num Namespace Reference

Namespaces
namespace	backends
namespace	cuda
namespace	detail
namespace	kernel
namespace	markov
namespace	mpi
namespace	ode
namespace	operators
namespace	pde
namespace	plt
namespace	spectral
namespace	viz

Classes
struct	BackwardEuler
class	BandedMatrix
	LAPACK-style band storage. More...
struct	BandedSolverResult
class	BasicMatrix
	Dense row-major owning matrix. More...
class	BasicVector
	Dense owning vector. More...
class	CellList2D
class	CellList3D
struct	ClusterResult
struct	ComplexTriDiag
struct	CrankNicolsonADI
struct	EigenResult
	Symmetric eigendecomposition \(A=V\Lambda V^T\). More...
struct	Euler
class	EulerSteps
class	FieldSolver
struct	GivensRotation
class	Gnuplot
struct	Grid2D
class	Grid3D
struct	Histogram
	Fixed-bin histogram over \([\ell,h)\). More...
struct	IntRange
struct	LanczosResult
struct	LUResult
	Packed factorization \(PA=LU\). More...
class	MagneticSolver
struct	MCMCProblem
struct	Metropolis
struct	ODEParams
struct	ODEProblem
struct	ODEResult
struct	PBCLattice2D
	4-neighbor periodic-boundary index arrays for an NxN lattice. More...
struct	PowerResult
	Result of a single-eigenvalue iteration. More...
struct	QRResult
	QR factorization \(A=QR\). More...
struct	RK4
struct	RK45
class	RK45Steps
class	RK4_2ndSteps
class	RK4Steps
struct	Rng
	Seeded pseudo-random number generator (Mersenne Twister). Pass a pointer to rng_* functions to draw samples. More...
struct	RootResult
struct	RunningStats
	Welford updates for mean and variance. More...
class	ScalarField2D
class	ScalarField3D
struct	Series
struct	SmallMatrix
struct	SmallVec
struct	SolverResult
class	SparseMatrix
	Sparse matrix in Compressed Sparse Row (CSR) format. More...
struct	SPHKernel
struct	Step
struct	StepEnd
struct	SVDResult
struct	SymplecticResult
struct	SymplecticStep
struct	Vec2ConstView
struct	Vec2View
	Non-owning view of a flat vector as \((x_i,y_i)\) pairs. More...
struct	VectorField3D
class	VerletList2D
class	VerletSteps
class	Yoshida4Steps

Concepts
concept	VectorLike
	Indexed real-valued vector interface.
concept	MutableVectorLike
	Mutable indexed real-valued vector interface.
concept	ContiguousVectorLike
	Contiguous real-valued vector storage.
concept	DenseMatrixLike
	Dense row-major real matrix interface.
concept	MutableDenseMatrixLike
	Mutable dense row-major real matrix interface.
concept	ContiguousDenseMatrixLike
	Contiguous dense row-major real matrix storage.
concept	VecField
concept	IsODEProblem
concept	IsExplicitODEAlg
concept	IsImplicitODEAlg
concept	IsMCMCAlg

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	Matrix = BasicMatrix< real >
	Double-precision dense matrix with full backend dispatch (CPU + GPU).
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	LinearSolver = std::function< SolverResult(const Vector &rhs, Vector &x)>
	Callable that solves \(Ax=\mathrm{rhs}\).
using	ODERhsFn = std::function< void(real t, const Vector &y, Vector &dydt)>
using	AccelFn = std::function< void(const Vector &q, Vector &acc)>
using	ObserverFn = std::function< void(real t, const Vector &y)>
using	SympObserverFn = std::function< void(real t, const Vector &q, const Vector &v)>
using	Point = std::pair< double, double >

Enumerations
enum class	Backend {   seq , blocked , simd , blas ,   omp , gpu , lapack }

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 = alpha*A + beta*B.
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<class T >
constexpr idx	to_idx (T x) noexcept
	Cast any integer to idx without a verbose static_cast.
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).
real	gaussian2d (real x, real y, real cx, real cy, real sigma)
	2D isotropic Gaussian centred at \((c_x, c_y)\) with width \(\sigma\): \(\exp\!\bigl(-\tfrac{(x-c_x)^2+(y-c_y)^2}{2\sigma^2}\bigr)\)
void	scale (Vector &v, real alpha, Backend b=default_backend)
	Compute \(v \leftarrow \alpha v\).
void	add (const Vector &x, const Vector &y, Vector &z, Backend b=default_backend)
	Compute \(z=x+y\).
void	axpy (real alpha, const Vector &x, Vector &y, Backend b=default_backend)
	Compute \(y \leftarrow y+\alpha x\).
real	dot (const Vector &x, const Vector &y, Backend b=default_backend)
	Compute \(x^T y\).
real	norm (const Vector &x, Backend b=default_backend)
	Compute \(\|x\|_2\).
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)
	In-place banded \(PA=LU\) factorization.
void	banded_lu_solve (const BandedMatrix &A, const idx *ipiv, Vector &b)
	Solve \(Ax=b\) using a precomputed banded LU factorization.
void	banded_lu_solve_multi (const BandedMatrix &A, const idx *ipiv, real *B, idx nrhs)
	Solve \(AX=B\) using a precomputed banded LU factorization.
BandedSolverResult	banded_solve (const BandedMatrix &A, const Vector &b, Vector &x)
	Factor and solve \(Ax=b\).
void	banded_matvec (const BandedMatrix &A, const Vector &x, Vector &y, Backend backend=default_backend)
	Compute \(y=Ax\).
void	banded_gemv (real alpha, const BandedMatrix &A, const Vector &x, real beta, Vector &y, Backend backend=default_backend)
	Compute \(y=\alpha Ax+\beta y\).
real	banded_rcond (const BandedMatrix &A, const idx *ipiv, real anorm)
	Estimate \(1/\kappa_1(A)\).
real	banded_norm1 (const BandedMatrix &A)
	Compute \(\|A\|_1\).
EigenResult	eig_sym (const Matrix &A, real tol=1e-12, idx max_sweeps=100, Backend backend=lapack_backend)
template<class Op > requires operators::LinearOperator<Op, Vector, Vector>
LanczosResult	lanczos (const Op &A, idx k, real tol=1e-10, idx max_steps=0, Backend backend=Backend::seq)
	Operator Lanczos for any symmetric \(y=A x\) adapter.
LanczosResult	lanczos (const Matrix &A, idx k, real tol=1e-10, idx max_steps=0, Backend backend=Backend::seq)
LanczosResult	lanczos (const SparseMatrix &A, idx k, real tol=1e-10, idx max_steps=0, Backend backend=Backend::seq)
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.
template<class Op > requires operators::LinearOperator<Op, Vector, Vector>
Vector	expv (real t, const Op &A, const Vector &v, int m_max=30, real tol=1e-8)
	Compute \(\exp(tA)v\) for any \(y=A x\) adapter.
Vector	expv (real t, const SparseMatrix &A, const Vector &v, int m_max=30, real tol=1e-8)
LUResult	lu (const Matrix &A, Backend backend=lapack_backend)
void	lu_solve (const LUResult &f, const Vector &b, Vector &x)
	Solve \(Ax=b\) from a precomputed \(PA=LU\) factorization.
void	lu_solve (const LUResult &f, const Matrix &B, Matrix &X)
	Solve \(AX=B\) from a precomputed \(PA=LU\) factorization.
real	lu_det (const LUResult &f)
	Compute \(\det(A)=\det(P)^{-1}\prod_i U_{ii}\).
Matrix	lu_inv (const LUResult &f)
	Compute \(A^{-1}\) by solving \(AX=I\).
QRResult	qr (const Matrix &A, Backend backend=lapack_backend)
	Factor \(A\in\mathbb{R}^{m\times n}\) as \(A=QR\).
void	qr_solve (const QRResult &f, const Vector &b, Vector &x)
	Solve \(\min_x \|Ax-b\|_2\).
void	thomas (const Vector &a, const Vector &b, const Vector &c, const Vector &d, Vector &x, Backend backend=lapack_backend)
SolverResult	cg (const Matrix &A, const Vector &b, Vector &x, real tol=1e-10, idx max_iter=1000, Backend backend=default_backend)
template<class Op > requires operators::LinearOperator<Op, Vector, Vector>
SolverResult	cg (const Op &A, const Vector &b, Vector &x, real tol=1e-10, idx max_iter=1000, Backend backend=default_backend)
	Operator CG for any \(y=A x\) adapter.
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.
template<class Op > requires operators::LinearOperator<Op, Vector, Vector>
SolverResult	gmres (const Op &A, const Vector &b, Vector &x, real tol=1e-6, idx max_iter=1000, idx restart=30)
	Operator GMRES for any \(y=A x\) adapter.
SolverResult	gmres (const SparseMatrix &A, const Vector &b, Vector &x, real tol=1e-6, idx max_iter=1000, idx restart=30)
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)
void	sparse_matvec (const SparseMatrix &A, const Vector &x, Vector &y)
	y = A * x
SVDResult	svd (const Matrix &A, Backend backend=lapack_backend, real tol=1e-12, idx max_sweeps=100)
SVDResult	svd_truncated (const Matrix &A, idx k, Backend backend=default_backend, idx oversampling=10, Rng *rng=nullptr)
EulerSteps	euler (ODERhsFn f, Vector y0, ODEParams p={})
RK4Steps	rk4 (ODERhsFn f, Vector y0, ODEParams p={})
RK45Steps	rk45 (ODERhsFn f, Vector y0, ODEParams p={})
VerletSteps	verlet (AccelFn accel, Vector q0, Vector v0, ODEParams p={})
Yoshida4Steps	yoshida4 (AccelFn accel, Vector q0, Vector v0, ODEParams p={})
RK4_2ndSteps	rk4_2nd (AccelFn accel, Vector q0, Vector v0, ODEParams p={})
ODEResult	ode_euler (ODERhsFn f, Vector y0, ODEParams p={}, ObserverFn obs=nullptr)
	Forward Euler, 1st-order, fixed step.
ODEResult	ode_rk4 (ODERhsFn f, Vector y0, ODEParams p={}, ObserverFn obs=nullptr)
	Classic 4th-order Runge-Kutta, fixed step.
ODEResult	ode_rk45 (ODERhsFn f, Vector y0, ODEParams p={}, ObserverFn obs=nullptr)
	Adaptive Dormand-Prince RK45 with FSAL and PI step-size control.
SymplecticResult	ode_verlet (AccelFn accel, Vector q0, Vector v0, ODEParams p={}, SympObserverFn obs=nullptr)
	Velocity Verlet, 2nd-order symplectic, 1 force evaluation per step.
SymplecticResult	ode_yoshida4 (AccelFn accel, Vector q0, Vector v0, ODEParams p={}, SympObserverFn obs=nullptr)
	Yoshida 4th-order symplectic, 3 force evaluations per step.
SymplecticResult	ode_rk4_2nd (AccelFn accel, Vector q0, Vector v0, ODEParams p={}, SympObserverFn obs=nullptr)
	RK4 for second-order systems q'' = accel(q), Nystrom form.
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)
	Periodic second-order 2D Laplacian stencil.
template<typename T >
void	laplacian_stencil_2d_4th (const BasicVector< T > &x, BasicVector< T > &y, int N)
	Fourth-order 2D Laplacian cross stencil.
real	sample_2d_periodic (const Vector &field, idx N, real h, real px, real py, real ox, real oy)
template<typename T , typename F >
void	col_fiber_sweep (BasicVector< T > &data, int N, F &&f)
	Apply a mutable 1D operation to each column fiber.
template<typename T , typename F >
void	row_fiber_sweep (BasicVector< T > &data, int N, F &&f)
	Apply a mutable 1D operation to each row fiber.
template<typename F >
void	fill_grid (Vector &u, int N, double h, F &&f)
	Fill grid values at \(x_i=(i+1)h,\ y_j=(j+1)h\).
Series	row_slice (const Vector &u, int N, double h, int row)
	Extract one row as \((x_j,u_j)\) plot data.
Series	col_slice (const Vector &u, int N, double h, int col)
	Extract one column as \((y_i,u_i)\) plot data.
template<typename F >
void	fill_grid (ScalarField2D &g, F &&f)
Series	row_slice (const ScalarField2D &g, int row)
Series	col_slice (const ScalarField2D &g, int col)
void	laplacian_stencil_2d_periodic (const ScalarField2D &x, ScalarField2D &y)
void	laplacian_stencil_2d_4th (const ScalarField2D &x, ScalarField2D &y)
real	sample_2d_periodic (const ScalarField2D &g, real px, real py, real ox=0.0, real oy=0.0)
void	neg_laplacian_3d (const Vector &x, Vector &y, int nx, int ny, int nz, double inv_dx2)
	Compute \(-\Delta_h x\) on a 3D grid.
void	gradient_3d (const Grid3D &phi, Grid3D &gx, Grid3D &gy, Grid3D &gz)
	Compute \(\nabla\phi\) with central differences.
void	divergence_3d (const Grid3D &fx, const Grid3D &fy, const Grid3D &fz, Grid3D &out)
	Compute \(\nabla\cdot f\) with central differences.
void	curl_3d (const Grid3D &ax, const Grid3D &ay, const Grid3D &az, Grid3D &bx, Grid3D &by, Grid3D &bz)
	Compute \(\nabla\times A\) with central differences.
void	apply_siam_style (Gnuplot &gp)
	Apply SIAM-style theme to a raw Gnuplot pipe.
void	set_loglog (Gnuplot &gp)
void	set_logx (Gnuplot &gp)
void	save_png (Gnuplot &gp, const std::string &filename, int w=900, int h=600)
template<IsODEProblem P>
ODEResult	solve (const P &prob, const RK45 &alg, ObserverFn obs=nullptr)
template<IsODEProblem P>
ODEResult	solve (const P &prob, const RK4 &alg, ObserverFn obs=nullptr)
template<IsODEProblem P>
ODEResult	solve (const P &prob, const Euler &alg, ObserverFn obs=nullptr)
template<VecField F>
void	solve (F &u, const BackwardEuler &alg)
template<VecField F, typename Observer >
void	solve (F &u, const BackwardEuler &alg, Observer &&obs)
template<IsMCMCAlg A, typename MeasureFn , typename RNG >
double	solve (const MCMCProblem &prob, const A &alg, MeasureFn &&measure, RNG &rng)
	Run equilibration and measurement sweeps, then return the mean.
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)

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 Backend	lapack = Backend::lapack
constexpr bool	has_blas
constexpr bool	has_lapack
constexpr bool	has_omp
constexpr bool	has_simd
constexpr Backend	default_backend
constexpr Backend	best_backend = default_backend
constexpr Backend	lapack_backend
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)>

Definition at line 13 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 132 of file vector.hpp.

idx

using num::idx = typedef std::size_t

Definition at line 11 of file types.hpp.

LinearSolver

using num::LinearSolver = typedef std::function<SolverResult(const Vector& rhs, Vector& x)>

Callable that solves \(Ax=\mathrm{rhs}\).

Definition at line 12 of file linear_solver.hpp.

Matrix

using num::Matrix = typedef BasicMatrix<real>

Double-precision dense matrix with full backend dispatch (CPU + GPU).

Definition at line 127 of file matrix.hpp.

ObserverFn

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

Definition at line 14 of file ode.hpp.

ODERhsFn

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

Definition at line 12 of file ode.hpp.

Point

using num::Point = typedef std::pair<double, double>

Definition at line 13 of file plot.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.

SympObserverFn

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

Definition at line 15 of file ode.hpp.

Vector

using num::Vector = typedef BasicVector<real>

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

Definition at line 129 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

Enumerator
seq
blocked
simd
blas
omp
gpu
lapack

Definition at line 7 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 103 of file quadrature.cpp.

add()

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

Compute \(z=x+y\).

Definition at line 36 of file vector.cpp.

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

apply_siam_style()

void num::apply_siam_style ( Gnuplot &  gp )

inline

Apply SIAM-style theme to a raw Gnuplot pipe.

Definition at line 52 of file plot.hpp.

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 160 of file math.hpp.

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 124 of file math.hpp.

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.

axpy() [1/2]

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

y += alpha * x

Definition at line 106 of file vector.cpp.

References num::BasicVector< T >::size().

axpy() [2/2]

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

Compute \(y \leftarrow y+\alpha x\).

Definition at line 44 of file vector.cpp.

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

Referenced by cg(), cg(), num::pde::diffusion_step_2d(), num::pde::diffusion_step_2d_4th_dirichlet(), num::pde::diffusion_step_2d_dirichlet(), expv(), lanczos(), and num::kernel::subspace::mgs_orthogonalize().

banded_gemv()

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

Compute \(y=\alpha Ax+\beta y\).

Definition at line 304 of file banded.cpp.

References beta(), num::BasicVector< T >::data(), num::BandedMatrix::data(), gpu, num::BandedMatrix::kl(), num::BandedMatrix::ku(), num::BandedMatrix::ldab(), num::BandedMatrix::size(), and num::BasicVector< T >::size().

Referenced by banded_matvec().

banded_lu()

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

In-place banded \(PA=LU\) factorization.

Definition at line 135 of file banded.cpp.

References num::BandedMatrix::data(), num::BandedMatrix::kl(), num::BandedMatrix::ku(), num::BandedMatrix::ldab(), and num::BandedMatrix::size().

Referenced by banded_solve().

banded_lu_solve()

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

Solve \(Ax=b\) using a precomputed banded LU factorization.

Definition at line 207 of file banded.cpp.

References num::BasicVector< T >::data(), num::BandedMatrix::data(), num::BandedMatrix::kl(), num::BandedMatrix::ku(), num::BandedMatrix::ldab(), num::BandedMatrix::size(), and num::BasicVector< T >::size().

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 \(AX=B\) using a precomputed banded LU factorization.

Definition at line 247 of file banded.cpp.

References num::BandedMatrix::data(), num::BandedMatrix::kl(), num::BandedMatrix::ku(), num::BandedMatrix::ldab(), and num::BandedMatrix::size().

banded_matvec()

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

Compute \(y=Ax\).

Definition at line 300 of file banded.cpp.

References banded_gemv().

banded_norm1()

real num::banded_norm1

(

const BandedMatrix &

A

)

Compute \(\|A\|_1\).

Definition at line 348 of file banded.cpp.

References num::BandedMatrix::data(), num::BandedMatrix::kl(), num::BandedMatrix::ku(), num::BandedMatrix::ldab(), and num::BandedMatrix::size().

banded_rcond()

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

Estimate \(1/\kappa_1(A)\).

Definition at line 364 of file banded.cpp.

References banded_lu_solve(), and num::BandedMatrix::size().

banded_solve()

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

Factor and solve \(Ax=b\).

Definition at line 281 of file banded.cpp.

References banded_lu(), banded_lu_solve(), num::BandedMatrix::size(), num::BasicVector< T >::size(), and num::BandedSolverResult::success.

bessel_i()

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

inline

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

Definition at line 74 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 56 of file math.hpp.

bessel_k()

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

inline

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

Definition at line 83 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 65 of file math.hpp.

beta()

real num::beta ( real  a, real  b  )

inline

B(a, b) – beta function.

Definition at line 248 of file math.hpp.

Referenced by num::kernel::subspace::arnoldi_step(), banded_gemv(), num::markov::boltzmann_accept(), cg(), cg(), 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(), num::backends::seq::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.

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 61 of file roots.cpp.

References e.

cg() [1/2]

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

Definition at line 8 of file cg.cpp.

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

Referenced by num::pde::make_cg_solver(), num::FieldSolver::solve_poisson(), num::FieldSolver::solve_var_poisson(), and thomas().

cg() [2/2]

template<class Op >
requires operators::LinearOperator<Op, Vector, Vector>

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

Operator CG for any \(y=A x\) adapter.

Definition at line 27 of file cg.hpp.

References axpy(), beta(), dot(), e, num::SolverResult::iterations, scale(), and num::BasicVector< T >::size().

col_fiber_sweep()

template<typename T , typename F >

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

Apply a mutable 1D operation to each column fiber.

Definition at line 141 of file stencil.hpp.

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

col_slice() [1/2]

Series num::col_slice ( const ScalarField2D &  g, int  col  )

inline

Definition at line 205 of file stencil.hpp.

References col_slice(), num::ScalarField2D::h(), num::ScalarField2D::N(), and num::ScalarField2D::vec().

col_slice() [2/2]

Series num::col_slice ( const Vector &  u, int  N, double  h, int  col  )

inline

Extract one column as \((y_i,u_i)\) plot data.

Definition at line 187 of file stencil.hpp.

References num::Series::store().

Referenced by col_slice().

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 38 of file connected_components.hpp.

References num::ClusterResult::id, num::ClusterResult::largest_id, num::ClusterResult::largest_size, and num::ClusterResult::sizes.

curl_3d()

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

inline

Compute \(\nabla\times A\) with central differences.

Definition at line 294 of file stencil.hpp.

References num::Grid3D::dx(), 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 \(\nabla\cdot f\) with central differences.

Definition at line 274 of file stencil.hpp.

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

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 111 of file vector.cpp.

References num::BasicVector< T >::size().

dot() [2/2]

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

Compute \(x^T y\).

Definition at line 65 of file vector.cpp.

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

Referenced by cg(), cg(), num::mpi::dot(), inverse_iteration(), lanczos(), num::kernel::subspace::mgs_orthogonalize(), num::mpi::norm(), power_iteration(), and rayleigh_iteration().

eig_sym()

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

Definition at line 11 of file eig.cpp.

References num::backends::lapack::eig_sym(), num::backends::omp::eig_sym(), num::backends::seq::eig_sym(), lapack, and omp.

Referenced by lanczos().

ellint_E()

real num::ellint_E ( real  k )

inline

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

Definition at line 182 of file math.hpp.

ellint_Ei()

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

inline

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

Definition at line 209 of file math.hpp.

References 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 200 of file math.hpp.

References phi.

ellint_K()

real num::ellint_K ( real  k )

inline

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

Definition at line 173 of file math.hpp.

ellint_Pi()

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

inline

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

Definition at line 191 of file math.hpp.

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 218 of file math.hpp.

References phi.

euler()

EulerSteps num::euler	(	ODERhsFn	f,
		Vector	y0,
		ODEParams	p = {}
	)

Definition at line 372 of file ode.cpp.

Referenced by ode_euler().

expint()

real num::expint ( real  x )

inline

Ei(x) – exponential integral.

Definition at line 230 of file math.hpp.

expv() [1/2]

template<class Op >
requires operators::LinearOperator<Op, Vector, Vector>

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

Compute \(\exp(tA)v\) for any \(y=A x\) adapter.

Definition at line 27 of file expv.hpp.

References axpy(), beta(), num::detail::dense_expm_pade6(), e, num::kernel::subspace::mgs_orthogonalize(), norm(), scale(), and num::BasicVector< T >::size().

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
	)

Definition at line 95 of file expv.cpp.

References expv().

fill_grid() [1/2]

template<typename F >

void num::fill_grid	(	ScalarField2D &	g,
		F &&	f
	)

Definition at line 197 of file stencil.hpp.

References fill_grid(), num::ScalarField2D::h(), num::ScalarField2D::N(), and num::ScalarField2D::vec().

fill_grid() [2/2]

template<typename F >

void num::fill_grid	(	Vector &	u,
		int	N,
		double	h,
		F &&	f
	)

Fill grid values at \(x_i=(i+1)h,\ y_j=(j+1)h\).

Definition at line 167 of file stencil.hpp.

Referenced by fill_grid().

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 63 of file quadrature.cpp.

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 num::BasicMatrix< T >::cols(), e, num::BasicMatrix< T >::rows(), and num::BasicVector< T >::size().

gaussian2d()

real num::gaussian2d ( real  x, real  y, real  cx, real  cy, real  sigma  )

inline

2D isotropic Gaussian centred at \((c_x, c_y)\) with width \(\sigma\): \(\exp\!\bigl(-\tfrac{(x-c_x)^2+(y-c_y)^2}{2\sigma^2}\bigr)\)

Definition at line 312 of file math.hpp.

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
	)

Definition at line 22 of file krylov.cpp.

References num::BasicMatrix< T >::cols(), gmres(), and num::BasicMatrix< T >::rows().

gmres() [2/3]

template<class Op >
requires operators::LinearOperator<Op, Vector, Vector>

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

Operator GMRES for any \(y=A x\) adapter.

Definition at line 24 of file krylov.hpp.

References beta(), e, num::kernel::subspace::mgs_orthogonalize(), norm(), scale(), and num::BasicVector< T >::size().

Referenced by gmres(), and gmres().

gmres() [3/3]

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

Definition at line 8 of file krylov.cpp.

References gmres(), num::SparseMatrix::n_cols(), and num::SparseMatrix::n_rows().

gradient_3d()

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

inline

Compute \(\nabla\phi\) with central differences.

Definition at line 258 of file stencil.hpp.

References 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 142 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 270 of file math.hpp.

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 46 of file power.cpp.

References num::BasicMatrix< T >::cols(), dot(), e, lu(), lu_solve(), matvec(), num::detail::normalise(), and num::BasicMatrix< T >::rows().

ipow()

template<int N, typename T >

constexpr T num::ipow ( T  x )

constexprnoexcept

Compute x^N at compile time via repeated squaring.

Definition at line 9 of file integer_pow.hpp.

References ipow().

Referenced by 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 num::BasicMatrix< T >::cols(), e, num::BasicMatrix< T >::rows(), and num::BasicVector< T >::size().

laguerre()

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

inline

L_n(x) – Laguerre polynomial.

Definition at line 151 of file math.hpp.

lanczos() [1/3]

LanczosResult num::lanczos	(	const Matrix &	A,
		idx	k,
		real	tol = 1e-10,
		idx	max_steps = 0,
		Backend	backend = Backend::seq
	)

Definition at line 10 of file lanczos.cpp.

References num::BasicMatrix< T >::cols(), lanczos(), and num::BasicMatrix< T >::rows().

lanczos() [2/3]

template<class Op >
requires operators::LinearOperator<Op, Vector, Vector>

LanczosResult num::lanczos	(	const Op &	A,
		idx	k,
		real	tol = 1e-10,
		idx	max_steps = 0,
		Backend	backend = Backend::seq
	)

Operator Lanczos for any symmetric \(y=A x\) adapter.

Definition at line 31 of file lanczos.hpp.

References axpy(), beta(), dot(), e, eig_sym(), num::kernel::subspace::mgs_orthogonalize(), num::EigenResult::values, and num::EigenResult::vectors.

Referenced by lanczos(), and lanczos().

lanczos() [3/3]

LanczosResult num::lanczos	(	const SparseMatrix &	A,
		idx	k,
		real	tol = 1e-10,
		idx	max_steps = 0,
		Backend	backend = Backend::seq
	)

Definition at line 18 of file lanczos.cpp.

References lanczos(), num::SparseMatrix::n_cols(), and num::SparseMatrix::n_rows().

laplacian_stencil_2d()

template<typename T >

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

Definition at line 22 of file stencil.hpp.

Referenced by num::pde::diffusion_step_2d_dirichlet().

laplacian_stencil_2d_4th() [1/2]

template<typename T >

void num::laplacian_stencil_2d_4th	(	const BasicVector< T > &	x,
		BasicVector< T > &	y,
		int	N
	)

Fourth-order 2D Laplacian cross stencil.

\[ y_{ij} = \frac{1}{12}\bigl( -x_{i-2,j} + 16x_{i-1,j} - 30x_{i,j} + 16x_{i+1,j} - x_{i+2,j} -x_{i,j-2} + 16x_{i,j-1} + 16x_{i,j+1} - x_{i,j+2} \bigr) \]

Definition at line 66 of file stencil.hpp.

Referenced by num::pde::diffusion_step_2d_4th_dirichlet(), and laplacian_stencil_2d_4th().

laplacian_stencil_2d_4th() [2/2]

void num::laplacian_stencil_2d_4th ( const ScalarField2D &  x, ScalarField2D &  y  )

inline

Definition at line 213 of file stencil.hpp.

References laplacian_stencil_2d_4th(), num::ScalarField2D::N(), and num::ScalarField2D::vec().

laplacian_stencil_2d_periodic() [1/2]

template<typename T >

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

Periodic second-order 2D Laplacian stencil.

Definition at line 42 of file stencil.hpp.

References num::BasicVector< T >::data().

Referenced by num::pde::diffusion_step_2d(), and laplacian_stencil_2d_periodic().

laplacian_stencil_2d_periodic() [2/2]

void num::laplacian_stencil_2d_periodic ( const ScalarField2D &  x, ScalarField2D &  y  )

inline

Definition at line 209 of file stencil.hpp.

References laplacian_stencil_2d_periodic(), num::ScalarField2D::N(), and num::ScalarField2D::vec().

legendre()

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

inline

P_n(x) – Legendre polynomial of degree n.

Definition at line 115 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 260 of file math.hpp.

lu()

LUResult num::lu	(	const Matrix &	A,
		Backend	backend = lapack_backend
	)

Definition at line 10 of file lu.cpp.

References lapack, num::backends::lapack::lu(), and num::backends::seq::lu().

Referenced by num::detail::dense_expm_pade6(), inverse_iteration(), and rayleigh_iteration().

lu_det()

real num::lu_det

(

const LUResult &

f

)

Compute \(\det(A)=\det(P)^{-1}\prod_i U_{ii}\).

Definition at line 57 of file lu.cpp.

References num::LUResult::LU, num::LUResult::piv, and num::BasicMatrix< T >::rows().

lu_inv()

Matrix num::lu_inv

(

const LUResult &

f

)

Compute \(A^{-1}\) by solving \(AX=I\).

Definition at line 71 of file lu.cpp.

References e, num::LUResult::LU, lu_solve(), and num::BasicMatrix< T >::rows().

lu_solve() [1/2]

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

Solve \(AX=B\) from a precomputed \(PA=LU\) factorization.

Definition at line 42 of file lu.cpp.

References num::BasicMatrix< T >::cols(), lu_solve(), and num::BasicMatrix< T >::rows().

lu_solve() [2/2]

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

Solve \(Ax=b\) from a precomputed \(PA=LU\) factorization.

Definition at line 19 of file lu.cpp.

References num::LUResult::LU, num::LUResult::piv, and num::BasicMatrix< T >::rows().

Referenced by num::detail::dense_expm_pade6(), 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 68 of file matrix.cpp.

References beta(), blas, blocked, gpu, lapack, 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 20 of file matrix.cpp.

References blas, blocked, gpu, lapack, 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 num::detail::dense_expm_pade6(), and 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 94 of file matrix.cpp.

References 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 98 of file matrix.cpp.

References 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 106 of file matrix.cpp.

References 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 45 of file matrix.cpp.

References blas, blocked, gpu, lapack, 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 num::operators::DenseOp::apply(), cg(), inverse_iteration(), power_iteration(), rayleigh_iteration(), num::FieldSolver::solve_poisson(), and num::FieldSolver::solve_var_poisson().

matvec_simd()

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

y = A * x (SIMD-accelerated)

Definition at line 110 of file matrix.cpp.

References 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 \(-\Delta_h x\) on a 3D grid.

Definition at line 232 of file stencil.hpp.

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 30 of file roots.cpp.

References e.

norm() [1/2]

real num::norm

(

const CVector &

x

)

Euclidean norm sqrt(Sigma |v_i|^2)

Definition at line 118 of file vector.cpp.

References num::BasicVector< T >::size().

norm() [2/2]

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

Compute \(\|x\|_2\).

Definition at line 83 of file vector.cpp.

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

Referenced by expv(), gmres(), num::kernel::subspace::mgs_orthogonalize(), num::kernel::subspace::mgs_orthogonalize(), and num::Histogram::pdf().

ode_euler()

ODEResult num::ode_euler	(	ODERhsFn	f,
		Vector	y0,
		ODEParams	p = {},
		ObserverFn	obs = nullptr
	)

Forward Euler, 1st-order, fixed step.

Definition at line 392 of file ode.cpp.

References euler().

Referenced by solve().

ode_rk4()

ODEResult num::ode_rk4	(	ODERhsFn	f,
		Vector	y0,
		ODEParams	p = {},
		ObserverFn	obs = nullptr
	)

Classic 4th-order Runge-Kutta, fixed step.

Definition at line 399 of file ode.cpp.

References rk4().

Referenced by solve().

ode_rk45()

ODEResult num::ode_rk45	(	ODERhsFn	f,
		Vector	y0,
		ODEParams	p = {},
		ObserverFn	obs = nullptr
	)

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

Definition at line 406 of file ode.cpp.

References rk45().

Referenced by solve().

ode_rk4_2nd()

SymplecticResult num::ode_rk4_2nd	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		ODEParams	p = {},
		SympObserverFn	obs = nullptr
	)

RK4 for second-order systems q'' = accel(q), Nystrom form.

Note

Not symplectic. Prefer ode_verlet or ode_yoshida4 for long Hamiltonian runs.

Definition at line 435 of file ode.cpp.

References rk4_2nd().

ode_verlet()

SymplecticResult num::ode_verlet	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		ODEParams	p = {},
		SympObserverFn	obs = nullptr
	)

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

Definition at line 413 of file ode.cpp.

References verlet().

ode_yoshida4()

SymplecticResult num::ode_yoshida4	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		ODEParams	p = {},
		SympObserverFn	obs = nullptr
	)

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

Definition at line 424 of file ode.cpp.

References yoshida4().

operator*()

template<idx N>

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

constexprnoexcept

Definition at line 57 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 52 of file small_matrix.hpp.

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 10 of file power.cpp.

References num::BasicMatrix< T >::cols(), dot(), e, matvec(), num::detail::normalise(), and num::BasicMatrix< T >::rows().

qr()

QRResult num::qr	(	const Matrix &	A,
		Backend	backend = lapack_backend
	)

Factor \(A\in\mathbb{R}^{m\times n}\) as \(A=QR\).

Definition at line 10 of file qr.cpp.

References lapack, num::backends::lapack::qr(), and num::backends::seq::qr().

Referenced by svd_truncated().

qr_solve()

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

Solve \(\min_x \|Ax-b\|_2\).

Definition at line 19 of file qr.cpp.

References num::BasicMatrix< T >::cols(), num::QRResult::Q, num::QRResult::R, and num::BasicMatrix< T >::rows().

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 = v^T*A*v at every step -> fresh LU each iteration.

Parameters

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

Definition at line 95 of file power.cpp.

References num::BasicMatrix< T >::cols(), dot(), lu(), lu_solve(), matvec(), num::detail::normalise(), num::BasicMatrix< T >::rows(), num::LUResult::singular, and num::BasicVector< T >::size().

rk4()

RK4Steps num::rk4	(	ODERhsFn	f,
		Vector	y0,
		ODEParams	p = {}
	)

Definition at line 375 of file ode.cpp.

Referenced by ode_rk4().

rk45()

RK45Steps num::rk45	(	ODERhsFn	f,
		Vector	y0,
		ODEParams	p = {}
	)

Definition at line 378 of file ode.cpp.

Referenced by ode_rk45().

rk4_2nd()

RK4_2ndSteps num::rk4_2nd	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		ODEParams	p = {}
	)

Definition at line 388 of file ode.cpp.

Referenced by ode_rk4_2nd().

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 303 of file math.hpp.

References num::Rng::engine.

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 298 of file math.hpp.

References num::Rng::engine.

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 293 of file math.hpp.

References num::Rng::engine.

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 109 of file quadrature.cpp.

row_fiber_sweep()

template<typename T , typename F >

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

Apply a mutable 1D operation to each row fiber.

Definition at line 154 of file stencil.hpp.

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

row_slice() [1/2]

Series num::row_slice ( const ScalarField2D &  g, int  row  )

inline

Definition at line 201 of file stencil.hpp.

References num::ScalarField2D::h(), num::ScalarField2D::N(), row_slice(), and num::ScalarField2D::vec().

row_slice() [2/2]

Series num::row_slice ( const Vector &  u, int  N, double  h, int  row  )

inline

Extract one row as \((x_j,u_j)\) plot data.

Definition at line 177 of file stencil.hpp.

References num::Series::store().

Referenced by row_slice().

sample_2d_periodic() [1/2]

real num::sample_2d_periodic ( const ScalarField2D &  g, real  px, real  py, real  ox = 0.0, real  oy = 0.0  )

inline

Definition at line 217 of file stencil.hpp.

References num::ScalarField2D::h(), num::ScalarField2D::N(), sample_2d_periodic(), and num::ScalarField2D::vec().

sample_2d_periodic() [2/2]

real num::sample_2d_periodic ( const Vector &  field, idx  N, real  h, real  px, real  py, real  ox, real  oy  )

inline

Bilinear interpolation on a periodic NxN grid with configurable stagger offset.

field[i,j] is defined at physical position ((i + ox/h)*h, (j + oy/h)*h). Returns the interpolated field value at physical point (px, py).

Parameters

ox	x-axis origin offset in physical units (0 for unstaggered, h/2 for v-face)
oy	y-axis origin offset in physical units (0 for unstaggered, h/2 for u-face)

MAC grid usage:

\[ \text{interp\_u}(px,py) = \texttt{sample\_2d\_periodic}(u, N, h,\; px, py,\; 0,\; h/2) \]

\[ \text{interp\_v}(px,py) = \texttt{sample\_2d\_periodic}(v, N, h,\; px, py,\; h/2,\; 0) \]

Definition at line 116 of file stencil.hpp.

Referenced by sample_2d_periodic().

save_png()

void num::save_png ( Gnuplot &  gp, const std::string &  filename, int  w = 900, int  h = 600  )

inline

Definition at line 73 of file plot.hpp.

scale() [1/2]

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

v *= alpha

Definition at line 101 of file vector.cpp.

References num::BasicVector< T >::size().

scale() [2/2]

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

Compute \(v \leftarrow \alpha v\).

Definition at line 15 of file vector.cpp.

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

Referenced by num::kernel::subspace::arnoldi_step(), cg(), cg(), num::detail::dense_expm_pade6(), expv(), gmres(), and num::VectorField3D::scale().

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 44 of file roots.cpp.

References e.

set_loglog()

void num::set_loglog ( Gnuplot &  gp )

inline

Definition at line 67 of file plot.hpp.

set_logx()

void num::set_logx ( Gnuplot &  gp )

inline

Definition at line 70 of file plot.hpp.

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 26 of file quadrature.cpp.

solve() [1/6]

template<IsMCMCAlg A, typename MeasureFn , typename RNG >

double num::solve	(	const MCMCProblem &	prob,
		const A &	alg,
		MeasureFn &&	measure,
		RNG &	rng
	)

Run equilibration and measurement sweeps, then return the mean.

Definition at line 71 of file solve.hpp.

References num::MCMCProblem::accept_prob, num::markov::metropolis_sweep_prob(), num::MCMCProblem::n_sites, and num::MCMCProblem::propose.

solve() [2/6]

template<IsODEProblem P>

ODEResult num::solve	(	const P &	prob,
		const Euler &	alg,
		ObserverFn	obs = nullptr
	)

Definition at line 55 of file solve.hpp.

References ode_euler().

solve() [3/6]

template<IsODEProblem P>

ODEResult num::solve	(	const P &	prob,
		const RK4 &	alg,
		ObserverFn	obs = nullptr
	)

Definition at line 50 of file solve.hpp.

References ode_rk4().

solve() [4/6]

template<IsODEProblem P>

ODEResult num::solve	(	const P &	prob,
		const RK45 &	alg,
		ObserverFn	obs = nullptr
	)

Definition at line 39 of file solve.hpp.

References num::RK45::atol, num::RK45::h, num::RK45::max_steps, ode_rk45(), num::RK45::rtol, and num::ODEParams::t0.

solve() [5/6]

template<VecField F>

void num::solve	(	F &	u,
		const BackwardEuler &	alg
	)

Definition at line 60 of file solve.hpp.

References num::ode::advance(), and num::BackwardEuler::solver.

solve() [6/6]

template<VecField F, typename Observer >

void num::solve	(	F &	u,
		const BackwardEuler &	alg,
		Observer &&	obs
	)

Definition at line 65 of file solve.hpp.

References num::ode::advance(), and num::BackwardEuler::solver.

sparse_matvec()

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

y = A * x

Definition at line 122 of file sparse.cpp.

References num::SparseMatrix::col_idx(), num::SparseMatrix::n_cols(), num::SparseMatrix::n_rows(), num::SparseMatrix::row_ptr(), num::BasicVector< T >::size(), and num::SparseMatrix::values().

Referenced by num::operators::SparseOp::apply().

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 94 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 104 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 133 of file math.hpp.

svd()

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

Definition at line 11 of file svd.cpp.

References lapack, num::backends::lapack::svd(), and num::backends::seq::svd().

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
	)

Definition at line 20 of file svd.cpp.

References num::BasicMatrix< T >::cols(), matmul(), num::QRResult::Q, qr(), rng_normal(), num::BasicMatrix< T >::rows(), num::SVDResult::S, svd(), num::SVDResult::U, and num::SVDResult::Vt.

thomas()

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

Definition at line 12 of file thomas.cpp.

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

to_idx()

template<class T >

constexpr idx num::to_idx ( T  x )

constexprnoexcept

Cast any integer to idx without a verbose static_cast.

Definition at line 16 of file types.hpp.

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.

verlet()

VerletSteps num::verlet	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		ODEParams	p = {}
	)

Definition at line 382 of file ode.cpp.

Referenced by ode_verlet().

yoshida4()

Yoshida4Steps num::yoshida4	(	AccelFn	accel,
		Vector	q0,
		Vector	v0,
		ODEParams	p = {}
	)

Definition at line 385 of file ode.cpp.

Referenced by ode_yoshida4().

zeta()

real num::zeta ( real  x )

inline

zeta(x) – Riemann zeta function

Definition at line 239 of file math.hpp.

Referenced by num::backends::seq::svd().

Variable Documentation

best_backend

constexpr Backend num::best_backend = default_backend

inlineconstexpr

Definition at line 64 of file policy.hpp.

blas

constexpr Backend num::blas = Backend::blas

inlineconstexpr

Definition at line 20 of file policy.hpp.

blocked

constexpr Backend num::blocked = Backend::blocked

inlineconstexpr

Definition at line 18 of file policy.hpp.

default_backend

constexpr Backend num::default_backend

inlineconstexpr

Initial value:

=
 
 
 
 
 
 
 
                    Backend::blocked

Definition at line 53 of file policy.hpp.

e

constexpr real num::e = 2.71828182845904523536

constexpr

Definition at line 44 of file math.hpp.

Referenced by autocorr_time(), brent(), cg(), cg(), num::backends::lapack::eig_sym(), num::backends::omp::eig_sym(), num::backends::seq::eig_sym(), expv(), gauss_seidel(), gmres(), inverse_iteration(), jacobi(), lanczos(), num::backends::seq::lu(), lu_inv(), newton(), num::detail::normalise(), num::viz::phase_hsv_color(), power_iteration(), num::backends::seq::qr(), secant(), num::FieldSolver::solve_var_poisson(), num::backends::lapack::svd(), and num::backends::seq::svd().

gpu

constexpr Backend num::gpu = Backend::gpu

inlineconstexpr

Definition at line 22 of file policy.hpp.

half_pi

constexpr real num::half_pi = 1.57079632679489661923

constexpr

pi/2

Definition at line 51 of file math.hpp.

has_blas

constexpr bool num::has_blas

inlineconstexpr

Initial value:

=
 
 
 
    false

Definition at line 25 of file policy.hpp.

has_lapack

constexpr bool num::has_lapack

inlineconstexpr

Initial value:

=
 
 
 
    false

Definition at line 32 of file policy.hpp.

has_omp

constexpr bool num::has_omp

inlineconstexpr

Initial value:

=
 
 
 
    false

Definition at line 39 of file policy.hpp.

has_simd

constexpr bool num::has_simd

inlineconstexpr

Initial value:

=
 
 
 
    false

Definition at line 46 of file policy.hpp.

inv_pi

constexpr real num::inv_pi = 0.31830988618379067154

constexpr

1/pi

Definition at line 49 of file math.hpp.

lapack

constexpr Backend num::lapack = Backend::lapack

inlineconstexpr

Definition at line 23 of file policy.hpp.

lapack_backend

constexpr Backend num::lapack_backend

inlineconstexpr

Initial value:

=
 
 
 
 
 
    Backend::seq

Definition at line 66 of file policy.hpp.

ln2

constexpr real num::ln2 = 0.69314718055994530942

constexpr

Definition at line 48 of file math.hpp.

omp

constexpr Backend num::omp = Backend::omp

inlineconstexpr

Definition at line 21 of file policy.hpp.

phi

constexpr real num::phi = 1.61803398874989484820

constexpr

Golden ratio.

Definition at line 45 of file math.hpp.

Referenced by num::MagneticSolver::current_density(), ellint_Ei(), ellint_F(), ellint_Pi_inc(), num::FieldSolver::gradient(), gradient_3d(), num::FieldSolver::solve_poisson(), and num::FieldSolver::solve_var_poisson().

pi

constexpr real num::pi = 3.14159265358979323846

constexpr

Definition at line 43 of file math.hpp.

Referenced by num::pde::poisson2d(), and num::pde::poisson2d_fd().

seq

constexpr Backend num::seq = Backend::seq

inlineconstexpr

Definition at line 17 of file policy.hpp.

simd

constexpr Backend num::simd = Backend::simd

inlineconstexpr

Definition at line 19 of file policy.hpp.

sqrt2

constexpr real num::sqrt2 = 1.41421356237309504880

constexpr

Definition at line 46 of file math.hpp.

sqrt3

constexpr real num::sqrt3 = 1.73205080756887729353

constexpr

Definition at line 47 of file math.hpp.

two_pi

constexpr real num::two_pi = 6.28318530717958647692

constexpr

2pi

Definition at line 50 of file math.hpp.