Calculus and differential equations: API index

Find an operation by its purpose. Entries link to the complete signature and contract.

HMath.Calculus

API	Purpose
Estimate	Estimated value, absolute error estimate and callback evaluation count.
Estimate	Declaration and type contract below.
estimateValue	Returned numerical derivative/integral estimate.
estimateError	Absolute error estimate from the selected approximation/refinement rule; not a universal rigorous bound.
evaluations	Number of function evaluations used by the calculation.
Tolerance	Positive absolute tolerance, nonnegative relative tolerance and positive budget.
Tolerance	Declaration and type contract below.
absoluteTolerance	Requested finite nonnegative absolute error threshold.
relativeTolerance	Requested finite nonnegative error threshold relative to estimate magnitude.
maximumEvaluations	Positive caller-supplied function-evaluation budget.
defaultTolerance	Default adaptive tolerance: 1e-10 absolute, 1e-8 relative, 100000 evaluations.
Difference	One-sided or central first differences.
Forward	Declaration and type contract below.
Backward	Declaration and type contract below.
Central	Declaration and type contract below.
derivative	First derivative with explicit positive finite step; rounded-away steps fail.
derivativeN	Arbitrary nonnegative derivative order by recursively centered differences.
complexDerivative	Complex-step first derivative.
richardson	Richardson extrapolation of a central first difference.
ridders	Ridders extrapolation with factor 1.4 and a best-error stopping criterion.
limit	One-sided numerical limit, approaching geometrically from the requested side.
partialDerivative	Partial derivative at a zero-based coordinate.
gradient	Gradient using one central difference per coordinate.
jacobian	Row-major Jacobian; callback output dimensions must remain constant.
hessian	Hessian with centered diagonal and mixed differences.
directionalDerivative	Directional derivative; the direction is used as supplied, without normalization.
jvp	Jacobian-vector product by directional central differences.
vjp	Vector-Jacobian product as the gradient of a scalar contraction.
hvp	Hessian-vector product with four directional/coordinate samples per component.
divergence	Divergence of a vector field with equal input and output dimensions.
curl	Curl of a three-dimensional vector field.
laplacian	Laplacian as the sum of centered second coordinate differences.
Quadrature	Composite Newton-Cotes formulas.
Trapezoidal	Declaration and type contract below.
Simpson	Declaration and type contract below.
ThreeEighths	Declaration and type contract below.
integrateComposite	Composite finite-interval integration with exact panel constraints.
romberg	Romberg integration using trapezoidal refinement and Richardson diagonals.
gaussLegendre	n-point Gauss-Legendre quadrature; nodes are refined by Newton iteration.
gaussHermite	Weighted Gaussian quadrature on the real line for weight exp(-x^2).
gaussLaguerre	Weighted Gaussian quadrature on [0,infinity) for weight exp(-x).
gaussChebyshev	Weighted first-kind Chebyshev rule on [-1,1], weight 1/sqrt(1-x^2).
gaussKronrod	Embedded 7-point Gauss / 15-point Kronrod estimate.
adaptiveIntegrate	Adaptive Gauss-Kronrod with global summed error and a strict evaluation budget.
clenshawCurtis	Clenshaw-Curtis quadrature on n cosine-spaced panels using exact moments of the cosine interpolant. n must be positive and at most 4096.
tanhSinh	Tanh-sinh quadrature with geometric step refinement on a finite interval.
integrateInfinite	Integrate on finite, half-infinite or doubly-infinite intervals by an open coordinate transform followed by adaptive Gauss-Kronrod.
integrateOscillatory	Integrate f(x)cos(omegax) or f(x)sin(omegax), splitting at least once per half-period to avoid accepting an aliased embedded quadrature estimate.
integrateND	Tensor-product midpoint integration over a box.
monteCarlo	Reproducible Monte Carlo box integral with an explicit 64-bit seed.
quasiMonteCarlo	Deterministic Halton-box quadrature, up to 16 dimensions.
cumulativeIntegral	Cumulative trapezoidal integrals at strictly increasing sample coordinates.
pathIntegral	Scalar line integral f(r(t))*|r'(t)| along an explicitly parameterized path.
lineIntegral	Oriented vector-field line integral along a parameterized path.
surfaceIntegral	Scalar surface integral over a rectangular parameter domain in R3.
volumeIntegral	Volume integral over a three-dimensional rectangular box.

HMath.Interpolation

API	Purpose
Extrapolation	Reject out-of-domain queries, clamp to an endpoint, or evaluate the endpoint polynomial beyond its interval.
Reject	Declaration and type contract below.
Clamp	Declaration and type contract below.
Extrapolate	Declaration and type contract below.
InterpolationMethod	Independent scalar nearest, linear, Lagrange, Newton divided-difference or barycentric formula.
Nearest	Declaration and type contract below.
Linear	Declaration and type contract below.
Lagrange	Declaration and type contract below.
Newton	Declaration and type contract below.
Barycentric	Declaration and type contract below.
interpolate	Interpolate ordered (x,y) samples using the selected formula and extrapolation policy; nearest ties choose the earlier node.
hermite	Global confluent Hermite polynomial, distinct nodes and supplied first derivatives.
cubicHermite	Piecewise cubic Hermite interpolation with supplied endpoint derivatives.
SplineBoundary	Natural zero curvature, supplied clamped endpoint slopes, or periodic value/derivative matching.
Natural	Declaration and type contract below.
Clamped	Declaration and type contract below.
Periodic	Declaration and type contract below.
PiecewisePolynomial	Opaque ascending local polynomial coefficients and strictly increasing breakpoints.
piecewisePolynomial	Validate breakpoints and nonempty coefficient rows.
piecewiseKnots	Return the immutable piecewise breakpoints, including both outer endpoints.
evaluatePiecewise	Evaluate by Horner rule in the containing interval; use the final interval at the right endpoint.
differentiatePiecewise	Differentiate each local coefficient row without changing the breakpoints.
integratePiecewise	Exact polynomial antiderivative across all crossed intervals; both limits must lie inside the domain.
cubicSpline	Cubic C2 interpolation.
pchip	Fritsch--Carlson/PCHIP weighted harmonic interior slopes and limited one- sided endpoints preserve monotonicity on monotone data.
akima	Akima cubic interpolation using slope-difference weights and linearly extended endpoint secants; at least three samples.
smoothingSpline	Natural cubic smoothing spline minimizing sum (y_i-s(x_i))² plus lambda times integral s''².
BSpline	Opaque rational B-spline with positive weights and a validated full knot vector.
bSpline	Nondecreasing full knot vector, degree 0..16, positive rational weights, control points of equal dimension.
evaluateBSpline	Evaluate rational Cox--de Boor basis functions and divide homogeneous coordinates by their positive total weight.
derivativeBSpline	Analytic first derivative of a rational B-spline using basis derivatives and the homogeneous quotient rule.
insertKnot	One Boehm knot insertion in homogeneous coordinates; shape is preserved.
bezier	Polynomial Bezier curve with unit weights; controls are vectors with 1..16 coordinates.
rationalBezier	Positive-weight rational Bezier curve on [0,1], represented exactly as an open clamped B-spline (degree <=16).
cardinalSpline	Kochanek--Bartels interpolation with zero continuity/bias and supplied tension in [-1,1].
catmullRom	Cardinal cubic interpolation with zero tension and repeated endpoint secants.
kochanekBartels	Interpolating cubic Hermite segment sequence with Kochanek--Bartels tension, continuity and bias in [-1,1]; endpoint secants are repeated.
curveIntegral	Componentwise adaptive integral of a B-spline between in-domain limits, with estimated quadrature errors.
arcLength	Adaptive integral of the Euclidean analytic derivative norm; limits must increase and remain in the spline domain.
gridInterpolate	Rectilinear multilinear interpolation; values are row-major/last axis fast.
RadialKernel	Radial interpolation kernels; the first three accept a positive inverse-length shape parameter.
Gaussian	Declaration and type contract below.
Multiquadric	Declaration and type contract below.
InverseMultiquadric	Declaration and type contract below.
ThinPlate	Declaration and type contract below.
Cubic	Declaration and type contract below.
RBF	Opaque fitted radial model with an affine polynomial tail.
fitRBF	Solve the augmented radial/affine system with nonnegative diagonal smoothing; singular or affinely degenerate systems fail.
evaluateRBF	Evaluate a fitted scattered model at one finite point with the original coordinate dimension.
inverseDistance	Normalized inverse-distance interpolation with positive power; coincident queries return the exact observation.
shepard	Classical inverse-square Shepard interpolation in the supplied Euclidean coordinates.
kriging	Ordinary kriging with stationary exponential covariance exp(-distance/range), optional independent nugget, unbiased constant mean.
Approximation	Coefficients or parameters, observed absolute residual, and iteration/evaluation count; error is an estimate.
Approximation	Declaration and type contract below.
approximationCoefficients	Fitted coefficients in the approximation routine's documented basis/order.
approximationError	Reported approximation residual/error measure, not a general analytic bound.
approximationIterations	Number of fitting/refinement iterations performed.
polynomialFit	Full-column-rank polynomial least squares via pivoted QR, returning ascending coefficients and maximum observation error.
chebyshevApproximate	Chebyshev-root interpolation on a finite interval; report maximum error on a separate 513-point uniform validation grid.
evaluateChebyshev	Evaluate ascending Chebyshev coefficients after mapping the original interval to [-1,1].
pade	Padé [m/n] from ascending Taylor coefficients, q_0=1.
rationalFit	Linearized rational least squares y*q(x)-p(x)=0 with q_0=1; poles at observations are rejected.
remez	Discrete Remez exchange on an explicit dense grid, 1..64 degree.
nonlinearFit	Damped Gauss--Newton least squares with central-difference Jacobian, strict residual/step convergence and finite total callbacks; returns fitted parameters.
interpolationError	Maximum absolute discrepancy at the supplied nonempty observations; nonfinite callback results fail.

HMath.ODE

API	Purpose
ODE	Time and state callback, returning a derivative of exactly the same length.
Method	Explicit, multistep, implicit, and linearly implicit methods.
Euler	Declaration and type contract below.
Heun	Declaration and type contract below.
Midpoint	Declaration and type contract below.
RK2	Declaration and type contract below.
RK4	Declaration and type contract below.
RK23	Declaration and type contract below.
RK45	Declaration and type contract below.
DormandPrince	Declaration and type contract below.
DOP853	Declaration and type contract below.
AdamsBashforth	Declaration and type contract below.
AdamsMoulton	Declaration and type contract below.
PredictorCorrector	Declaration and type contract below.
BackwardEuler	Declaration and type contract below.
BDF	Declaration and type contract below.
Radau	Declaration and type contract below.
Rosenbrock	Declaration and type contract below.
ODEOptions	Positive step bounds and tolerances; maximumSteps bounds accepted and rejected attempts, and newtonIterations bounds each implicit correction.
ODEOptions	Declaration and type contract below.
odeAbsoluteTolerance	Finite positive absolute local error tolerance.
odeRelativeTolerance	Finite nonnegative local error tolerance relative to state scale.
initialStep	Requested positive initial step magnitude; integration direction follows time bounds.
minimumStep	Positive minimum permitted step magnitude before reporting failure.
maximumStep	Positive maximum permitted step magnitude.
maximumSteps	Maximum integration-step work budget.
newtonIterations	Iteration budget for implicit-method nonlinear solves.
defaultODEOptions	Conservative default adaptive settings for Double callbacks.
Trajectory	Strictly monotonic time samples, states and endpoint derivatives.
Trajectory	Declaration and type contract below.
trajectoryTimes	Accepted sample times in integration order.
trajectoryStates	State vector at each trajectory time.
trajectoryDerivatives	ODE derivative vector at each trajectory sample, used by dense output.
solveIVP	Integrate on n equal panels.
solveAdaptive	Adaptive explicit RK integration.
interpolateTrajectory	Cubic Hermite interpolation inside the trajectory's closed time range.
Event	A located event time and its dense-interpolated state.
Event	Declaration and type contract below.
eventTime	Located event crossing time.
eventState	Interpolated state vector at eventTime.
EventDirection	Event direction in increasing physical time, even for backward integration.
AnyDirection	Declaration and type contract below.
Rising	Declaration and type contract below.
Falling	Declaration and type contract below.
locateEvents	Locate bracketed sign-changing events by dense-output bisection.
solveBoundary	Shooting boundary-value solver: all initial-state components are unknown.
solveDAE	Index-one implicit DAE driver F(t,y,y')=0 with backward differences.
solveHamiltonian	Fixed-step symplectic integration; output states concatenate q then p.
HamiltonianMethod	Separable Hamiltonian methods for q'=velocity(p), p'=force(q).
SymplecticEuler	Declaration and type contract below.
Verlet	Declaration and type contract below.
Leapfrog	Declaration and type contract below.
solveNystrom	Fourth-order explicit Runge-Kutta-Nystrom method for q''=a(t,q), with acceleration independent of velocity.
sensitivities	Terminal-state parameter sensitivities by centered perturbation of the complete IVP solve.
equilibrium	Find an autonomous equilibrium by a locally convergent Newton solve.
phaseTrajectories	Compute autonomous phase-space trajectories for independently supplied starts.
lyapunovExponents	Full Lyapunov spectrum by the Benettin tangent-space method: integrate variational equations with RK4, then modified Gram-Schmidt at each sample.

HMath.PDE

API	Purpose
Grid	Opaque Cartesian grid with positive representable spacings and 1..3 axes.
grid	Construct a grid with >=3 nodes per axis; reject overflowing counts and rounded-away duplicate coordinates.
gridShape	Number of nodes along each axis, preserving last-axis-fastest storage order.
gridCoordinates	Physical coordinates in flat storage order; periodic upper endpoints are omitted.
gridSpacing	Positive uniform distance between adjacent nodes on each axis.
flattenIndex	Checked conversion of zero-based multidimensional node coordinates to a row-major flat index.
unflattenIndex	Inverse checked row-major index conversion.
Boundary	Constant essential value, outward derivative, au+bdu/dn=c, or paired periodic faces.
Dirichlet	Declaration and type contract below.
Neumann	Declaration and type contract below.
Robin	Declaration and type contract below.
Periodic	Declaration and type contract below.
Discretization	Nodal nondivergence FD, conservative face-flux FV, or tensor-product Q1 Galerkin FEM.
FiniteDifference	Declaration and type contract below.
FiniteVolume	Declaration and type contract below.
FiniteElement	Declaration and type contract below.
PDEOperator	Opaque assembled stiffness, source mass matrix, boundary load and essential constraints.
assemblePDE	Assemble -kappa*Laplacian (FD) or -div(kappa grad) (FV/FEM) plus constant reaction; coefficient field is positive at every node.
pdeStiffness	Read the canonical sparse stiffness and source mass operators; essential rows are identity/zero respectively.
pdeMass	Read the canonical sparse stiffness and source mass operators; essential rows are identity/zero respectively.
pdeBoundaryLoad	Read boundary contributions, including prescribed values at essential rows.
PDEResult	Nodal solution with independently recomputed infinity residual; errorEstimate reports backward residual only.
PDEResult	Declaration and type contract below.
pdeValues	Computed grid solution values in the solver's documented spatial order.
pdeResidual	Residual measure at the returned grid solution.
pdeErrorEstimate	Solver-specific discretization/error estimate; not a universal certified bound.
solveElliptic	Dense pivoted reference solve and independently recomputed infinity residual.
poisson	Solve a unit-diffusivity zero-reaction Poisson problem with the selected spatial discretization.
laplace	Solve the corresponding homogeneous-source boundary-value problem.
helmholtz	Solve -Laplacian(u)-k²u=f; resonances/singular systems fail.
spatialGradient	Second-order central spatial gradients; physical endpoints use second-order one-sided stencils.
laplacianStencil	Apply the FD Laplacian with ghost normal/Robin boundary data.
spectralDerivative	Periodic Fourier collocation derivative of order 0..8 along one axis; odd real Nyquist derivatives are zero.
spectralLaplacian	Sum periodic second Fourier derivatives over all spatial axes.
spectralPoisson	Solve (-Laplacian+reaction)u=f spectrally; reaction>=0, and zero reaction requires a mean-free source with zero-mean solution gauge.
pseudoSpectralDiffusion	Pseudospectral divergence(kappa*gradient u).
methodOfLines	Method of lines with a constant assembled operator and checked time-source callback.
heat	Explicit heat/diffusion uses RK4, with a conservative forward-Euler CFL check (therefore valid for RK4 too).
diffusion	Variable positive nodal diffusivity, chosen spatial method and increasing output times; FEM uses a stricter consistent-mass CFL bound.
wave	Periodic velocity-Verlet wave equation u_tt=c² Laplacian(u); returns u at requested equally or unequally spaced increasing times.
advection	Conservative first-order upwind periodic advection, central conservative variable-coefficient diffusion.
advectionDiffusion	Periodic upwind advection plus central diffusion using explicit Euler and the combined positivity CFL bound.
stabilityLimit	Conservative explicit positivity bound 1sum(|v|h+2*kappa/h^2); infinity denotes a zero spatial operator.
cflNumber	Requested nonnegative timestep divided by the conservative combined stability limit.
manufacturedError	Maximum absolute nodal difference from a finite exact-solution callback; this measures continuum error when an exact solution is known.

HMath.Root

API	Purpose
RootOptions	Root-solver iterationevaluation budgets and residualstep tolerances, initialized with defaultRootOptions.
RootOptions	Declaration and type contract below.
rootMaxIterations	Maximum scalar-root iteration count.
rootMaxEvaluations	Maximum scalar function/derivative evaluation budget.
rootTolerance	Requested finite positive residual tolerance.
rootStepTolerance	Requested finite positive parameter-step tolerance.
defaultRootOptions	Default root-solving limits: 200 iterations, 4000 evaluations, residual tolerance 1e-10 and step tolerance 1e-12.
RootResult	Scalar root estimate with residual,optional certified bracket error bound,and work counts.
RootResult	Declaration and type contract below.
rootValue	Returned scalar root estimate.
rootResidual	Absolute function residual at rootValue.
rootErrorBound	Optional bracket-based absolute root error bound; Nothing when the method cannot certify one.
rootIterations	Completed scalar-root iterations.
rootEvaluations	Recorded function/derivative evaluation count.
Bracket	Validated scalar bracketing state used by checked bracketed root solvers; construct through the bracketing API.
bracket	Validate finite a<b and endpoint roots or opposite signs; construction never assumes a discontinuity is a root.
BracketMethod	Bracket-preserving root strategy such as bisection,Brent,Ridder,or regula falsi variants.
Bisection	Declaration and type contract below.
RegulaFalsi	Declaration and type contract below.
Illinois	Declaration and type contract below.
Brent	Declaration and type contract below.
Ridder	Declaration and type contract below.
solveBracket	Brackets are re-evaluated with the supplied function, preventing stale or mismatched endpoint caches from certifying a different function's root.
bracketScan	Evaluate a uniform finite grid under the evaluation budget and return every endpoint/sign-change bracket.
isolateIntervals	Refine sign-change scan brackets and deduplicate endpoint roots.
DerivativeMethod	Derivative-based scalar-root strategy, including Newton,Halley,and Householder methods with their required derivatives.
Newton	Declaration and type contract below.
Halley	Declaration and type contract below.
Householder3	Declaration and type contract below.
solveDerivative	Derivatives [f', f'', f'''] are required through order 1, 2, or 3.
secant	Unbracketed two-point secant iteration; zero slopes, stagnation, and exhausted budgets fail explicitly.
steffensen	Derivative-free Steffensen root update using f(x+f(x)); termination checks the original residual.
FixedPointMethod	Fixed-point acceleration strategy, selecting Picard,Aitken,or Steffensen iteration.
Picard	Declaration and type contract below.
Aitken	Declaration and type contract below.
SteffensenFixed	Declaration and type contract below.
fixedPoint	Picard iteration, non-restarted Aitken sequence acceleration, or restarted Steffensen acceleration.
anderson	Anderson mixing with depth m and Tikhonov regularization.
SystemMethod	Multivariate root strategy, selecting checked Newton or Broyden updates.
MultivariateNewton	Declaration and type contract below.
NewtonTrustRegion	Declaration and type contract below.
Broyden	Declaration and type contract below.
SystemRootResult	Returned multivariate root vector with residual norm and iteration/evaluation counts.
SystemRootResult	Declaration and type contract below.
systemRoot	Returned multivariate root parameter vector.
systemResidual	Norm of the residual function at systemRoot.
systemIterations	Completed nonlinear-system iterations.
systemEvaluations	Recorded system function/evaluation count.
solveSystem	Solve a square analytic-Jacobian system using Newton, trust-radius Newton, or rank-one Broyden updates.
polynomialRoots	Reuse the checked Aberth-Ehrlich simultaneous complex solver for an ascending-coefficient polynomial.
realPolynomialRoots	Use exact square-free factorization and Sturm isolation; return rational intervals and integer multiplicities.