backends/default Tutorial

Small Case: Accept The Default Dense Backend Without Hard-Wiring The Algorithm

///|
fn[M : @algebra.MatrixShape] matrix_shape(matrix : M) -> (Int, Int) {
  @default.shape_of(matrix)
}

///|
test "shape_of dispatches through MatrixShape" {
  let matrix : @default.DenseMatrix[Int] = @default.DenseMatrix::from_2d_array([
    [1, 2],
    [3, 4],
  ])
  let (rows, cols) = matrix_shape(matrix)
  inspect(rows, content="2")
  inspect(cols, content="2")
}

This case demonstrates the intended role of the default backend layer:

1. Construct a concrete dense matrix with DenseMatrix.
2. Pass it into a helper that asks only for MatrixShape.
3. Let @default.shape_of bridge the concrete wrapper and the public trait-oriented view.

That gives users a ready-to-use dense backend while still keeping new algorithms portable.

Suggested Flow

1. Start with DenseMatrix, DenseVector, ImmutableDenseMatrix, or ImmutableDenseVector when you want the repository's reference backend immediately.
2. Write new reusable helpers against algebra traits such as MatrixShape.
3. Keep the backend wrapper at the boundary where concrete storage is actually chosen.

Practical Guidance

• Use the default backend wrappers when you want a public dense implementation with straightforward constructors.
• Keep backend-specific products, norms, solve strategies, or decompositions in the backend or a dedicated algorithm layer unless they can be expressed as same-category structural operations.
• Treat backends/default as a bridge layer, not as the only place where generic linear algebra code should live.