General Linear System Solver

Description

Use bounded partial-pivot Gauss–Jordan elimination to classify a rectangular real linear system and return a particular solution, coefficient nullspace, and reduced augmented matrix under an explicit zero threshold, without a conditioning guarantee.

A real linear system has equations A*x=b. A consistent system can have one solution or a family of solutions; an inconsistent one has none. A particular solution and the nullspace of A describe the full family only when the system is consistent.

When to use this tool

Use this tool when you need the stated calculation or classification with explicit inputs, intermediate evidence, and boundary conventions.

How to use the tool and how it works

Supply coefficients, rightHandSide, absoluteTolerance in the field shapes below.
Run the calculation. Correct format, shape, range, or undefined-statistic errors before interpreting a result.
Read augmentedRank, coefficientRank, nullspaceBasis, particularSolution, reducedAugmentedMatrix, solutionType. Use the conventions below.

Apply partial-pivot Gauss-Jordan elimination to the augmented coefficient/right-hand-side matrix, treating magnitudes at most absoluteTolerance as zero. Compare coefficient and augmented ranks to classify consistency, set free variables to zero for a particular solution, and construct one nullspace vector per free column.^[1]

Input field	Meaning and units	Accepted input
coefficients	Rectangular real matrix A; each row is one equation and each column one unknown.	Required. Array of Array of Finite number.; minimum items 1; maximum items 32.; minimum items 1; maximum items 32.
rightHandSide	Vector b aligned with coefficient rows.	Required. Array of Finite number.; minimum items 1; maximum items 32.
absoluteTolerance	Absolute zero threshold applied at each elimination stage; default 0 tests floating-point zeros exactly. A positive threshold gives an approximate classification, not a certified rank.	Optional. Finite number; minimum 0; default 0.

Output field	Interpretation	Returned shape
augmentedRank	Augmented pivot count under that threshold.	Required. Safe integer; minimum 0; maximum 32.
coefficientRank	Coefficient pivot count under the selected absolute zero threshold.	Required. Safe integer; minimum 0; maximum 32.
nullspaceBasis	Vectors spanning the computed homogeneous coefficient nullspace.	Required. Array of Array of Finite number.; maximum items 32.; maximum items 32.
particularSolution	One value per unknown with free variables set to zero, or empty when inconsistent.	Required. Array of Finite number.; maximum items 32.
reducedAugmentedMatrix	Eliminated matrix including its final right-hand-side column.	Required. Array of Array of Finite number.; minimum items 2; maximum items 33.; minimum items 1; maximum items 32.
solutionType	unique, infinite, or inconsistent under the performed arithmetic and tolerance.	Required. Text; one of "inconsistent", "infinite", "unique".

json

{
  "input": {
    "coefficients": [
      [
        0,
        1,
        1
      ],
      [
        2,
        2,
        0
      ],
      [
        0,
        0,
        1
      ]
    ],
    "rightHandSide": [
      5,
      6,
      3
    ]
  },
  "output": {
    "augmentedRank": 3,
    "coefficientRank": 3,
    "nullspaceBasis": [],
    "particularSolution": [
      1,
      2,
      3
    ],
    "reducedAugmentedMatrix": [
      [
        1,
        0,
        0,
        1
      ],
      [
        0,
        1,
        0,
        2
      ],
      [
        0,
        0,
        1,
        3
      ]
    ],
    "solutionType": "unique"
  }
}

Limitations and assumptions

Rows must be rectangular and match rightHandSide length. Inconsistent output has an empty particularSolution but still returns coefficient nullspace evidence. For consistent systems every solution is the particular vector plus a linear combination of basis vectors. Default tolerance zero and finite arithmetic do not certify conditioning or numerical rank.
The field tables state structural bounds; cross-field constraints and bounded search or output budgets also apply. Unknown keys are rejected where the object contract disallows them. Input errors are distinct from null fields and deliberate invalid controls in a successful output.

Alternative or Complementary methods

Choose the statistic according to the question and data conventions. Inspect the supplied series or matrix and the stated assumptions before comparing outputs from implementations with different defaults.

References

Tool behavior and example: General Linear System Solver — tool.io

Similar or alternative tools

Inverse
Invert a square matrix with Gauss-Jordan elimination under partial pivoting.
Qr decomposition
Householder QR decomposition of a square matrix: orthogonal Q and upper-triangular R with A = Q·R.