Alphabeta MathNumerical Linear Algebra and Approximation
Pathway
Pages are grouped by how many dependency steps into this group they sit. Everything a page needs from this group appears above it.
Level 0
14 pages- Composite quadrature rules7 results
This page supplies exact uniform-panel midpoint and composite Simpson procedures, their derivative-norm error bounds, and Simpson exactness through cubic degree.
2 algorithms, 2 lemmas, 3 theoremsExamples & counterexamples → - Fast multipole expansions and error10 results
For planar charges, Finite planar quadtree and separated interaction lists fixes half-open dyadic cells, a fixed logarithm branch per far pair, direct near pairs, and the separated-cell ratio ρ=(a C+a D)/R CD<1.
2 definitions, 2 algorithms, 3 lemmas, 3 theoremsExamples & counterexamples → This page gives operationwise finite-precision contracts for pairwise and compensated summation, TwoSum exact-sum recovery and FMA-assisted product residual computation…
1 definition, 8 algorithms, 6 lemmas, 8 theoremsExamples & counterexamples →The page defines the entire phi family by its convergent series, proves its recurrence, and obtains every phi action (including the value at zero) from one augmented exponential.
3 definitions, 2 algorithms, 4 lemmas, 3 theoremsExamples & counterexamples →- Multigrid methods and complexity11 results
This page fixes a finite-dimensional real SPD convention for Galerkin multigrid.
2 definitions, 2 algorithms, 2 lemmas, 5 theoremsExamples & counterexamples → Using the published half-open, equal-weight point-multiset definition of star discrepancy, this page asks for the exact minimum order when the integer dimension d≥3 is fixed before n→∞.
This page develops finite dyadic endpoint formats and directed rounding before defining real intervals, complex rectangles, and midpoint-radius balls.
7 definitions, 3 algorithms, 19 lemmas, 6 theoremsExamples & counterexamples →This page fixes a grade and block orientation for square complex matrix polynomials, including homogeneous eigenpairs and the infinity point when the leading coefficient is singular.
4 definitions, 1 lemma, 1 proposition, 2 theoremsExamples & counterexamples →- Polynomial interpolation and remainder14 results
For distinct real nodes, this page defines the cardinal interpolation functions and proves that their interpolant is the unique polynomial of degree at most the node count minus one.
1 definition, 1 algorithm, 7 lemmas, 2 propositions, 3 theoremsExamples & counterexamples → - Radial basis and kernel interpolation17 results
This page defines strict positive-definite interpolation, strict conditional-positive-definite polynomial augmentation, site fill and separation, and an exact augmented RBF-FD procedure.
3 definitions, 1 algorithm, 9 lemmas, 3 theorems, 1 corollaryExamples & counterexamples → - Randomized low-rank approximation21 results
This page develops the exact Gaussian range finder over real and normalized complex fields and the QR-reorthogonalized power procedure.
1 definition, 2 algorithms, 13 lemmas, 5 theoremsExamples & counterexamples → This page defines a zero-based compressed sparse row (CSR) record, including empty rows, repeated column indices, and explicitly stored zeros.
2 definitions, 1 algorithm, 4 theoremsExamples & counterexamples →- Stationary iterations and relaxation9 results
This page adds finite-budget Richardson, Jacobi, fixed-order Gauss–Seidel, and SOR procedures using the published stationary-splitting definition.
1 definition, 4 algorithms, 1 lemma, 3 theoremsExamples & counterexamples → This page develops exact-arithmetic compression for finite tensors over ℝ or ℂ, with fixed lexicographic unfolding conventions.
4 definitions, 4 algorithms, 2 lemmas, 5 theoremsExamples & counterexamples →
Level 1
18 pagesThis page separates adaptive Simpson's finite procedure and its heuristic indicator from a conditional certified mode.
1 definition, 2 algorithms, 1 lemma, 3 theoremsExamples & counterexamples →This page specifies normalized finite real or complex matrix-product states together with concrete open-boundary or periodic presentations, finite Hermitian Hamiltonian inputs, and exact one-site or two-site sweeps.
1 definition, 1 algorithm, 5 lemmas, 1 proposition, 2 theoremsExamples & counterexamples →- Certified geometric predicates16 results
This page defines exact determinant predicates for orientation and lifted circle/sphere tests on finite dyadic coordinates.
1 definition, 5 algorithms, 6 lemmas, 4 theoremsExamples & counterexamples → - Computed fast transforms and convolution19 results
This page extends the canonical finite Fourier transform and radix-two FFT with computed error bounds, arbitrary-length reductions, trigonometric transforms, and convolution.
1 definition, 4 algorithms, 5 lemmas, 9 theoremsExamples & counterexamples → This page develops the computed SVD in two stages: Householder reduction to bidiagonal form, followed by finite-budget implicit shifted QR sweeps with explicit deflation…
5 algorithms, 6 lemmas, 1 proposition, 5 theorems, 1 corollaryExamples & counterexamples →For a finite matrix, the page first fixes ordered cluster trees, geometric admissibility, and a recursive block partition.
3 definitions, 7 algorithms, 3 lemmas, 7 theoremsExamples & counterexamples →This page gives two certified root-isolation routes. Exact rational polynomials use gcd squarefree decomposition, Sturm variation at exact rational endpoints, and dyadic…
1 definition, 4 algorithms, 5 lemmas, 5 theoremsExamples & counterexamples →For a real symmetric matrix, the Lanczos estimate keeps the top-eigenspace projection of the starting vector and the gap separating the wanted eigenvalue from the unwanted spectrum.
1 lemma, 1 proposition, 4 theoremsExamples & counterexamples →For a<b, continuous complex amplitude f, real phase g, and nonzero real frequency ω, this page defines I ω(f,g;[a,b]).
1 definition, 4 algorithms, 6 lemmas, 5 theoremsExamples & counterexamples →For f∈ C[-1,1], E n(f) is the best degree-n uniform error. The first-kind root and Chebyshev–Lobatto interpolation operators use their specified n+1 angular nodes, with the…
1 definition, 3 lemmas, 2 theorems, 1 corollaryExamples & counterexamples →This page adds a fixed-order, fixed-pattern ILU(0) procedure and proves its stored-factor pattern and exact setup and application counts.
2 algorithms, 2 theoremsExamples & counterexamples →This page defines cubic Hermite pieces with explicit nodal values and slopes, proves a sufficient monotonicity region in normalized endpoint slopes, and gives the finite…
2 definitions, 1 algorithm, 2 lemmas, 6 theoremsExamples & counterexamples →- Sparse direct elimination and ordering18 results
This page builds on the structural graph, ordering, filled graph, and elimination-forest interfaces owned by the sparse-matrix-models-and-kernel-costs page.
2 definitions, 6 algorithms, 5 lemmas, 5 theoremsExamples & counterexamples → This page builds Smolyak cubature from nested composite-Simpson levels.
3 definitions, 2 algorithms, 4 lemmas, 3 theoremsExamples & counterexamples →This page evaluates a B-spline curve from its local coefficient list, proves the exact de Boor recurrence, its O(p²) arithmetic count, and a range-qualified forward-error bound.
1 definition, 3 algorithms, 2 lemmas, 9 theoremsExamples & counterexamples →This page gives a finite-horizon covariance-factor implementation of the canonical Kalman prediction and measurement maps.
1 algorithm, 7 lemmas, 2 theoremsExamples & counterexamples →This page defines finite, domain-specific enclosures for integer-order Bessel and modified Bessel functions, Airy functions, Fresnel integrals, the error function and a Gauss…
10 definitions, 6 algorithms, 24 lemmas, 6 theoremsExamples & counterexamples →This page separates finite periodic transforms from function approximation.
9 definitions, 2 lemmas, 1 proposition, 6 theorems, 1 corollaryExamples & counterexamples →
Level 2
11 pagesThis page develops barycentric rational quotients, the Floater–Hormann family, finite-sample AAA, and exact Padé coefficient matching.
5 definitions, 2 algorithms, 12 lemmas, 1 proposition, 5 theorems, 1 counterexampleExamples & counterexamples →Clenshaw–Curtis quadrature is defined by integrating the Chebyshev–Lobatto interpolant, with explicit DCT-I endpoint conventions and a midpoint case at degree zero.
5 definitions, 2 algorithms, 6 lemmas, 5 theorems, 2 corollariesExamples & counterexamples →This page defines real and complex Schur forms and a finite-budget Hessenberg QR procedure.
3 definitions, 3 algorithms, 10 lemmas, 5 theoremsExamples & counterexamples →This page develops exact Delaunay triangulations and their Voronoi duals for finite indexed dyadic sites.
5 definitions, 3 algorithms, 9 lemmas, 7 theoremsExamples & counterexamples →This page defines the neutral mean-zero Poisson problem on an arbitrary full-rank three-dimensional torus and proves the Gaussian Ewald split, its renormalized self term, and uniform real and reciprocal tail bounds.
3 definitions, 1 algorithm, 7 lemmas, 5 theoremsExamples & counterexamples →This page defines finite Cartesian Gaussian primitives and contractions, then proves their Gaussian product, moment, derivative, and kinetic recurrences.
4 definitions, 2 algorithms, 5 lemmas, 10 theoremsExamples & counterexamples →This page proves a strong-RRQR-selected column interpolative decomposition with coefficient and trailing-block error bounds.
2 definitions, 1 algorithm, 3 lemmas, 5 theoremsExamples & counterexamples →For a continuous real target on a nondegenerate compact interval, the page proves existence of a best polynomial approximant, the n+2 alternating-extrema criterion for nonzero error, and uniqueness.
2 definitions, 2 algorithms, 7 lemmas, 7 theorems, 1 corollaryExamples & counterexamples →- Nonuniform fast fourier transforms12 results
For type 1, finitely many weighted points on the period-one torus are mapped to integer Fourier modes.
3 definitions, 1 algorithm, 7 lemmas, 1 theoremExamples & counterexamples → This page develops squared-norm subspace embeddings for a fixed subspace, an iid Gaussian construction with an additive dimension and failure-probability bound, and…
3 definitions, 2 algorithms, 3 lemmas, 5 theoremsExamples & counterexamples →This page fixes the zero-based lag, lower-shift, reversal, and canonical DFT conventions used by the structured solvers.
2 definitions, 1 algorithm, 3 lemmas, 1 proposition, 3 theoremsExamples & counterexamples →
Level 3
3 pagesThis page develops regular finite matrix pencils, their homogeneous finite and infinite eigenvalues, and the generalized Schur form.
3 definitions, 2 algorithms, 7 lemmas, 2 propositions, 4 theoremsExamples & counterexamples →- Sylvester and lyapunov matrix equations19 results
This page develops the equation AX+XB=C over the real and complex fields, its Kronecker representation, the spectral criterion for unique solvability, and the Frobenius separation that controls solution error.
2 definitions, 2 algorithms, 6 lemmas, 9 theoremsExamples & counterexamples → This page develops verified point and family linear solves, Krawczyk inclusion, real determinant sign checks, Hermitian eigenvalue enclosures, inertia based eigenvalue bisection, and a complex resolvent cover validator.
3 definitions, 8 algorithms, 11 lemmas, 14 theoremsExamples & counterexamples →
Level 4
3 pages- Matrix Functions and Scaling Squaring25 results
An analytic matrix function can be defined by applying one scalar holomorphic branch and its derivatives to the Jordan blocks.
3 definitions, 7 algorithms, 3 lemmas, 11 theorems, 1 remarkExamples & counterexamples → This page fixes the real and complex CARE/DARE conventions, proves the closed-region PBH equivalences and stabilizing value solutions, and relates those solutions to Hamiltonian and symplectic-pencil stable graphs.
2 definitions, 4 algorithms, 5 lemmas, 7 propositions, 8 theoremsExamples & counterexamples →- Restarted and block eigensolvers10 results
This page defines finite-budget thick and implicit Arnoldi restarts, with normalized Ritz vectors, residual-based acceptance, retained locked directions, and explicit breakdown statuses.
3 algorithms, 2 lemmas, 5 theoremsExamples & counterexamples →
Level 5
1 page- Gaussian quadrature21 results
For a finite positive Borel measure on a compact interval whose support has at least m+1 points, the moment pairing is positive definite through degree m-1 under the…
5 definitions, 1 algorithm, 4 lemmas, 11 theoremsExamples & counterexamples →
Level 6
1 pageThis page develops four scoped tools for difficult quadrature domains.
3 definitions, 3 algorithms, 8 lemmas, 8 theorems, 1 corollaryExamples & counterexamples →