	print_state (const quantum::Statevector &sv, int n_qubits, real threshold=1e-6)
	Pretty-print a statevector, hiding amplitudes below threshold.

Variables
constexpr Backend	seq = Backend::seq

constexpr Backend	blocked = Backend::blocked

constexpr Backend	simd = Backend::simd

constexpr Backend	blas = Backend::blas

constexpr Backend	omp = Backend::omp

constexpr Backend	gpu = Backend::gpu

constexpr bool	has_blas
	True when a BLAS/cblas library was found at configure time.

constexpr bool	has_omp
	True when OpenMP was found at configure time.

constexpr Backend	default_backend

constexpr Backend	best_backend
	Best backend for memory-bound vector ops: blas > omp > blocked.

constexpr real	pi = 3.14159265358979323846

constexpr real	e = 2.71828182845904523536

constexpr real	phi = 1.61803398874989484820
	Golden ratio.

constexpr real	sqrt2 = 1.41421356237309504880

constexpr real	sqrt3 = 1.73205080756887729353

constexpr real	ln2 = 0.69314718055994530942

constexpr real	inv_pi = 0.31830988618379067154
	1/pi

constexpr real	two_pi = 6.28318530717958647692
	2pi

constexpr real	half_pi = 1.57079632679489661923
	pi/2

Typedef Documentation

◆ AccelFn

using num::AccelFn = typedef std::function<void(const Vector& q, Vector& acc)>

Acceleration function for symplectic integrators. Fills acc = −∇V(q) / m (generalised force per unit mass) from positions q.

Definition at line 37 of file ode.hpp.

◆ cplx

using num::cplx = typedef std::complex<real>

Definition at line 12 of file types.hpp.

◆ CVector

using num::CVector = typedef BasicVector<cplx>

Complex-valued dense vector (sequential; no GPU)

Definition at line 125 of file vector.hpp.

◆ idx

using num::idx = typedef std::size_t

Definition at line 11 of file types.hpp.

◆ MatVecFn

using num::MatVecFn = typedef std::function<void(const Vector&, Vector&)>

Callable type for matrix-free matvec: computes y = A*x.

Definition at line 13 of file cg.hpp.

◆ ODERhsFn

using num::ODERhsFn = typedef std::function<void(real t, const Vector& y, Vector& dydt)>

Right-hand side callable: fills dydt = f(t, y) in-place.

Definition at line 28 of file ode.hpp.

◆ real

using num::real = typedef double

Definition at line 10 of file types.hpp.

◆ ScalarFn

using num::ScalarFn = typedef std::function<real(real)>

Real-valued scalar function f(x)

Definition at line 11 of file types.hpp.

◆ StepCallback

using num::StepCallback = typedef std::function<void(real t, const Vector& y)>

Called after each accepted step. Use to record trajectories or detect events.

Parameters

t	Time at end of step
y	State at end of step

Definition at line 33 of file ode.hpp.

◆ SymplecticCallback

using num::SymplecticCallback = typedef std::function<void(real t, const Vector& q, const Vector& v)>

Per-step callback for symplectic integrators.

Parameters

t	Time at end of step
q	Generalised positions
v	Generalised velocities

Definition at line 43 of file ode.hpp.

◆ Vector

using num::Vector = typedef BasicVector<real>

Real-valued dense vector with full backend dispatch (CPU + GPU)

Definition at line 122 of file vector.hpp.

◆ VectorFn

using num::VectorFn = typedef std::function<void(real, real*, real*)>

Real-valued multivariate function f(x) where x is a scalar parameter and the function returns a vector of residuals – used in nonlinear systems.

Definition at line 15 of file types.hpp.

Enumeration Type Documentation

◆ Backend

enum class num::Backend

strong

Selects which backend handles a linalg operation.

Enumerator
seq	Naive textbook loops – always available.
blocked	Cache-blocked; compiler auto-vectorizes inner loops.
simd	Hand-written SIMD intrinsics (AVX2 or NEON)
blas	cblas/LAPACKE – OpenBLAS, MKL, Apple Accelerate
omp	OpenMP parallel blocked loops.
gpu	CUDA – custom kernels or cuBLAS.

Definition at line 19 of file policy.hpp.

Function Documentation

◆ adaptive_simpson()

real num::adaptive_simpson	(	ScalarFn	f,
		real	a,
		real	b,
		real	tol = `1e-8`,
		idx	max_depth = `50`
	)

Adaptive Simpson quadrature.

Parameters

f	Integrand
a	Lower bound
b	Upper bound
tol	Absolute error tolerance
max_depth	Maximum recursion depth

Definition at line 92 of file quadrature.cpp.

References ipow().

◆ add()

void num::add	(	const Vector &	x,
		const Vector &	y,
		Vector &	z,
		Backend	b = `default_backend`
	)

z = x + y

Definition at line 38 of file vector.cpp.

References num::cuda::add(), num::backends::seq::add(), gpu, num::BasicVector< T >::gpu_data(), ipow(), and num::BasicVector< T >::size().

◆ assoc_laguerre()

real num::assoc_laguerre	(	unsigned int	n,
		unsigned int	m,
		real	x
	)

inline

L_n^m(x) – associated Laguerre polynomial.

Definition at line 156 of file math.hpp.

References ipow().

◆ assoc_legendre()

real num::assoc_legendre	(	unsigned int	n,
		unsigned int	m,
		real	x
	)

inline

P_n^m(x) – associated Legendre polynomial.

Definition at line 120 of file math.hpp.

References ipow().

◆ autocorr_time()

real num::autocorr_time	(	const real *	data,
		idx	n,
		real	c = `6.0`
	)

Integrated autocorrelation time tau_int from a stored time series.

Uses the automatic windowing criterion (Madras & Sokal 1988): accumulate C(t)/C(0) until W > c*tau_int (c = 6 default). Returns tau_int >= 0.5; returns 0.5 on failure.

Parameters

data	Pointer to time series of length n
n	Length of the series
c	Window parameter (default 6)

Definition at line 6 of file stats.cpp.

References e, and ipow().

◆ axpy() [1/2]

void num::axpy	(	cplx	alpha,
		const CVector &	x,
		CVector &	y
	)

y += alpha * x

Definition at line 87 of file vector.cpp.

References ipow(), and num::BasicVector< T >::size().

◆ axpy() [2/2]

void num::axpy	(	real	alpha,
		const Vector &	x,
		Vector &	y,
		Backend	b = `default_backend`
	)

y += alpha * x

Definition at line 46 of file vector.cpp.

References num::backends::blas::axpy(), num::backends::gpu::axpy(), num::backends::omp::axpy(), num::backends::seq::axpy(), blas, blocked, gpu, ipow(), omp, seq, and simd.

Referenced by cg(), cg_matfree(), num::pde::diffusion_step_2d(), num::pde::diffusion_step_2d_dirichlet(), and expv().

◆ banded_gemv()

void num::banded_gemv	(	real	alpha,
		const BandedMatrix &	A,
		const Vector &	x,
		real	beta,
		Vector &	y,
		Backend	backend = `default_backend`
	)

Banded matrix-vector product y = alpha*A*x + beta*y.

Generalized matrix-vector multiply with scaling factors.

Parameters

alpha	Scalar multiplier for A*x
A	Banded matrix
x	Input vector
beta	Scalar multiplier for y
y	Input/output vector
backend	Execution backend (Backend::gpu uses CUDA path when available)

Definition at line 268 of file banded.cpp.

References beta(), num::BasicVector< T >::data(), gpu, ipow(), and num::BasicVector< T >::size().

Referenced by banded_matvec().

◆ banded_lu()

BandedSolverResult num::banded_lu	(	BandedMatrix &	A,
		idx *	ipiv
	)

LU factorization of banded matrix with partial pivoting.

Computes A = P * L * U where:

P is a permutation matrix
L is lower triangular with unit diagonal
U is upper triangular

The factorization is performed in-place. After factorization:

U is stored in rows 0 to kl+ku (including fill-in)
L multipliers are stored in rows kl+ku+1 to 2*kl+ku
ipiv contains the pivot indices

Parameters

A	Banded matrix (modified in place with LU factors)
ipiv	Pivot indices (size n, allocated by caller)

Returns: Result indicating success or location of zero pivot

Complexity: O(n * kl * (kl + ku)) operations

Definition at line 111 of file banded.cpp.

References ipow().

Referenced by banded_solve().

◆ banded_lu_solve()

void num::banded_lu_solve	(	const BandedMatrix &	A,
		const idx *	ipiv,
		Vector &	b
	)

Solve banded system using precomputed LU factorization.

Solves A*x = b where A has been factored by banded_lu(). The solution overwrites the right-hand side vector.

Parameters

A	LU-factored banded matrix from banded_lu()
ipiv	Pivot indices from banded_lu()
b	Right-hand side (overwritten with solution)

Complexity: O(n * (kl + ku)) operations

Definition at line 175 of file banded.cpp.

References ipow().

Referenced by banded_rcond(), and banded_solve().

◆ banded_lu_solve_multi()

void num::banded_lu_solve_multi	(	const BandedMatrix &	A,
		const idx *	ipiv,
		real *	B,
		idx	nrhs
	)

Solve multiple right-hand sides using LU factorization.

Solves A*X = B where B has nrhs columns stored contiguously.

Parameters

A	LU-factored banded matrix
ipiv	Pivot indices
B	Right-hand sides (n x nrhs, column-major, overwritten with solutions)
nrhs	Number of right-hand sides

Definition at line 213 of file banded.cpp.

References ipow().

◆ banded_matvec()

void num::banded_matvec	(	const BandedMatrix &	A,
		const Vector &	x,
		Vector &	y,
		Backend	backend = `default_backend`
	)

Banded matrix-vector product y = A*x.

Optimized for cache efficiency with loop ordering that accesses the band storage in column-major order.

Parameters

A	Banded matrix
x	Input vector
y	Output vector (overwritten)
backend	Execution backend (Backend::gpu uses CUDA path when available)

Definition at line 264 of file banded.cpp.

References banded_gemv(), and ipow().

◆ banded_norm1()

real num::banded_norm1 ( const BandedMatrix & A )

Compute 1-norm of banded matrix.

Parameters

A	Banded matrix

Returns: Maximum absolute column sum

Definition at line 305 of file banded.cpp.

References ipow().

◆ banded_rcond()

real num::banded_rcond	(	const BandedMatrix &	A,
		const idx *	ipiv,
		real	anorm
	)

Estimate reciprocal condition number of banded matrix.

Uses 1-norm condition number estimation after LU factorization.

Parameters

A	LU-factored banded matrix
ipiv	Pivot indices
anorm	1-norm of original matrix (before factorization)

Returns: Reciprocal condition number (small = ill-conditioned)

Definition at line 321 of file banded.cpp.

References banded_lu_solve(), and ipow().

◆ banded_solve()

BandedSolverResult num::banded_solve	(	const BandedMatrix &	A,
		const Vector &	b,
		Vector &	x
	)

Solve banded system Ax = b (convenience function)

Performs LU factorization and solve in one call. For solving multiple systems with the same matrix, use banded_lu() and banded_lu_solve() separately to avoid redundant factorization.

Parameters

A	Banded matrix (NOT modified - internal copy made)
b	Right-hand side
x	Solution vector (output)

Returns: Result indicating success or failure

Definition at line 247 of file banded.cpp.

References banded_lu(), banded_lu_solve(), ipow(), and num::BasicVector< T >::size().

◆ bell_pair()

Circuit num::bell_pair	(	int	q0 = `0`,
		int	q1 = `1`
	)

Bell state |Phi+> = (|00> + |11>)/sqrt2 on qubits q0, q1.

Definition at line 389 of file circuit.cpp.

References num::Circuit::cx(), and num::Circuit::h().

◆ bessel_i()

real num::bessel_i	(	real	nu,
		real	x
	)

inline

I_nu(x) – modified Bessel function of the first kind.

Definition at line 71 of file math.hpp.

◆ bessel_j()

real num::bessel_j	(	real	nu,
		real	x
	)

inline

J_nu(x) – Bessel function of the first kind.

Definition at line 53 of file math.hpp.

Referenced by tdse::TDSESolver::compute_modes().

◆ bessel_k()

real num::bessel_k	(	real	nu,
		real	x
	)

inline

K_nu(x) – modified Bessel function of the second kind.

Definition at line 80 of file math.hpp.

◆ bessel_y()

real num::bessel_y	(	real	nu,
		real	x
	)

inline

Y_nu(x) – Bessel function of the second kind (Neumann function)

Definition at line 62 of file math.hpp.

◆ beta()

real num::beta	(	real	a,
		real	b
	)

inline

B(a, b) – beta function.

Definition at line 242 of file math.hpp.

References ipow().

Referenced by banded_gemv(), num::markov::boltzmann_accept(), cg(), cg_matfree(), expv(), gmres(), lanczos(), num::markov::make_boltzmann_table(), num::backends::blas::matadd(), num::backends::omp::matadd(), num::backends::seq::matadd(), matadd(), num::markov::metropolis_sweep(), svd(), and num::markov::umbrella_sweep().

◆ bisection()

RootResult num::bisection	(	ScalarFn	f,
		real	a,
		real	b,
		real	tol = `1e-10`,
		idx	max_iter = `1000`
	)

Bisection method.

Parameters

f	Continuous function
a,b	Bracket: f(a) and f(b) must have opposite signs
tol	Convergence tolerance on \|f(root)\| and interval width
max_iter	Maximum iterations

Definition at line 8 of file roots.cpp.

References ipow().

◆ brent()

RootResult num::brent	(	ScalarFn	f,
		real	a,
		real	b,
		real	tol = `1e-10`,
		idx	max_iter = `1000`
	)

Brent's method (bisection + secant + inverse quadratic interpolation)

Guaranteed to converge if f(a) and f(b) have opposite signs. Superlinear convergence near smooth roots. Preferred method when a reliable bracket is available.

Parameters

f	Function
a,b	Bracket: f(a) and f(b) must have opposite signs
tol	Convergence tolerance
max_iter	Maximum iterations

Definition at line 54 of file roots.cpp.

References e, and ipow().

Referenced by tdse::TDSESolver::compute_modes().

◆ cg()

SolverResult num::cg	(	const Matrix &	A,
		const Vector &	b,
		Vector &	x,
		real	tol = `1e-10`,
		idx	max_iter = `1000`,
		Backend	backend = `default_backend`
	)

Conjugate gradient solver for Ax = b.

Parameters

A	Symmetric positive definite matrix
b	Right-hand side vector
x	Solution vector (initial guess on input, solution on output)
tol	Convergence tolerance on residual norm (default 1e-10)
max_iter	Maximum iterations (default 1000)
backend	Backend for internal matvec/dot/axpy/scale (default: default_backend)

Returns: SolverResult with convergence info

Definition at line 8 of file cg.cpp.

References axpy(), beta(), dot(), e, gpu, ipow(), num::SolverResult::iterations, matvec(), scale(), num::BasicVector< T >::size(), num::BasicVector< T >::to_cpu(), num::cuda::to_device(), and num::BasicVector< T >::to_gpu().

Referenced by thomas().

◆ cg_matfree()

SolverResult num::cg_matfree	(	MatVecFn	matvec,
		const Vector &	b,
		Vector &	x,
		real	tol = `1e-6`,
		idx	max_iter = `1000`
	)

Matrix-free conjugate gradient for Ax = b where A is SPD.

Parameters

matvec	Callable computing y = A*x
b	Right-hand side
x	Initial guess (modified in-place -> solution)
tol	Convergence tolerance on residual norm
max_iter	Maximum CG iterations

Definition at line 66 of file cg.cpp.

References axpy(), beta(), dot(), e, ipow(), num::SolverResult::iterations, scale(), and seq.

Referenced by num::FieldSolver::solve_poisson(), and physics::ElectricSolver::solve_potential().

◆ col_fiber_sweep()

template<typename T , typename F >

void num::col_fiber_sweep	(	BasicVector< T > &	data,
		int	N,
		F &&	f
	)

For each column j in [0,N), extract the x-direction fiber data[0..N-1, j] into a std::vector<T>, call f(fiber), then write back.

f must have signature: void(std::vector<T>&) and modifies in-place.

Use for ADI / Crank-Nicolson sweeps along x.

Definition at line 77 of file stencil.hpp.

References ipow().

Referenced by num::CrankNicolsonADI::sweep().

◆ connected_components()

template<typename InCluster , typename Neighbors >

ClusterResult num::connected_components	(	int	n_sites,
		InCluster &&	in_cluster,
		Neighbors &&	neighbors
	)

BFS connected-component labelling with pre-allocated flat queue (no heap per call).

Parameters

n_sites	Total number of sites
in_cluster	bool(int i) – include site i?
neighbors	void(int i, auto&& visit) – call visit(nb) per neighbor of i

Definition at line 34 of file connected_components.hpp.

References num::ClusterResult::id, and ipow().

Referenced by IsingLattice::sweep_umbrella().

◆ curl_3d()

void num::curl_3d	(	const Grid3D &	ax,
		const Grid3D &	ay,
		const Grid3D &	az,
		Grid3D &	bx,
		Grid3D &	by,
		Grid3D &	bz
	)

inline

Compute the central-difference curl ∇×A of a vector field. One-sided differences at domain boundaries. bx, by, bz must already be allocated with the same dimensions as ax, ay, az.

Definition at line 170 of file stencil.hpp.

References num::Grid3D::dx(), ipow(), num::Grid3D::nx(), num::Grid3D::ny(), and num::Grid3D::nz().

Referenced by num::FieldSolver::curl().

◆ divergence_3d()

void num::divergence_3d	(	const Grid3D &	fx,
		const Grid3D &	fy,
		const Grid3D &	fz,
		Grid3D &	out
	)

inline

Compute the central-difference divergence of a vector field (fx, fy, fz). One-sided differences at domain boundaries. out must already be allocated with the same dimensions.

Definition at line 151 of file stencil.hpp.

References ipow(), and num::Grid3D::nx().

Referenced by num::FieldSolver::divergence().

◆ dot() [1/2]

cplx num::dot	(	const CVector &	x,
		const CVector &	y
	)

Conjugate inner product <x, y> = Sigma conj(x_i) * y_i.

Definition at line 91 of file vector.cpp.

References ipow(), and num::BasicVector< T >::size().

◆ dot() [2/2]

real num::dot	(	const Vector &	x,
		const Vector &	y,
		Backend	b = `default_backend`
	)

dot product

Definition at line 57 of file vector.cpp.

References blas, blocked, num::backends::blas::dot(), num::backends::gpu::dot(), num::backends::omp::dot(), num::backends::seq::dot(), gpu, ipow(), omp, seq, and simd.

Referenced by cg(), cg_matfree(), num::mpi::dot(), IsingLattice::energy_per_spin(), expv(), inverse_iteration(), IsingLattice::magnetization(), num::mpi::norm(), power_iteration(), and rayleigh_iteration().

◆ eig_sym() [1/2]

EigenResult num::eig_sym	(	const Matrix &	A,
		real	tol = `1e-12`,
		idx	max_sweeps = `100`,
		Backend	backend = `Backend::seq`
	)

inline

Full eigendecomposition of a real symmetric matrix.

The rotation accumulation loop is parallelised when backend == Backend::omp.

Parameters

A	Real symmetric nxn matrix (upper/lower triangle used)
backend	Execution backend (Backend::omp parallelises the rotation loop)
tol	Convergence: stop when Frobenius norm of off-diagonal < tol
max_sweeps	Safety cap on Jacobi sweeps (typically < 20 for n < 500)

Returns: EigenResult with eigenvalues in ascending order and corresponding eigenvectors as columns of the V matrix (A = V diag(lambda) Vᵀ)

Definition at line 50 of file jacobi_eig.hpp.

References e, and ipow().

Referenced by lanczos().

◆ eig_sym() [2/2]

EigenResult num::eig_sym	(	const Matrix &	A_in,
		real	tol,
		idx	max_sweeps
	)

Definition at line 33 of file eig.cpp.

References e, and ipow().

◆ ellint_E()

real num::ellint_E ( real k )

inline

E(k) – complete elliptic integral of the second kind.

Definition at line 177 of file math.hpp.

References ipow().

◆ ellint_Ei()

real num::ellint_Ei	(	real	k,
		real	phi
	)

inline

E(k, phi) – incomplete elliptic integral of the second kind.

Definition at line 204 of file math.hpp.

References ipow(), and phi.

◆ ellint_F()

real num::ellint_F	(	real	k,
		real	phi
	)

inline

F(k, phi) – incomplete elliptic integral of the first kind.

Definition at line 195 of file math.hpp.

References ipow(), and phi.

◆ ellint_K()

real num::ellint_K ( real k )

inline

K(k) – complete elliptic integral of the first kind.

Definition at line 168 of file math.hpp.

References ipow().

◆ ellint_Pi()

real num::ellint_Pi	(	real	n,
		real	k
	)

inline

Pi(n, k) – complete elliptic integral of the third kind.

Definition at line 186 of file math.hpp.

References ipow().

◆ ellint_Pi_inc()

real num::ellint_Pi_inc	(	real	n,
		real	k,
		real	phi
	)

inline

Pi(n, k, phi) – incomplete elliptic integral of the third kind.

Definition at line 213 of file math.hpp.

References ipow(), and phi.

◆ expint()

real num::expint ( real x )

inline

Ei(x) – exponential integral.

Definition at line 224 of file math.hpp.

◆ expv() [1/2]

Vector num::expv	(	real	t,
		const MatVecFn &	matvec,
		idx	n,
		const Vector &	v,
		int	m_max = `30`,
		real	tol = `1e-8`
	)

Compute exp(t*A)*v via Krylov-Padé approximation (matrix-free)

Parameters

t	Scalar multiplier on A
matvec	Callable y = A*x
n	Dimension of the state space (size of v)
v	Input vector
m_max	Maximum Krylov dimension (default 30)
tol	Breakdown tolerance for Arnoldi (default 1e-8)

Returns: Approximation of exp(t*A)*v

Definition at line 110 of file expv.cpp.

References axpy(), beta(), dot(), e, ipow(), matvec(), and norm().

Referenced by expv().

◆ expv() [2/2]

Vector num::expv	(	real	t,
		const SparseMatrix &	A,
		const Vector &	v,
		int	m_max = `30`,
		real	tol = `1e-8`
	)

Compute exp(t*A)*v via Krylov-Padé approximation (sparse matrix)

Parameters

t	Scalar multiplier on A
A	Sparse matrix in CSR format
v	Input vector
m_max	Maximum Krylov dimension (default 30)
tol	Breakdown tolerance for Arnoldi (default 1e-8)

Returns: Approximation of exp(t*A)*v

Definition at line 180 of file expv.cpp.

References expv(), ipow(), and sparse_matvec().

◆ gauss_legendre()

real num::gauss_legendre	(	ScalarFn	f,
		real	a,
		real	b,
		idx	p = `5`
	)

Gauss-Legendre quadrature (exact for polynomials up to degree 2p-1)

Parameters

f	Integrand
a	Lower bound
b	Upper bound
p	Number of quadrature points (1 to 5 supported)

Definition at line 59 of file quadrature.cpp.

References ipow().

◆ gauss_seidel()

SolverResult num::gauss_seidel	(	const Matrix &	A,
		const Vector &	b,
		Vector &	x,
		real	tol = `1e-10`,
		idx	max_iter = `1000`,
		Backend	backend = `default_backend`
	)

Gauss-Seidel iterative solver for Ax = b.

Updates each component x[i] in-place using the latest values of all other components. Converges for strictly diagonally dominant or symmetric positive definite A.

With Backend::omp the residual computation is parallelised; the update sweep remains sequential to preserve convergence properties.

Parameters

A	Square matrix
b	Right-hand side vector
x	Solution vector (initial guess on input, solution on output)
tol	Convergence tolerance on residual norm (default 1e-10)
max_iter	Maximum iterations (default 1000)
backend	Execution backend (default: default_backend)

Returns: SolverResult with convergence info

Definition at line 13 of file gauss_seidel.cpp.

References e, ipow(), and num::BasicVector< T >::size().

◆ ghz_state()

Circuit num::ghz_state ( int n_qubits )

n-qubit GHZ state (|00...0> + |11...1>)/sqrt2

Definition at line 395 of file circuit.cpp.

References num::Circuit::cx(), and num::Circuit::h().

◆ gmres() [1/3]

SolverResult num::gmres	(	const Matrix &	A,
		const Vector &	b,
		Vector &	x,
		real	tol = `1e-6`,
		idx	max_iter = `1000`,
		idx	restart = `30`,
		Backend	backend = `default_backend`
	)

Restarted GMRES with a dense matrix.

Parameters

backend Backend for the internal matvec at each Arnoldi step

Definition at line 126 of file krylov.cpp.

References gmres(), ipow(), and matvec().

◆ gmres() [2/3]

SolverResult num::gmres	(	const SparseMatrix &	A,
		const Vector &	b,
		Vector &	x,
		real	tol = `1e-6`,
		idx	max_iter = `1000`,
		idx	restart = `30`
	)

Restarted GMRES with a sparse (CSR) matrix.

Definition at line 113 of file krylov.cpp.

References gmres(), ipow(), and sparse_matvec().

◆ gmres() [3/3]

SolverResult num::gmres	(	MatVecFn	matvec,
		idx	n,
		const Vector &	b,
		Vector &	x,
		real	tol = `1e-6`,
		idx	max_iter = `1000`,
		idx	restart = `30`
	)

Restarted GMRES(restart) – matrix-free interface.

Works for any invertible A (symmetric or non-symmetric, indefinite). The Krylov subspace is restarted every restart steps to bound memory.

Parameters

matvec	Callable computing y = A*x
n	System dimension
b	Right-hand side
x	Initial guess (modified in-place -> solution)
tol	Convergence tolerance on residual norm
max_iter	Maximum total matrix-vector products
restart	Krylov subspace size before restart (default 30)

Returns: SolverResult with convergence info

Definition at line 11 of file krylov.cpp.

References beta(), e, ipow(), and num::BasicVector< T >::size().

Referenced by gmres(), and gmres().

◆ gradient_3d()

void num::gradient_3d	(	const Grid3D &	phi,
		Grid3D &	gx,
		Grid3D &	gy,
		Grid3D &	gz
	)

inline

Compute the central-difference gradient of a scalar field. One-sided differences at domain boundaries. gx, gy, gz must already be allocated with the same dimensions as phi.

Definition at line 132 of file stencil.hpp.

References ipow(), and phi.

Referenced by num::FieldSolver::gradient().

◆ hermite()

real num::hermite	(	unsigned int	n,
		real	x
	)

inline

H_n(x) – (physicists') Hermite polynomial.

Definition at line 138 of file math.hpp.

◆ int_range()

std::vector< int > num::int_range	(	int	start,
		int	n
	)

inline

Integer sequence [start, start+1, ..., start+n-1]. Wraps std::iota.

Definition at line 263 of file math.hpp.

References ipow().

◆ inverse_iteration()

PowerResult num::inverse_iteration	(	const Matrix &	A,
		real	sigma,
		real	tol = `1e-10`,
		idx	max_iter = `1000`,
		Backend	backend = `default_backend`
	)

Inverse iteration – finds the eigenvalue closest to a shift sigma.

Factorizes (A - sigmaI) once then solves repeatedly.

Parameters

A	Square matrix (symmetric recommended)
sigma	Shift – should be near the target eigenvalue
tol	Tolerance on eigenvalue change between iterations
max_iter	Maximum iterations
backend	Backend forwarded to matvec and dot

Definition at line 48 of file power.cpp.

References dot(), e, ipow(), lu(), lu_solve(), matvec(), and num::detail::normalise().

◆ ipow()

template<int N, typename T >

constexpr T num::ipow ( T x )

constexprnoexcept

Compute x^N at compile time via repeated squaring.

Template Parameters

N	Non-negative integer exponent (must be a compile-time constant).
T	Arithmetic type (float, double, int, ...).

Definition at line 34 of file integer_pow.hpp.

References ipow().

◆ jacobi()

SolverResult num::jacobi	(	const Matrix &	A,
		const Vector &	b,
		Vector &	x,
		real	tol = `1e-10`,
		idx	max_iter = `1000`,
		Backend	backend = `default_backend`
	)

Jacobi iterative solver for Ax = b.

Updates all components simultaneously using only values from the previous iteration. Converges for strictly diagonally dominant A. Trivially parallelisable with Backend::omp.

Parameters

A	Square matrix
b	Right-hand side vector
x	Solution vector (initial guess on input, solution on output)
tol	Convergence tolerance on residual norm (default 1e-10)
max_iter	Maximum iterations (default 1000)
backend	Execution backend (default: default_backend)

Returns: SolverResult with convergence info

Definition at line 7 of file jacobi.cpp.

References e, ipow(), and num::BasicVector< T >::size().

◆ laguerre()

real num::laguerre	(	unsigned int	n,
		real	x
	)

inline

L_n(x) – Laguerre polynomial.

Definition at line 147 of file math.hpp.

◆ lanczos()

LanczosResult num::lanczos	(	MatVecFn	matvec,
		idx	n,
		idx	k,
		real	tol = `1e-10`,
		idx	max_steps = `0`,
		Backend	backend = `Backend::seq`
	)

Lanczos eigensolver for large sparse symmetric matrices.

Parameters

matvec	Callable computing w = A*v (matrix-free)
n	Dimension of the problem
k	Number of eigenvalues requested (k << n)
tol	Convergence on \|\|Au - lambdau\|\| for each Ritz pair
max_steps	Maximum Lanczos steps (default: min(3k, n))
backend	Backend for reorthogonalisation inner products (default: seq)

Definition at line 30 of file lanczos.cpp.

References beta(), e, eig_sym(), ipow(), and matvec().

Referenced by tdse::TDSESolver::compute_modes().

◆ laplacian_stencil_2d()

template<typename T >

void num::laplacian_stencil_2d	(	const BasicVector< T > &	x,
		BasicVector< T > &	y,
		int	N
	)

Compute y[i,j] = x[i+1,j] + x[i-1,j] + x[i,j+1] + x[i,j-1] - 4*x[i,j] Dirichlet: out-of-bounds neighbors contribute 0. Works for T = real or cplx.

Definition at line 32 of file stencil.hpp.

References ipow().

Referenced by tdse::TDSESolver::compute_energy(), tdse::TDSESolver::compute_modes(), and num::pde::diffusion_step_2d_dirichlet().

◆ laplacian_stencil_2d_periodic()

template<typename T >

void num::laplacian_stencil_2d_periodic	(	const BasicVector< T > &	x,
		BasicVector< T > &	y,
		int	N
	)

Same as laplacian_stencil_2d but with periodic (wrap-around) boundaries.

The j boundaries (j=0 and j=N-1) are peeled out of the inner loop so the interior range j=1..N-2 contains no modulo arithmetic and is auto-vectorizable. Row wrapping (i±1) uses precomputed indices.

Definition at line 52 of file stencil.hpp.

References num::BasicVector< T >::data(), and ipow().

Referenced by num::pde::diffusion_step_2d().

◆ legendre()

real num::legendre	(	unsigned int	n,
		real	x
	)

inline

P_n(x) – Legendre polynomial of degree n.

Definition at line 111 of file math.hpp.

◆ linspace()

std::vector< real > num::linspace	(	real	start,
		real	stop,
		idx	n
	)

inline

Evenly spaced values from start to stop, inclusive. MATLAB/NumPy linspace.

Definition at line 253 of file math.hpp.

References ipow().

◆ lu()

LUResult num::lu ( const Matrix & A )

LU factorization of a square matrix A with partial pivoting.

Computes P, L, U such that P*A = L*U where: P = permutation matrix encoded in piv L = unit lower triangular (diagonal = 1, stored below diag of LU) U = upper triangular (stored on and above diag of LU)

Partial pivoting (swap largest element to pivot position) ensures |L(i,j)| <= 1 for all i > j, which bounds round-off growth.

Definition at line 21 of file lu.cpp.

References e, ipow(), and num::LUResult::LU.

Referenced by inverse_iteration(), and rayleigh_iteration().

◆ lu_det()

real num::lu_det ( const LUResult & f )

Determinant of A from its LU factorization.

det(A) = det(P)^{-1} * prod(U[i,i]) = (-1)^{swaps} * prod(U[i,i])

Overflow is possible for large n – use log-determinant for stability.

Definition at line 109 of file lu.cpp.

References ipow(), and num::Matrix::rows().

◆ lu_inv()

Matrix num::lu_inv ( const LUResult & f )

Inverse of A from its LU factorization.

Solves A * X = I column by column. O(n^3) – only use when the full inverse is genuinely needed (prefer lu_solve for specific right-hand sides).

Definition at line 122 of file lu.cpp.

References e, ipow(), lu_solve(), and num::Matrix::rows().

◆ lu_solve() [1/2]

void num::lu_solve	(	const LUResult &	f,
		const Matrix &	B,
		Matrix &	X
	)

Solve A*X = B for multiple right-hand sides.

Each column of B is solved independently using the same factorization. B and X are nxnrhs matrices (column-major access pattern).

Definition at line 90 of file lu.cpp.

References ipow(), and lu_solve().

◆ lu_solve() [2/2]

void num::lu_solve	(	const LUResult &	f,
		const Vector &	b,
		Vector &	x
	)

Solve A*x = b using a precomputed LU factorization.

Three steps:

Apply permutation: y = P*b
Forward substitution: solve L*z = y
Backward substitution: solve U*x = z

Definition at line 67 of file lu.cpp.

References ipow().

Referenced by inverse_iteration(), lu_inv(), lu_solve(), and rayleigh_iteration().

◆ matadd()

void num::matadd	(	real	alpha,
		const Matrix &	A,
		real	beta,
		const Matrix &	B,
		Matrix &	C,
		Backend	b = `default_backend`
	)

C = alpha*A + beta*B.

Definition at line 105 of file matrix.cpp.

References beta(), blas, blocked, gpu, ipow(), num::backends::blas::matadd(), num::backends::omp::matadd(), num::backends::seq::matadd(), omp, seq, and simd.

◆ matmul()

void num::matmul	(	const Matrix &	A,
		const Matrix &	B,
		Matrix &	C,
		Backend	b = `default_backend`
	)

C = A * B.

Definition at line 83 of file matrix.cpp.

References blas, blocked, gpu, ipow(), num::backends::blas::matmul(), num::backends::gpu::matmul(), num::backends::omp::matmul(), num::backends::seq::matmul(), num::backends::simd::matmul(), num::backends::seq::matmul_blocked(), omp, seq, and simd.

Referenced by svd_truncated().

◆ matmul_blocked()

void num::matmul_blocked	(	const Matrix &	A,
		const Matrix &	B,
		Matrix &	C,
		idx	block_size = `64`
	)

C = A * B (cache-blocked)

Divides A, B, C into BLOCKxBLOCK tiles so the working set fits in L2 cache.

Parameters

block_size Tile edge length (default 64).

Definition at line 117 of file matrix.cpp.

References ipow(), and num::backends::seq::matmul_blocked().

◆ matmul_register_blocked()

void num::matmul_register_blocked	(	const Matrix &	A,
		const Matrix &	B,
		Matrix &	C,
		idx	block_size = `64`,
		idx	reg_size = `4`
	)

C = A * B (register-blocked)

Extends cache blocking with a REGxREG register tile inside each cache tile.

Parameters

block_size	Cache tile edge (default 64).
reg_size	Register tile edge (default 4).

Definition at line 121 of file matrix.cpp.

References ipow(), and num::backends::seq::matmul_register_blocked().

◆ matmul_simd()

void num::matmul_simd	(	const Matrix &	A,
		const Matrix &	B,
		Matrix &	C,
		idx	block_size = `64`
	)

C = A * B (SIMD-accelerated)

Dispatches at compile time: AVX-256 + FMA on x86, NEON on AArch64, falls back to matmul_blocked if neither is available.

Definition at line 126 of file matrix.cpp.

References ipow(), and num::backends::simd::matmul().

◆ matvec()

void num::matvec	(	const Matrix &	A,
		const Vector &	x,
		Vector &	y,
		Backend	b = `default_backend`
	)

y = A * x

Definition at line 94 of file matrix.cpp.

References blas, blocked, gpu, ipow(), num::backends::blas::matvec(), num::backends::gpu::matvec(), num::backends::omp::matvec(), num::backends::simd::matvec(), num::backends::seq::matvec(), omp, seq, and simd.

Referenced by cg(), expv(), gmres(), inverse_iteration(), lanczos(), power_iteration(), rayleigh_iteration(), and num::FieldSolver::solve_poisson().

◆ matvec_simd()

void num::matvec_simd	(	const Matrix &	A,
		const Vector &	x,
		Vector &	y
	)

y = A * x (SIMD-accelerated)

Definition at line 130 of file matrix.cpp.

References ipow(), and num::backends::simd::matvec().

◆ neg_laplacian_3d()

void num::neg_laplacian_3d	(	const Vector &	x,
		Vector &	y,
		int	nx,
		int	ny,
		int	nz,
		double	inv_dx2
	)

inline

Compute the negative Laplacian: y = -∆x = (6x[i,j,k] - Σ6 nbrs) / dx² Dirichlet BC: boundary nodes satisfy y[bdry] = x[bdry] (identity row, so that the system is SPD when used with a b=0 boundary RHS).

x and y are flat vectors in Grid3D layout: idx = k*ny*nx + j*nx + i.

Definition at line 109 of file stencil.hpp.

References ipow().

Referenced by num::FieldSolver::solve_poisson().

◆ newton()

RootResult num::newton	(	ScalarFn	f,
		ScalarFn	df,
		real	x0,
		real	tol = `1e-10`,
		idx	max_iter = `1000`
	)

Newton-Raphson method.

Parameters

f	Function
df	Derivative of f
x0	Initial guess
tol	Convergence tolerance
max_iter	Maximum iterations

Definition at line 25 of file roots.cpp.

References e, and ipow().

Referenced by IsingLattice::mean_field_m().

◆ norm() [1/2]

real num::norm ( const CVector & x )

Euclidean norm sqrt(Sigma |v_i|^2)

Definition at line 97 of file vector.cpp.

References ipow(), and num::BasicVector< T >::size().

◆ norm() [2/2]

real num::norm	(	const Vector &	x,
		Backend	b = `default_backend`
	)

Euclidean norm.

Definition at line 69 of file vector.cpp.

References blas, blocked, gpu, ipow(), num::backends::blas::norm(), num::backends::gpu::norm(), num::backends::seq::norm(), omp, seq, and simd.

Referenced by tdse::TDSESolver::compute_norm(), expv(), num::quantum::norm(), num::quantum::normalize(), num::Histogram::pdf(), and tdse::TDSESolver::renormalize().

◆ ode_euler()

ODEResult num::ode_euler	(	ODERhsFn	f,
		Vector	y0,
		real	t0,
		real	t1,
		real	h,
		StepCallback	on_step = `nullptr`
	)

Forward Euler (1st order, fixed step): y_{n+1} = y_n + h · f(t_n, y_n)

Definition at line 15 of file ode.cpp.

References e, and ipow().

◆ ode_rk4()

ODEResult num::ode_rk4	(	ODERhsFn	f,
		Vector	y0,
		real	t0,
		real	t1,
		real	h,
		StepCallback	on_step = `nullptr`
	)

Classic 4th-order Runge-Kutta (fixed step).

Definition at line 33 of file ode.cpp.

References e, and ipow().

Referenced by nbody::NBodySim::step().

◆ ode_rk45()

ODEResult num::ode_rk45	(	ODERhsFn	f,
		Vector	y0,
		real	t0,
		real	t1,
		real	rtol = `1e-6`,
		real	atol = `1e-9`,
		real	h0 = `1e-3`,
		idx	max_steps = `1000000`,
		StepCallback	on_step = `nullptr`
	)

Adaptive Dormand-Prince RK45 with FSAL and PI step-size control.

Advances from t0 to t1, adjusting h so that the mixed absolute/relative error estimate stays below tolerance: err / (atol + |y|·rtol) ≤ 1

Parameters

rtol	Relative tolerance (default 1e-6)
atol	Absolute tolerance (default 1e-9)
h0	Initial step-size hint (default 1e-3)
max_steps	Hard limit on accepted steps (default 1e6)
on_step	Optional callback after each accepted step

Definition at line 67 of file ode.cpp.

References e, and ipow().

◆ ode_verlet()

SymplecticResult num::ode_verlet	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		real	t0,
		real	t1,
		real	h,
		SymplecticCallback	on_step = `nullptr`
	)

Velocity Verlet — 2nd-order symplectic, 1 force evaluation per step.

Algorithm (kick-drift-kick): a_n = accel(q_n) q_{n+1} = q_n + h·v_n + h²/2 · a_n a_{n+1} = accel(q_{n+1}) v_{n+1} = v_n + h/2 · (a_n + a_{n+1})

Acceleration is cached between steps (FSAL — first same as last).

Definition at line 154 of file ode.cpp.

References e, ipow(), and num::BasicVector< T >::size().

Referenced by nbody::NBodySim::step().

◆ ode_yoshida4()

SymplecticResult num::ode_yoshida4	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		real	t0,
		real	t1,
		real	h,
		SymplecticCallback	on_step = `nullptr`
	)

Yoshida 4th-order symplectic — 3 force evaluations per step.

Composes three leapfrog sub-steps with Yoshida (1990) coefficients: w₁ = 1 / (2 − ∛2), w₀ = 1 − 2w₁ c₁=c₄=w₁/2, c₂=c₃=(w₀+w₁)/2, d₁=d₃=w₁, d₂=w₀

Drift-kick sequence per step: q += c₁h·v → v += d₁h·a(q) → q += c₂h·v → v += d₂h·a(q) → q += c₃h·v → v += d₃h·a(q) → q += c₄h·v

Preferred over Verlet when accuracy matters and force is cheap.

Definition at line 195 of file ode.cpp.

References e, ipow(), and num::BasicVector< T >::size().

◆ operator*()

template<idx N>

constexpr SmallVec< N > num::operator*	(	real	s,
		SmallVec< N >	v
	)

constexprnoexcept

Definition at line 68 of file small_matrix.hpp.

◆ operator+()

template<idx N>

constexpr SmallVec< N > num::operator+	(	SmallVec< N >	a,
		const SmallVec< N > &	b
	)

constexprnoexcept

Definition at line 64 of file small_matrix.hpp.

References ipow().

◆ power_iteration()

PowerResult num::power_iteration	(	const Matrix &	A,
		real	tol = `1e-10`,
		idx	max_iter = `1000`,
		Backend	backend = `default_backend`
	)

Power iteration – finds the eigenvalue largest in absolute value.

Parameters

A	Square matrix (need not be symmetric)
tol	Tolerance on eigenvalue change between iterations
max_iter	Maximum iterations
backend	Backend forwarded to matvec and dot

Definition at line 11 of file power.cpp.

References dot(), e, ipow(), matvec(), and num::detail::normalise().

◆ print_state()

void num::print_state	(	const quantum::Statevector &	sv,
		int	n_qubits,
		real	threshold = `1e-6`
	)

Pretty-print a statevector, hiding amplitudes below threshold.

Definition at line 416 of file circuit.cpp.

References ipow().

◆ qft_circuit()

Circuit num::qft_circuit ( int n_qubits )

n-qubit Quantum Fourier Transform circuit Applied to |0...0> unless you prepend state-preparation gates.

Definition at line 403 of file circuit.cpp.

References num::Circuit::cp(), num::Circuit::h(), ipow(), and num::Circuit::swap().

◆ qr()

QRResult num::qr ( const Matrix & A )

QR factorization of an mxn matrix A (m >= n) via Householder reflections.

Each Householder step k:

Extract column k of the current working matrix from row k downward.
Compute v = x + sign(x[0]) * ||x|| * e_1 (sign chosen to avoid cancellation).
Normalise v.
Apply H_k = I - 2*v*v^T to the trailing submatrix.

After min(m-1, n) steps, the matrix is upper triangular. Q is reconstructed by accumulating the reflectors in reverse order.

Householder QR is backward-stable (||deltaA||/||A|| = O(u) for machine unit u), unlike classical Gram-Schmidt which is only forward-stable.

Definition at line 41 of file qr.cpp.

References e, and ipow().

Referenced by svd_truncated().

◆ qr_solve()

void num::qr_solve	(	const QRResult &	f,
		const Vector &	b,
		Vector &	x
	)

Solve the least-squares problem min ||A*x - b||_2.

Algorithm:

y = Q^T * b
Back-substitute: solve R[:n,:n] * x = y[:n]

For square non-singular A this returns the exact solution. For overdetermined A (m > n) this returns the minimum-residual solution.

Definition at line 119 of file qr.cpp.

References ipow().

◆ rayleigh_iteration()

PowerResult num::rayleigh_iteration	(	const Matrix &	A,
		const Vector &	x0,
		real	tol = `1e-10`,
		idx	max_iter = `50`,
		Backend	backend = `default_backend`
	)

Rayleigh quotient iteration – cubically convergent.

Updates the shift sigma = vᵀAv at every step -> fresh LU each iteration.

Parameters

A	Symmetric matrix
x0	Starting vector (determines which eigenvalue is found)
tol	Tolerance on residual \|\|Av - lambdav\|\|
max_iter	Maximum iterations
backend	Backend forwarded to matvec, dot, axpy, norm

Definition at line 93 of file power.cpp.

References dot(), ipow(), lu(), lu_solve(), matvec(), and num::detail::normalise().

◆ rng_int()

int num::rng_int	(	Rng *	r,
		int	lo,
		int	hi
	)

inline

Uniform integer in [lo, hi] (inclusive on both ends).

Definition at line 294 of file math.hpp.

References ipow().

◆ rng_normal()

real num::rng_normal	(	Rng *	r,
		real	mean,
		real	stddev
	)

inline

Normal (Gaussian) sample with given mean and standard deviation.

Definition at line 289 of file math.hpp.

References ipow().

Referenced by svd_truncated().

◆ rng_uniform()

real num::rng_uniform	(	Rng *	r,
		real	lo,
		real	hi
	)

inline

Uniform real in [lo, hi).

Definition at line 284 of file math.hpp.

References ipow().

◆ romberg()

real num::romberg	(	ScalarFn	f,
		real	a,
		real	b,
		real	tol = `1e-10`,
		idx	max_levels = `12`
	)

Romberg integration (Richardson extrapolation on trapezoidal rule)

Parameters

f	Integrand
a	Lower bound
b	Upper bound
tol	Convergence tolerance
max_levels	Maximum refinement levels

Definition at line 98 of file quadrature.cpp.

References ipow().

◆ row_fiber_sweep()

template<typename T , typename F >

void num::row_fiber_sweep	(	BasicVector< T > &	data,
		int	N,
		F &&	f
	)

For each row i in [0,N), extract the y-direction fiber data[i, 0..N-1] into a std::vector<T>, call f(fiber), then write back.

Use for ADI / Crank-Nicolson sweeps along y.

Definition at line 91 of file stencil.hpp.

References ipow().

Referenced by num::CrankNicolsonADI::sweep().

◆ scale() [1/2]

void num::scale	(	CVector &	v,
		cplx	alpha
	)

v *= alpha

Definition at line 83 of file vector.cpp.

References ipow(), and num::BasicVector< T >::size().

◆ scale() [2/2]

void num::scale	(	Vector &	v,
		real	alpha,
		Backend	b = `default_backend`
	)

v *= alpha

Definition at line 27 of file vector.cpp.

References blas, blocked, gpu, ipow(), omp, num::backends::blas::scale(), num::backends::gpu::scale(), num::backends::omp::scale(), num::backends::seq::scale(), seq, and simd.

Referenced by cg(), cg_matfree(), tdse::TDSESolver::renormalize(), num::VectorField3D::scale(), and tdse::TDSESolver::set_potential().

◆ secant()

RootResult num::secant	(	ScalarFn	f,
		real	x0,
		real	x1,
		real	tol = `1e-10`,
		idx	max_iter = `1000`
	)

Secant method (Newton without derivative)

Parameters

f	Function
x0,x1	Two distinct initial guesses
tol	Convergence tolerance
max_iter	Maximum iterations

Definition at line 39 of file roots.cpp.

References e, and ipow().

◆ simpson()

real num::simpson	(	ScalarFn	f,
		real	a,
		real	b,
		idx	n = `100`,
		Backend	backend = `Backend::seq`
	)

Simpson's 1/3 rule with n panels (n must be even)

Parameters

backend Backend::omp parallelises the panel sum

Definition at line 25 of file quadrature.cpp.

References ipow().

◆ sparse_matvec()

void num::sparse_matvec	(	const SparseMatrix &	A,
		const Vector &	x,
		Vector &	y
	)

y = A * x

Definition at line 110 of file sparse.cpp.

References ipow(), and num::BasicVector< T >::size().

Referenced by IsingLattice::energy_per_spin(), expv(), and gmres().

◆ sph_bessel_j()

real num::sph_bessel_j	(	unsigned int	n,
		real	x
	)

inline

j_n(x) – spherical Bessel function of the first kind

Definition at line 91 of file math.hpp.

◆ sph_bessel_y()

real num::sph_bessel_y	(	unsigned int	n,
		real	x
	)

inline

y_n(x) – spherical Neumann function (spherical Bessel of the second kind)

Definition at line 100 of file math.hpp.

◆ sph_legendre()

real num::sph_legendre	(	unsigned int	l,
		unsigned int	m,
		real	theta
	)

inline

Y_l^m(theta) – spherical harmonic (real part, theta in radians)

Definition at line 129 of file math.hpp.

References ipow().

◆ svd()

SVDResult num::svd	(	const Matrix &	A,
		Backend	backend = `Backend::seq`,
		real	tol = `1e-12`,
		idx	max_sweeps = `100`
	)

Full SVD of an mxn matrix via one-sided Jacobi.

Returns the economy SVD: r = min(m,n), U is mxr, S has r elements, Vt is rxn.

Parameters

A	Input matrix (not modified)
backend	Backend for column-update inner loops (Backend::omp parallelises)
tol	Convergence: stop when max \|col_p * col_q\| / (\|\|col_p\|\| \|\|col_q\|\|) < tol
max_sweeps	Safety cap on Jacobi sweeps

Definition at line 39 of file svd.cpp.

References beta(), e, ipow(), and zeta().

Referenced by svd_truncated().

◆ svd_truncated()

SVDResult num::svd_truncated	(	const Matrix &	A,
		idx	k,
		Backend	backend = `default_backend`,
		idx	oversampling = `10`,
		Rng *	rng = `nullptr`
	)

Randomized truncated SVD – top-k singular triplets.

Efficient when k << min(m,n). The two dominant costs – Y = A*Omega and B = Qᵀ*A – are dispatched via the given backend.

Parameters

A	Input matrix
k	Number of singular values/vectors to compute
backend	Backend for internal matmul calls (default: default_backend)
oversampling	Extra random vectors for accuracy (default 10)
rng	Random number generator (default: seeded from hardware)

Definition at line 143 of file svd.cpp.

References ipow(), matmul(), qr(), rng_normal(), and svd().

◆ thomas()

void num::thomas	(	const Vector &	a,
		const Vector &	b,
		const Vector &	c,
		const Vector &	d,
		Vector &	x,
		Backend	backend = `Backend::seq`
	)

Thomas algorithm (LU for tridiagonal systems), O(n).

Parameters

a	Sub-diagonal, size n-1
b	Main diagonal, size n
c	Super-diagonal, size n-1
d	Right-hand side vector, size n
x	Solution vector (output), size n
backend	Backend::seq (CPU) or Backend::gpu (CUDA batched, batch=1)

Definition at line 7 of file thomas.cpp.

References cg(), gpu, num::BasicVector< T >::gpu_data(), ipow(), num::BasicVector< T >::size(), num::cuda::thomas_batched(), num::BasicVector< T >::to_cpu(), and num::BasicVector< T >::to_gpu().

◆ trapz()

real num::trapz	(	ScalarFn	f,
		real	a,
		real	b,
		idx	n = `100`,
		Backend	backend = `Backend::seq`
	)

Trapezoidal rule with n panels.

Parameters

backend Backend::omp parallelises the panel sum

Definition at line 14 of file quadrature.cpp.

References ipow().

◆ zeta()

real num::zeta ( real x )

inline

zeta(x) – Riemann zeta function

Definition at line 233 of file math.hpp.

Referenced by svd().

Variable Documentation

◆ best_backend

constexpr Backend num::best_backend

inlineconstexpr

Initial value:

=
 
    Backend::blocked

Best backend for memory-bound vector ops: blas > omp > blocked.

Definition at line 65 of file policy.hpp.

◆ blas

constexpr Backend num::blas = Backend::blas

inlineconstexpr

Definition at line 33 of file policy.hpp.

◆ blocked

constexpr Backend num::blocked = Backend::blocked

inlineconstexpr

Definition at line 31 of file policy.hpp.

◆ default_backend

constexpr Backend num::default_backend

inlineconstexpr

Initial value:

=
 
    Backend::blocked

Automatically selected at configure time: blas > blocked. Used as the default parameter for all operations.

Definition at line 57 of file policy.hpp.

◆ e

constexpr real num::e = 2.71828182845904523536

constexpr

Definition at line 41 of file math.hpp.

Referenced by autocorr_time(), brent(), cg(), cg_matfree(), eig_sym(), eig_sym(), num::quantum::entanglement_entropy(), expv(), gauss_seidel(), gmres(), inverse_iteration(), jacobi(), lanczos(), lu(), lu_inv(), newton(), num::detail::normalise(), ode_euler(), ode_rk4(), ode_rk45(), ode_verlet(), ode_yoshida4(), power_iteration(), qr(), secant(), IsingLattice::set_temperature(), svd(), and IsingLattice::temperature().

◆ gpu

constexpr Backend num::gpu = Backend::gpu

inlineconstexpr

Definition at line 35 of file policy.hpp.

◆ half_pi

constexpr real num::half_pi = 1.57079632679489661923

constexpr

pi/2

Definition at line 48 of file math.hpp.

◆ has_blas

constexpr bool num::has_blas

inlineconstexpr

Initial value:

=
 
    false

True when a BLAS/cblas library was found at configure time.

Definition at line 39 of file policy.hpp.

◆ has_omp

constexpr bool num::has_omp

inlineconstexpr

Initial value:

=
 
    false

True when OpenMP was found at configure time.

Definition at line 47 of file policy.hpp.

◆ inv_pi

constexpr real num::inv_pi = 0.31830988618379067154

constexpr

1/pi

Definition at line 46 of file math.hpp.

◆ ln2

constexpr real num::ln2 = 0.69314718055994530942

constexpr

Definition at line 45 of file math.hpp.

◆ omp

constexpr Backend num::omp = Backend::omp

inlineconstexpr

Definition at line 34 of file policy.hpp.

◆ phi

constexpr real num::phi = 1.61803398874989484820

constexpr

Golden ratio.

Definition at line 42 of file math.hpp.

Referenced by num::MagneticSolver::current_density(), ellint_Ei(), ellint_F(), ellint_Pi_inc(), num::quantum::gate_u(), num::FieldSolver::gradient(), gradient_3d(), num::FieldSolver::solve_poisson(), and num::Circuit::u().

◆ pi

constexpr real num::pi = 3.14159265358979323846

constexpr

Definition at line 40 of file math.hpp.

Referenced by num::quantum::gate_t(), and num::quantum::gate_tdg().

◆ seq

constexpr Backend num::seq = Backend::seq

inlineconstexpr

Definition at line 30 of file policy.hpp.

◆ simd

constexpr Backend num::simd = Backend::simd

inlineconstexpr

Definition at line 32 of file policy.hpp.

◆ sqrt2

constexpr real num::sqrt2 = 1.41421356237309504880

constexpr

Definition at line 43 of file math.hpp.

◆ sqrt3

constexpr real num::sqrt3 = 1.73205080756887729353

constexpr

Definition at line 44 of file math.hpp.

◆ two_pi

constexpr real num::two_pi = 6.28318530717958647692

constexpr

2pi

Definition at line 47 of file math.hpp.

Namespaces

Classes

Typedefs

Enumerations

Functions

Variables

Typedef Documentation

◆ AccelFn

◆ cplx

◆ CVector

◆ idx

◆ MatVecFn

◆ ODERhsFn

◆ real

◆ ScalarFn

◆ StepCallback

◆ SymplecticCallback

◆ Vector

◆ VectorFn

Enumeration Type Documentation

◆ Backend

Function Documentation

◆ adaptive_simpson()

◆ add()

◆ assoc_laguerre()

◆ assoc_legendre()

◆ autocorr_time()

◆ axpy() [1/2]

◆ axpy() [2/2]

◆ banded_gemv()

◆ banded_lu()

◆ banded_lu_solve()

◆ banded_lu_solve_multi()

◆ banded_matvec()

◆ banded_norm1()

◆ banded_rcond()

◆ banded_solve()

◆ bell_pair()

◆ bessel_i()

◆ bessel_j()

◆ bessel_k()

◆ bessel_y()

◆ beta()

◆ bisection()

◆ brent()

◆ cg()

◆ cg_matfree()

◆ col_fiber_sweep()

◆ connected_components()

◆ curl_3d()

◆ divergence_3d()

◆ dot() [1/2]

◆ dot() [2/2]

◆ eig_sym() [1/2]

◆ eig_sym() [2/2]

◆ ellint_E()

◆ ellint_Ei()

◆ ellint_F()

◆ ellint_K()

◆ ellint_Pi()

◆ ellint_Pi_inc()

◆ expint()

◆ expv() [1/2]

◆ expv() [2/2]

◆ gauss_legendre()

◆ gauss_seidel()

◆ ghz_state()

◆ gmres() [1/3]

◆ gmres() [2/3]

◆ gmres() [3/3]

◆ gradient_3d()

◆ hermite()

◆ int_range()

◆ inverse_iteration()

◆ ipow()

◆ jacobi()

◆ laguerre()

◆ lanczos()

◆ laplacian_stencil_2d()

◆ laplacian_stencil_2d_periodic()