Linear algebra: API index

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

HMath.Backend

API	Purpose
Provider	Pure reference, OpenBLAS, BLIS, Intel oneMKL SDL, or Apple's system framework.
Pure	Declaration and type contract below.
OpenBLAS	Declaration and type contract below.
BLIS	Declaration and type contract below.
OneMKL	Declaration and type contract below.
Accelerate	Declaration and type contract below.
IntegerABI	BLAS/LAPACK integer width; Haskell dimensions must additionally fit Int.
LP64	Declaration and type contract below.
ILP64	Declaration and type contract below.
BackendRequest	Optional path overrides the provider default; Accelerate only accepts its system framework.
AutoBackend	Declaration and type contract below.
RequireBackend	Declaration and type contract below.
Backend	Opaque selected reference/native backend owner.
BackendInfo	Actual selected BLAS provider and integer-width ABI, available through backendInfo.
BackendInfo	Declaration and type contract below.
selectedProvider	Provider actually selected, including Pure when reference fallback is used.
selectedIntegerABI	Selected BLAS integer width: LP64 uses 32-bit indices and ILP64 uses 64-bit indices.
selectBackend	Given AutoBackend or explicit providerinteger-ABIlibrary request, select a reference or native BLAS backend in IO.
backendInfo	Provider identity is immutable for the lifetime of a handle.
pureBackend	Construct the deterministic reference backend without dynamically loading BLAS.
setBackendThreads	Positive thread count; Pure supports one thread.
lapackAvailable	Availability of the once-initialized independent LAPACKE loader.
BackendLayout	Physical matrix storage order: RowMajor or ColumnMajor, independent of mathematical transpose.
RowMajor	Declaration and type contract below.
ColumnMajor	Declaration and type contract below.
Transpose	BLAS operand transformation: unchanged,transposed,or conjugate-transposed.
NoTranspose	Declaration and type contract below.
Transpose	Declaration and type contract below.
ConjugateTranspose	Declaration and type contract below.
matrixFromLayout	Layout conversion is a checked, explicit copy; zero shapes retain rank.
matrixToLayout	Given RowMajor/ColumnMajor layout then Matrix, return values in that physical order without changing the logical matrix.
axpy	Given backend,alpha,x,y, return alpha*x+y as a new Double list.
dot	Real inner product.
nrm2	Given backend then finite Double vector, return Euclidean norm through the selected provider or reference path.
gemv	y := alpha op(A) x + beta y.
gemm	C := alpha op(A) op(B) + beta C.
complexAxpy	Given backend,complex alpha,x,y, compute alpha*x+y, requiring equal lengths and finite realimaginary components.
complexDot	Set the Boolean to True for the Hermitian dot product (conjugate x).
complexNrm2	Given backend then complex vector, return real Euclidean norm of all real/imaginary components.
complexGemv	Given backend,transpose selector,alpha,A,x,beta,y, return alphaop(A)x+beta*y as a new complex list.
complexGemm	Given backend,transpose A,transpose B,alpha,A,B,beta,C, return alphaop(A)op(B)+beta*C as a new complex matrix.
batchedGemm	Uniform scalar/transpose options, independently checked matrix triples.
stridedBatchedGemm	Row-major batches of A(m,k) and B(k,n).
backendLU	Given backend then real matrix, compute checked column-independent LU using the reference pivoted LU implementation.
backendQR	Given backend then real matrix, compute reference column-pivoted Householder QR; selection does not change this routine's provider.
backendSolve	Given backend,square real matrix,then RHS list, solve via reference LU for Pure or the independent LAPACKE solve bridge for a native backend.
backendCholesky	Given backend then finite symmetric positive-definite matrix, return reference lower Cholesky factor.
backendLeastSquares	Given backend,design matrix,then RHS list, solve using reference pivoted QR.
backendSVD	Given backend then finite real matrix, use bounded pure reference SVD for Pure or LAPACK thin SVD for native selection.
backendSymmetricEigen	Given backend then finite symmetric square matrix, use bounded cyclic-Jacobi reference solver for Pure or LAPACK otherwise.
pureSVD	Thin real SVD by one-sided Jacobi orthogonalization, at most 100 sweeps, dimensions <=128.
pureSymmetricEigen	Cyclic Jacobi symmetric eigensolver, at most 100 sweeps, ascending eigenvalues.
lapackStatus	Convert the LAPACK convention without losing the distinction between an invalid argument and failed numerical convergence.

HMath.Linear

API	Purpose
dot	Inner product.
dotChecked	Inner product, or VectorLengthMismatch with the left and right lengths.
magnitude	Euclidean magnitude.
normalize	Normalize by Euclidean magnitude.
add	Elementwise addition.
addChecked	Elementwise addition, requiring equal lengths.
subtractChecked	Elementwise left-minus-right subtraction, requiring equal lengths.
scale	Multiply every component by a scalar, preserving length.

HMath.Matrix

API	Purpose
Matrix	An abstract dense matrix with explicit dimensions and flat row-major storage.
fromList	Construct a matrix from a finite list in row-major order.
fromListChecked	Checked fromList.
toList	Return all elements in row-major order.
shape	Return (rows, columns), including either dimension of an empty matrix.
zeroMatrix	Construct a matrix filled with zeros, subject to the same shape checks as fromList.
zeroMatrixChecked	Checked zeroMatrix.
diagonal	Construct a square matrix with the finite input list on its diagonal and zeros elsewhere.
diagonalChecked	Checked diagonal.
at	Look up an element using zero-based row and column indices.
atChecked	Checked at.
sliceMatrix	Extract a slice given its (startRow, startColumn) and (rowCount, columnCount).
sliceMatrixChecked	Checked sliceMatrix.
reshape	Change the shape while preserving row-major element order.
reshapeChecked	Checked reshape.
addMatrix	Add matrices with equal shapes.
addMatrixChecked	Checked addMatrix.
subtractMatrix	Subtract matrices with equal shapes.
subtractMatrixChecked	Checked subtractMatrix.
hadamard	Multiply corresponding elements of matrices with equal shapes.
hadamardChecked	Checked hadamard.
scaleMatrix	Multiply every element by the scalar, preserving the shape.
multiply	Matrix product when the left column count equals the right row count and the output element count fits in Int.
multiplyChecked	Checked multiply.
transposeMatrix	Swap rows and columns.
identity	Construct a square identity matrix of nonnegative order.
identityChecked	Checked identity.

HMath.Matrix.Advanced

API	Purpose
generateMatrix	Given row count, column count, then zero-based (row,column) generator, construct a row-major matrix.
triangularMatrix	Given lower flag then matrix, preserve entries onbelow the diagonal if True, onabove if False, and zero the rest.
symmetricMatrix	Return (A+transpose(A))/2 for a square matrix.
hermitianMatrix	Return (A+conjugateTranspose(A))/2 for a square complex matrix.
permutationMatrix	Given a permutation p of [0..n-1], construct n-by-n P with P[i,p[i]]=1.
blockMatrix	Assemble a rectangular grid of matrix blocks.
bandedMatrix	Given rows, columns, then (diagonal offset,values) pairs, construct a matrix with zeros elsewhere.
conjugateTranspose	Transpose a complex matrix and conjugate every entry, exchanging its row/column dimensions.
zipMatrixWith	Given elementwise function and two identically shaped matrices, apply it to corresponding entries and return a new matrix.
batchedMultiply	Multiply pairs of matrices from equally long lists, preserving batch order.
kronecker	Return the block Kronecker product of matrices A and B, of shape (ArowsBrows,AcolsBcols).
khatriRao	Return columnwise Kronecker product, requiring equal column counts.
matrixPower	Given Integer exponent then square real matrix, compute power by squaring.
trace	Return sum of main diagonal of a square matrix; non-square input returns NonSquareMatrix.
matrixNorm1	Return maximum absolute column sum (induced one norm).
matrixNormInfinity	Return maximum absolute row sum (induced infinity norm).
frobeniusNorm	Return scaled Euclidean norm of all entries, reducing intermediate overflow; empty matrix norm is zero.
spectralNorm	Return largest singular value via SVD, or zero for an empty singular spectrum.
pseudoinverse	Given relative singular-value tolerance then m-by-n matrix, return n-by-m Moore-Penrose pseudoinverse via thresholded SVD.
complexPseudoinverse	Complex128 Moore-Penrose pseudoinverse, thresholded relative to the largest singular value.
matrixRank	Given finite nonnegative relative tolerance then matrix, count positive singular values whose ratio to the largest exceeds tolerance.
conditionNumber	Return largest/smallest singular-value ratio; singular matrices yield positive infinity, empty spectrum yields 1.
nullSpace	Given relative singular-value tolerance then m-by-n matrix, return an n-by-(n-rank) basis for its numerical right null space using full SVD.
columnSpace	Given relative SVD rank tolerance then matrix, return columns of left singular vectors spanning its numerical column space.
rowSpace	Given relative rank tolerance then matrix, return a column basis for its row space by applying columnSpace to its transpose.
changeCoordinates	Given source basis matrix, target square basis matrix, then source coordinate list, map to physical coordinates and solve in target basis.
matrixExp	Given positive finite tolerance, positive iteration budget, then square finite matrix, approximate exp(A) by scaling, Taylor series and squaring.
matrixLog	Compute principal real matrix logarithm of a symmetric positive-definite Double matrix using symmetric eigendecomposition.
matrixSqrt	Compute symmetric positive-semidefinite matrix square root by spectral decomposition; tolerate eigenvalues down to -1e-12 and clamp them to zero.
matrixLogGeneral	Real principal matrix logarithm on the same spectral domain as matrixSqrtGeneral.
matrixSqrtGeneral	Principal square root for invertible real matrices whose spectra avoid the nonpositive real axis.
polar	Return (U,P) from SVD so A=U*P, with P positive semidefinite; rectangular input uses the thin polar factors.

HMath.Matrix.Decomposition

API	Purpose
gaussianElimination	Given finite nonnegative relative tolerance then finite matrix, return normalized row-echelon form and zero-based pivot columns.
gaussJordan	Given relative tolerance then finite matrix, return reduced row-echelon form and zero-based pivot columns using row pivoting.
cholesky	Factor a finite symmetric positive-definite matrix as L*transpose(L), returning lower-triangular L.
ldlt	Factor a finite symmetric matrix as Ldiag(d)transpose(L), returning unit lower L and diagonal list d.
givensQR	Factor finite m-by-n matrix as Q*R using Givens rotations, returning square m-by-m Q and m-by-n R without column pivoting.
Bidiagonal	Orthogonal factors A=UBtranspose(V), retaining a rectangular bidiagonal B.
Bidiagonal	Declaration and type contract below.
bidiagonalU	Left orthogonal factor in A=UBtranspose(V).
bidiagonalB	Bidiagonal matrix in the factorization, retaining input shape.
bidiagonalV	Right orthogonal factor in A=UBtranspose(V).
bidiagonalize	Reduce a finite rectangular matrix to bidiagonal form with orthogonal factors, returning Bidiagonal fields.
tridiagonalize	Reduce a symmetric finite square matrix to (Q,T), satisfying A = Q * T * transpose Q with tridiagonal T.
Iteration	Accepted eigenpair estimate with residual and iteration count.
Iteration	Declaration and type contract below.
iterationValue	Returned real eigenvalue estimate.
iterationVector	Returned normalized eigenvector estimate.
iterationResidual	Eigenpair residual norm used to assess convergence.
iterationCount	Number of eigen-iteration updates performed.
powerIteration	Given positive finite tolerance,positive budget,square finite matrix,then nonzero starting vector, estimate a dominant eigenpair.
inverseIteration	Given tolerance,budget,shift,square matrix,then starting vector, estimate an eigenpair near shift by repeated shifted solves.
rayleighIteration	Given tolerance,budget,square matrix,then nonzero initial vector, refine an eigenpair with Rayleigh-quotient shifts.
Arnoldi	Krylov basis columns and upper-Hessenberg projection returned by arnoldi; exact breakdown may shorten the basis.
Arnoldi	Declaration and type contract below.
arnoldiBasis	Orthonormal Krylov basis vectors as matrix columns.
arnoldiHessenberg	Hessenberg projection relating the matrix action to the Arnoldi basis.
arnoldi	Given positive finite tolerance,requested basis size,square matrix,then nonzero initial vector, build Arnoldi basis/projection.
lanczos	Symmetric Lanczos with the three-term recurrence and two full reorthogonalization passes.
qrIteration	Given positive finite tolerance,positive budget,then finite square matrix, iterate QR to return (Q,T): accumulated orthogonal vectors and the resulting iterate.
jordanChains	Verify supplied Jordan chains against A and lambda and construct J and P.
jordanForm	Exact Jordan construction over Rational with a caller-provided complete set of distinct rational eigenvalues (multiplicities are inferred).

HMath.Matrix.Iterative

API	Purpose
IterativeOptions	Absolute and relative residual tolerances and maximum iterations.
IterativeOptions	Declaration and type contract below.
absoluteTolerance	Finite nonnegative absolute residual tolerance.
relativeTolerance	Finite nonnegative relative residual tolerance.
maximumIterations	Nonnegative maximum iteration count.
defaultIterativeOptions	1000 iterations, zero absolute tolerance and relative tolerance 1e-10.
IterativeResult	Converged state with norms evaluated from the original matrix and rhs.
IterativeResult	Declaration and type contract below.
iterativeSolution	Solution in original variable order.
iterativeIterations	Number of completed updates.
iterativeResidualNorm	Euclidean norm of b-Ax.
iterativeNormalResidualNorm	Euclidean norm of A^T(b-Ax), useful for inconsistent least squares.
Preconditioner	A checked inverse action M^-1; constructors retain their input dimension.
applyPreconditioner	Apply an inverse action; wrong vector lengths and nonfinite values fail.
identityPreconditioner	Identity inverse action in a nonnegative dimension.
jacobiPreconditioner	Inverse diagonal preconditioner.
blockJacobiPreconditioner	Block Jacobi from contiguous positive block sizes summing to the dimension.
ilu0Preconditioner	Incomplete LU with zero fill and no pivoting.
ic0Preconditioner	Incomplete Cholesky with zero fill for a symmetric positive-definite input.
conjugateGradient	Conjugate gradient for symmetric positive-definite matrices.
preconditionedCG	Preconditioned conjugate gradient.
gmres	Restarted GMRES with modified Gram--Schmidt Arnoldi and a second orthogonalization pass.
minres	Minimum residual on the symmetric Lanczos Krylov space, including indefinite systems.
bicg	Bi-conjugate gradient for nonsymmetric square matrices; the initial shadow residual equals the initial residual.
bicgstab	Stabilized BiCG with a fixed initial shadow residual and a minimal-residual omega step.
lsqr	LSQR by Golub--Kahan bidiagonalization and plane-rotation recurrences.
lsmr	LSMR's minimum-normal-residual Krylov objective using explicit Golub--Kahan bases and QR on the projected tridiagonal normal operator.
jacobiSolve	Simultaneous Jacobi stationary iteration.
gaussSeidelSolve	Forward Gauss--Seidel stationary iteration.
sorSolve	Successive over-relaxation, requiring a finite relaxation 0<omega<2.
iterativeRefinement	Residual-correction refinement using a reusable partial-pivoted LU.

HMath.Matrix.LAPACK

API	Purpose
SVD	Real singular-value decomposition with U,descending nonnegative singular values,and VT.
SVD	Declaration and type contract below.
svdU	Left singular vectors as columns of the real thin/full SVD factor.
singularValues	Nonnegative real singular values in descending order.
svdVT	Transposed right singular-vector factor in A=Udiag(s)VT.
svd	Compute thin real SVD A=Udiag(s)VT via LAPACK, with descending nonnegative singular values and k=min(m,n).
fullSVD	Full square U and V^T, including null-space completion.
truncatedSVD	Given retained rank k then real matrix, compute thin SVD and retain its first k components.
randomizedSVD	Randomized range finder with an explicit seed, oversampling, and power iterations, followed by a small deterministic SVD.
Eigen	Complex eigenvalues with corresponding right eigenvector columns, including conjugate pairs from real input.
Eigen	Declaration and type contract below.
eigenvalues	Complex eigenvalues matching eigenvector column order.
eigenvectors	Right eigenvectors as complex matrix columns.
eigen	Compute complex eigenvalues and right eigenvectors of a finite real square matrix via LAPACK.
symmetricEigen	Real symmetric eigensolver with orthonormal columns, sorted ascending.
GeneralizedEigen	Projective eigenvalues alpha/beta preserve infinite eigenvalues and indeterminate singular pencils without dividing by a zero beta.
GeneralizedEigen	Declaration and type contract below.
eigenAlpha	Generalized-eigenvalue numerators alpha; retain with beta to represent infinite eigenvalues.
eigenBeta	Generalized-eigenvalue denominators beta; zero beta can represent an infinite eigenvalue.
generalizedEigenvectors	Right eigenvector columns satisfying betaAv=alphaBv.
generalizedEigen	Given equally sized square matrices A,B, compute generalized alpha/beta eigenvalue representation and right eigenvectors.
Schur	Real Schur factors A=QTtranspose(Q), with quasi-triangular T and complex eigenvalues.
Schur	Declaration and type contract below.
schurQ	Orthogonal basis in A=QTtranspose(Q).
schurT	Real quasi-upper-triangular Schur form, including 2-by-2 blocks for conjugate pairs.
schurEigenvalues	Complex eigenvalues decoded from the real Schur form.
schur	Compute real Schur form A=QTtranspose(Q), returning orthogonal Q, quasi-upper-triangular T and complex eigenvalues.
QZ	Generalized real Schur factors for a matrix pair, retaining alpha/beta eigenvalue representation to allow infinite eigenvalues.
QZ	Declaration and type contract below.
qzQ	Left orthogonal QZ transform, with A=QStranspose(Z).
qzZ	Right orthogonal QZ transform for both matrix factors.
qzS	First generalized real Schur factor S.
qzT	Second generalized real Schur factor T.
qzAlpha	Generalized Schur eigenvalue numerators alpha.
qzBeta	Generalized Schur eigenvalue denominators beta; do not divide blindly when zero.
qz	Compute generalized real Schur factors for a finite square matrix pair A,B, retaining Q,Z,S,T and alphabeta eigenvalues.
Hessenberg	Orthogonal reduction A=QHtranspose(Q), with H upper Hessenberg.
Hessenberg	Declaration and type contract below.
hessenbergQ	Orthogonal transform in A=QHtranspose(Q).
hessenbergH	Upper Hessenberg form with zeros below the first subdiagonal.
hessenberg	Reduce finite real square A to upper Hessenberg H with orthogonal Q, satisfying A=QHtranspose(Q).
Balance	D^-1 A D = balancedMatrix.
Balance	Declaration and type contract below.
balancedMatrix	Matrix after LAPACK diagonal balancing.
balanceScales	Diagonal basis scaling factors used by the balancing transformation.
balance	Balance a real square matrix by diagonal scaling, returning the balanced matrix and scales needed to interpret its basis.
ComplexSVD	Complex128 thin SVD: A = U diag(s) V, where complexVT already stores V.
ComplexSVD	Declaration and type contract below.
complexU	Complex left singular vectors as columns.
complexSingularValues	Descending nonnegative real singular values of the complex matrix.
complexVT	Conjugate-transposed right singular vectors in complex SVD.
complexSVD	Compute thin complex SVD with conjugate-transposed right factor VT and real nonnegative singular values.
complexEigen	Compute complex eigenvalues and right eigenvectors of a finite complex square matrix via LAPACK.

HMath.Matrix.LU

API	Purpose
LU	An immutable, reusable factorization.
luDecompose	Factor a square matrix using the default relative pivot tolerance.
luDecomposeWithTolerance	Factor with a finite, nonnegative relative tolerance.
luLower	Unit lower-triangular factor.
luUpper	Upper-triangular factor.
luPermutation	Final zero-based row order: row i of P A is row luPermutation factors !! i of A.
solveLU	Solve A x = b using an existing factorization.
solveMatrixLU	Solve A X = B for any number of right-hand-side columns.
solve	Factor A with the default tolerance and solve A x = b.
solveMatrix	Factor A with the default tolerance and solve A X = B.
determinantLU	Determinant from the permutation sign and diagonal of U.
determinant	Determinant using the default factorization policy.
inverse	Invert a nonsingular square matrix by solving against the identity.

HMath.Matrix.QR

API	Purpose
QR	Opaque column-pivoted Householder factors satisfying A[:,p]=Q*R, with numerical-rank estimate.
qrDecompose	Column-pivoted Householder QR of a finite m-by-n matrix using relative rank tolerance max(m,n)*machine epsilon.
qrDecomposeWithTolerance	Given finite nonnegative relative rank tolerance then finite matrix, compute column-pivoted Householder QR.
qrQ	Extract the square m-by-m orthogonal Q factor from a QR decomposition.
qrR	Extract the m-by-n upper trapezoidal R factor.
qrPermutation	Return zero-based original column indices in pivoted order, satisfying A[:,p]=Q*R.
qrRank	Return estimated numerical rank selected by the factorization's relative diagonal-pivot tolerance.
applyQ	Given QR factorization then a matrix with m rows, return Q times that matrix.
applyQT	Given QR factorization then a matrix with m rows, return transpose(Q) times it.
leastSquares	Given finite m-by-n design matrix then m-element RHS, factor with pivoted QR and return least-squares coefficients.
leastSquaresMatrix	Given finite m-by-n design and m-by-k RHS matrices, return n-by-k least-squares solution via QR.
solveQR	Given full-column-rank QR factors of an m-by-n matrix then m-element right-hand side, return n least-squares coefficients in original column order.
solveMatrixQR	Given QR factors and an m-by-k RHS, return n-by-k least-squares coefficients, undoing column pivoting.
ridgeSolve	Given finite nonnegative lambda, design matrix, then RHS list, minimize squared residual plus lambda times squared coefficient norm.
ridgeSolveMatrix	Given lambda, m-by-n design, then m-by-k RHS, return n-by-k ridge coefficients via augmented QR.

HMath.Sparse

API	Purpose
Sparse	Canonical immutable sparse matrix with summed duplicate coordinates and explicit zeros removed.
SparseFormat	Interchange representation selector carrying COO,CSR,CSC,DIA,BSR,or ELLPACK arrays; validate through fromFormat.
COO	Declaration and type contract below.
CSR	Declaration and type contract below.
CSC	Declaration and type contract below.
DIA	Declaration and type contract below.
BSR	Declaration and type contract below.
ELLPACK	Declaration and type contract below.
sparse	Given rows,columns,then (row,column,value) entries, validate coordinates, sum duplicates and remove explicit zeros.
sparseShape	Return declared (row count,column count), including empty dimensions.
nonzeros	Return stored nonzero count after duplicate coalescing and zero removal.
sparseEntries	Return canonical nonzero (row,column,value) triples in ascending coordinate order.
fromFormat	Validate COOCSRCSCDIABSRELLPACK dimensions, pointersindices and values, then construct canonical sparse storage.
toCOO	Return coordinate-list SparseFormat with explicit rowcolumnvalue arrays, preserving canonical nonzeros.
toCSR	Return compressed sparse row format with row offsets, column indices and nonzero values.
toCSC	Return compressed sparse column format with column offsets, row indices and values.
toDIA	Return diagonal sparse format; absent positions in represented diagonals are padded with zero.
toBSR	Given positive block rows,block columns,then sparse matrix, build block-sparse-row format with zero padding within edge blocks.
toELLPACK	Return ELLPACK padded row storage, using row width equal to maximum nonzeros per row.
sparseToDense	Return a dense Matrix with the same dimensions and zeros for missing entries.
denseToSparse	Convert a dense matrix to canonical sparse storage, discarding exact zeros and preserving dimensions.
sparseTranspose	Exchange row/column coordinates and dimensions without changing values, returning canonical sparse order.
sparseAdd	Add equal-shaped sparse matrices, coalescing entries and dropping resulting zeros.
sparseSubtract	Subtract right sparse values from left for equal shapes; coalesce and remove zeros.
spmv	Given sparse matrix then dense vector, return matrix-vector product.
spmm	Multiply compatible sparse matrices using sparse row contributions.
sparseKronecker	Return sparse Kronecker product, multiplying row/column dimensions and nonzero pairs.
sparseTriangularSolve	Given lower Bool,square finite sparse matrix,then RHS list, solve by forward/back substitution.
sparseDiagonalSolve	Given diagonal square sparse matrix then RHS, divide by each diagonal pivot.
SparseLU	Opaque sparse row-pivoted LU factors with reconstruction permutation; solve through sparseSolveLU.
sparseLU	Given finite nonnegative relative pivot tolerance then square finite sparse matrix, compute sparse LU with row pivoting.
sparseLULower	Extract unit lower-triangular sparse factor from a SparseLU result.
sparseLUUpper	Extract upper-triangular sparse factor from a SparseLU result.
sparseLUPermutation	Return LU row permutation in pivoted order, for reconstructing PA=LU.
sparseSolveLU	Given sparse LU factors then RHS list, apply row permutation and triangular solves.
sparseCholesky	Factor a symmetric positive-definite finite sparse matrix into a lower sparse factor.
SparseQR	Opaque sparse QR factors with implicit orthogonal transform; use sparseQRUpper and sparseQRApplyQT.
sparseQR	Sparse Householder QR retains only the R nonzeros and compact reflectors; fill is retained, never silently dropped.
sparseQRUpper	Extract upper-triangular/trapezoidal sparse R from a sparse QR factorization.
sparseQRApplyQT	Given sparse QR factors then compatible RHS list, apply transpose(Q) without constructing dense Q.
sparseGraph	For a square sparse matrix, return one adjacency row per vertex with stored (column,value) neighbors.
sparseStorageBytes	Logical payload bytes: two machine-width indices and caller-provided scalar width per nonzero.
SparseTensor	Canonical sparse multidimensional entries, constructed by sparseTensor with checked coordinates and duplicate coalescing.
sparseTensor	Given dimensions then (coordinate-list,value) entries, validate rank/bounds, sum duplicate coordinates and remove zeros.
sparseTensorEntries	Return canonical nonzero (coordinate list,value) pairs in lexicographic coordinate order.
sparseTensorShape	Return axis dimensions of a sparse tensor; [] denotes a scalar.
sparseTensorLookup	Given sparse tensor then full zero-based coordinates, return stored value or zero.
sparseEigenReference	Small-problem spectral reference backend.
sparseSVDReference	Given explicit dense-byte budget then sparse Double matrix, densify only if within budget and run checked dense SVD.

HMath.Sparse.Spectral

API	Purpose
SparseSpectralOptions	Residual tolerances, per-pair iteration budget and an explicit workspace limit measured in scalar elements.
SparseSpectralOptions	Declaration and type contract below.
sparseAbsoluteTolerance	Absolute norm tolerance, finite and nonnegative.
sparseRelativeTolerance	Relative tolerance against max(abs eigen/singular value,maxAbs input).
sparseIterationLimit	Nonnegative number of power updates per pair.
sparseWorkspaceLimit	Maximum estimated simultaneous vector elements, nonnegative.
defaultSparseSpectralOptions	Absolute tolerance 1e-12, relative tolerance 1e-10, 1000 iterations per pair and a workspace limit of ten million scalar elements.
SparseEigenResult	A normalized real symmetric eigenpair and its independently checked residual.
SparseEigenResult	Declaration and type contract below.
sparseEigenvalue	Rayleigh quotient; negative dominant eigenvalues retain their sign.
sparseEigenvector	Unit Euclidean norm eigenvector.
sparseEigenResidual	Original norm ||A v-lambda v||_2.
sparseEigenIterations	Number of completed power updates.
sparseDominantEigen	Symmetric power iteration targeting the eigenvalue of largest magnitude.
SparseSVD	Compact singular triplets.
SparseSVD	Declaration and type contract below.
sparseSingularValues	Nonnegative singular estimates in extraction order.
sparseLeftVectors	Left unit vectors, one list per singular triplet.
sparseRightVectors	Right unit vectors (columns of V, rather than rows of V^T).
sparseSingularResiduals	Original (||A v-s u||_2,||A^T u-s v||_2) residuals.
sparseSingularIterations	Power updates used by each triplet.
sparseTruncatedSVD	Truncated SVD by two-sided power iteration and two-pass orthogonal deflation.

HMath.Vector

API	Purpose
Vector	Immutable dynamically sized coordinate list.
vector	Wrap a list as an immutable dynamic vector without numeric validation.
vectorValues	Extract the immutable list of vector values in order; no shape conversion or numeric validation occurs.
vectorDimension	Return the number of coordinates in a dynamic vector.
FixedVector	Opaque vector whose length matches its KnownNat type index.
SparseVector	Opaque dynamic vector with canonical nonzero coordinate entries.
fixedVector	Construct a vector whose list length equals the type-level KnownNat size.
fixedValues	Extract the ordered coordinates of a type-indexed vector; its length is the type-level size.
sparseVector	Given nonnegative dimension and (zero-based index,value) entries, sum duplicate indices and remove explicit zeros.
sparseVectorValues	Expand a sparse vector to a dense list of its declared dimension, filling absent entries with zero.
vectorSlice	Given start, count, then vector, return the half-open slice [start,start+count).
vectorConcat	Concatenate a list of vectors in order.
vectorAdd	Add corresponding values of equally sized vectors, returning a new vector.
vectorSubtract	Subtract corresponding right-hand coordinates from left-hand coordinates.
vectorScale	Given scalar then vector, multiply every coordinate by the scalar.
vectorDot	Sum coordinatewise products of equally sized vectors.
cross	Compute the right-handed cross product of two three-dimensional vectors.
outer	Return the matrix of pairwise products a[i]*b[j], with shape (dimension a,dimension b).
triple	Given a,b,c, compute scalar triple product a dot (b cross c).
norm0	Count nonzero coordinates, returning Int.
norm1	Sum absolute coordinate values.
norm2	Compute Euclidean length using a scaled stable norm to reduce intermediate overflow/underflow.
normP	Given positive exponent p then finite vector, compute the scaled p-norm.
normInfinity	Return maximum absolute coordinate, or zero for an empty vector.
normalizeVector	Return a unit-length vector in the same direction.
project	Given a then b, return the projection of a onto b.
reflect	Given incident vector then surface normal, return incident minus twice its projection onto the normal.
refract	Given positive finite refractive-index ratio, incident vector, then normal, return the refracted unit-direction expression.
distance	Return Euclidean distance between equal-sized finite vectors.
angle	Return the angle in radians, acos of cosineSimilarity, between two nonzero finite equally sized vectors.
cosineSimilarity	Return normalized dot product clamped to [-1,1].
gramSchmidt	Given finite nonnegative relative tolerance then vectors, perform two-pass modified Gram-Schmidt, returning an orthonormal basis in input order.