dsh-linalg

dsh-linalg

Linear algebra toolbox for DeepSeek Harness (dsh): matrix multiply, determinant, inverse, trace, transpose, reduced row echelon form, rank, integer matrix powers, a linear system solver that classifies unique / infinitely many / no solutions, a symmetric eigen-decomposition with measured residuals, matrix factorizations (LU with partial pivoting, thin QR, Cholesky) with measured reconstruction residuals, a least-squares solver for overdetermined systems, and vector operations (dot / cross / norm / projection / angle).

Zero runtime dependencies, pure local arithmetic — no network, no processes, no eval.

Why

Language models frequently make arithmetic errors on matrix multiplication, determinants, inverses and linear systems. This plugin hands those computations to exact local code with Gaussian elimination and partial pivoting, so results are deterministic and correct to 10 decimal places.

Install

dsh plugin --profile web add github:TYEclipse/dsh-linalg

Replace web with your profile name. (Requires pnpm on your PATH.)

Tools

Input matrices are plain JSON arrays of arrays, e.g. [[1, 2], [3, 4]]. Maximum dimension is 20×20 by default (configurable).

Examples

Determinant:

matrix_compute(matrix=[[2,1,1],[1,3,2],[1,0,0]], op="determinant")  →  -1

Inverse:

matrix_compute(matrix=[[4,7],[2,6]], op="inverse")
→ [[0.6,-0.7],[-0.2,0.4]]

Solve 2x+3y=8, x−y=1:

solve_linear(a=[[2,3],[1,-1]], b=[8,1])
→ unique solution: [2.2, 1.2]

Under-determined system (1 equation, 3 variables) reports the full parametric form:

solve_linear(a=[[1,1,1],[1,2,3]], b=[6,14])
→ kind=infinite, particular=[-2,8,0], nullspaceBasis=[[1,-2,1]], freeVariableCount=1

Cross product:

vector_ops(op="cross", a=[1,2,3], b=[4,5,6])  →  [-3,6,-3]

Angle between [3,4] and the x-axis:

vector_ops(op="angle", a=[3,4], b=[4,0])  →  53.1301023542  (degrees)

Rank and integer powers:

matrix_compute(matrix=[[1,2,3],[4,5,6],[7,8,9]], op="rank")  →  2
matrix_compute(matrix=[[1,1],[1,0]], op="power", exponent=20)
→ [[10946, 6765], [6765, 4181]]

Eigenvalues of a symmetric matrix (2x2 by hand would be a quadratic; here it is exact):

matrix_eigen(matrix=[[2,1],[1,2]])
→ eigenvalues (descending): 3, 1
  sum = 4, max residual = 0
  eigenvectors (unit, one per row):
    lambda1 = 3: [0.7071067812, 0.7071067812]
    lambda2 = 1: [0.7071067812, -0.7071067812]

The 4x4 tridiagonal [[2,-1,0,0],[-1,2,-1,0],[0,-1,2,-1],[0,0,-1,2]] gives 3.6180339887, 2.6180339887, 1.3819660113, 0.3819660113 — the closed form 2 − 2·cos(kπ/5).

LU factorization (row permuted, one swap):

matrix_decompose(matrix=[[2,1,1],[4,-6,0],[-2,7,2]], kind="lu")
→ p = [1, 0, 2]  (row i of P·A is row p[i] of A)
  swaps = 1, sign = -1, singular = false, det from U = -16
  L = [[1,0,0],[0.5,1,0],[-0.5,1,1]]
  U = [[4,-6,0],[0,4,1],[0,0,1]]

Thin QR (R diagonal positive, so the factorization is unique):

matrix_decompose(matrix=[[3,0],[4,5]], kind="qr")
→ Q = [[0.6,-0.8],[0.8,0.6]]   R = [[5,4],[0,3]]

Cholesky of a symmetric positive definite matrix:

matrix_decompose(matrix=[[4,12,-16],[12,37,-43],[-16,-43,98]], kind="cholesky")
→ L = [[2,0,0],[6,1,0],[-8,5,3]], det from L = 36

Least squares — fit y = a + b·x through (1,1), (2,2), (3,2):

least_squares(a=[[1,1],[1,2],[1,3]], b=[1,2,2])
→ x1 = 0.6666666667, x2 = 0.5
  min ‖A·x − b‖ = 0.4082482905, max ‖Aᵀ(A·x − b)‖ = 0

Numerical behaviour

• Gaussian elimination with partial pivoting (largest-magnitude pivot per column).
• All output numbers are rounded to 10 decimal places (configurable roundPlaces); values below 5e-12 are snapped to exactly 0, so -0 never appears.
• Pivot tolerance is 1e-12: entries below that are treated as zero, so numerically singular matrices are reported as singular instead of producing garbage.
• Singular inverse requests fail cleanly with an error message; inconsistent systems return kind: "none" with the reducing row equation.
• matrix_eigen runs the cyclic Jacobi method in full double precision: every eigenvector is unit length, eigenvectors are signed so the leading non-zero component is positive, and the reply carries maxResidual = the largest |A·v − λ·v| entry actually measured. Non-square or non-symmetric input, and non-convergence, are reported as errors — never a guessed spectrum.
• power uses binary exponentiation; negative exponents invert first (and fail cleanly on singular input), exponents are limited to the integer range −64…64.
• matrix_decompose reports the residual it actually measured (max |P·A − L·U|, max |A − Q·R|, max |A − L·Lᵀ|). A singular matrix still gets an LU pair, but singular: true is set and the determinant from U is 0; linearly dependent columns make QR fail (no orthonormal completion is invented); an asymmetric or indefinite matrix is rejected by Cholesky with the offending entry / pivot spelled out.
• least_squares solves through the thin QR factorization (not the normal equations, which square the condition number) and returns both the minimum residual norm and max |Aᵀ(A·x − b)| — zero proves the answer really is the minimizer. Rank-deficient or underdetermined input is an error rather than one arbitrary minimizer.

Configuration

- name: 'github:TYEclipse/dsh-linalg'
  config:
    maxDimension: 20   # max rows/columns (1–50)
    roundPlaces: 10    # decimals in every output number (1–15)

Development

pnpm install
pnpm build
pnpm test
pnpm lint

License

MIT