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numojo.routines.linalg.products

Matrix and vector products (numojo.routines.linalg.products)

This module provides functions for computing products of vectors and matrices, such as cross product, dot product, and matrix multiplication.

Functions

cross

cross[dtype: DType = DType.float64](array1: NDArray[dtype], array2: NDArray[dtype]) -> NDArray[dtype]

Compute the cross product of two arrays.

Parameters dtype: The element type.

Constraints

array1 and array2 must be of shape (3,).

Parameters:

  • dtype (DType)

Args:

  • array1 (NDArray): A array.
  • array2 (NDArray): A array.

Returns:

  • NDArray

Raises

dot

dot[dtype: DType = DType.float64](array1: NDArray[dtype], array2: NDArray[dtype]) -> NDArray[dtype]

Compute the dot product of two arrays.

Parameters dtype: The element type.

Constraints

array1 and array2 must be 1 dimensional.

Parameters:

  • dtype (DType)

Args:

  • array1 (NDArray): A array.
  • array2 (NDArray): A array.

Returns:

  • NDArray

Raises

tile

tile[tiled_fn: Static2DTileUnitFunc, tile_x: Int, tile_y: Int](end_x: Int, end_y: Int)

Parameters:

  • tiled_fn (Static2DTileUnitFunc)
  • tile_x (Int)
  • tile_y (Int)

Args:

  • end_x (Int)
  • end_y (Int)

matmul_tiled_unrolled_parallelized

matmul_tiled_unrolled_parallelized[dtype: DType](A: NDArray[dtype], B: NDArray[dtype]) -> NDArray[dtype]

Matrix multiplication vectorized, tiled, unrolled, and parallelized.

Parameters:

  • dtype (DType)

Args:

  • A (NDArray)
  • B (NDArray)

Returns:

  • NDArray

Raises

matmul_1darray

matmul_1darray[dtype: DType](A: NDArray[dtype], B: NDArray[dtype]) -> NDArray[dtype]

Array multiplication for 1-d arrays (inner dot).

Parameters:

  • dtype (DType)

Args:

  • A (NDArray)
  • B (NDArray)

Returns:

  • NDArray

Raises

matmul_2darray

matmul_2darray[dtype: DType](A: NDArray[dtype], B: NDArray[dtype]) -> NDArray[dtype]

Array multiplication for 2-d arrays (inner dot).

Parameter: dtype: Data type.

Return: A multiplied by B.

Notes: The multiplication is vectorized and parallelized.

References: [1] https://docs.modular.com/mojo/notebooks/Matmul. resultompared to the reference, we increases the size of the SIMD vector from the default width to 16. The purpose is to increase the performance via SIMD. This reduces the execution time by ~50 percent compared to matmul_parallelized and matmul_tiled_unrolled_parallelized for large matrices.

Parameters:

  • dtype (DType)

Args:

  • A (NDArray): First array.
  • B (NDArray): Second array.

Returns:

  • NDArray

Raises

When the shape does not match.

matmul

Overload 1

matmul[dtype: DType](A: NDArray[dtype], B: NDArray[dtype]) -> NDArray[dtype]

Array multiplication for any dimensions.

Parameter: dtype: Data type.

Return: A multiplied by B.

Notes:

When A and B are 1darray, it is equal to dot of vectors: (i) @ (i) -> (1).

When A and B are 2darray, it is equal to inner products of matrices: (i,j) @ (j,k) -> (i,k).

When A and B are more than 2d, it is equal to a stack of 2darrays: (i,j,k) @ (i,k,l) -> (i,j,l) and (i,j,k,l) @ (i,j,l,m) -> (i,j,k,m).

Parameters:

  • dtype (DType)

Args:

  • A (NDArray): First array.
  • B (NDArray): Second array.

Returns:

  • NDArray

Raises

(1) The shapes of first n-2 dimensions do not match.

(2) The shape of -2 dimension of first array does not match the shape of -1 dimension of the second array.

Overload 2

matmul[dtype: DType](A: Matrix[dtype], B: Matrix[dtype]) -> Matrix[dtype]

Matrix multiplication.

Example: 

from numojo import Matrix
from numojo.routines.linalg import matmul
var A = Matrix.rand(shape=(1000, 1000))
var B = Matrix.rand(shape=(1000, 1000))
var result = matmul(A, B)

Parameters:

  • dtype (DType)

Args:

  • A (Matrix)
  • B (Matrix)

Returns:

  • Matrix

Raises

matmul_naive

matmul_naive[dtype: DType](A: NDArray[dtype], B: NDArray[dtype]) -> NDArray[dtype]

Matrix multiplication with three nested loops.

Parameters:

  • dtype (DType)

Args:

  • A (NDArray)
  • B (NDArray)

Returns:

  • NDArray

Raises