Matrix solve linear

matrix-solve-linear · version 1.0.0 · Vectors & matrices · free, no key needed

Solve the small square linear system A x = b for x using Gaussian elimination.

Use when you need to: solve linear system · solve Ax=b · linear equation solver.

Supported

• solve linear system
• solve Ax=b
• linear equation solver

Not supported

• non-square or rank-deficient (singular) systems
• systems larger than 12x12
• least-squares solutions for overdetermined systems

Behavior

• Input a is a square NxN coefficient matrix of finite JSON numbers, N from 1 to 12. Input b is a length-N finite JSON number vector. Magnitudes are bounded to at most 1000000.
• Computed via Gaussian elimination with partial (largest-magnitude) pivoting and back substitution, in IEEE-754 double precision.
• Optional input tolerance (default 1e-9, range 0 to 1) is the minimum acceptable pivot magnitude; if every remaining candidate pivot in a column is at or below tolerance, the system is rejected as singular or numerically singular (no unique solution).
• The result is the unique solution vector x such that a * x approximately equals b; a result entry of -0 is normalized to 0.
• Every result entry is checked to be a finite number before it is returned; a very small nonzero pivot (below tolerance-rejection but large enough to be accepted) can otherwise amplify through elimination to overflow, which is rejected as invalid input rather than silently serialized as null.

Input

• a (array of array of number, required): min items 1; max items 12
• b (array of number, required): min items 1; max items 12
• tolerance (number, optional): min 0; max 1

Output

• result (array of number, required)

Limits

• max dim: 12
• max abs value: 1000000

Example

Request input:

{
  "a": [
    [
      2,
      1
    ],
    [
      1,
      3
    ]
  ],
  "b": [
    3,
    5
  ]
}

Response:

{
  "result": {
    "result": [
      0.8,
      1.4
    ]
  }
}

How to call it

MCP

Connect https://computefirst.net/mcp (setup), then call execute with:

{
  "id": "matrix-solve-linear",
  "version": "1.0.0",
  "input": {
    "a": [
      [
        2,
        1
      ],
      [
        1,
        3
      ]
    ],
    "b": [
      3,
      5
    ]
  }
}

HTTP (no key)

curl -X POST https://computefirst.net/v1/tools/matrix-solve-linear/versions/1.0.0/execute \
  -H "Content-Type: application/json" \
  -d '{"a":[[2,1],[1,3]],"b":[3,5]}'

The machine-readable contract is at /v1/tools/matrix-solve-linear/versions/1.0.0.

CLI

node cli.mjs run matrix-solve-linear 1.0.0 --input input.json --base-url https://computefirst.net

Get the client at /clients/cli/.

Related tools

• Matrix vector multiply: Multiply an MxN finite-number matrix by a length-N finite-number vector, producing a length-M vector.
• Matrix add: Add two equal-shape finite-number matrices elementwise.
• Matrix determinant: Compute the determinant of a small square finite-number matrix.
• Matrix hadamard: Multiply two equal-shape finite-number matrices elementwise (Hadamard product).
• Matrix identity: Construct the NxN identity matrix for a given size.
• Matrix inverse: Compute the inverse of a small square finite-number matrix.