View Source Nx.LinAlg (Nx v0.8.0)

Solve the equation a x = b for x, assuming a is a batch of triangular matrices. Can also solve x a = b for x. See the :left_side option below.

Functions

adjoint(t)

Returns the adjoint of a given tensor.

If the input tensor is real it transposes it's two inner-most axes. If the input tensor is complex, it additionally applies Nx.conjugate/1 to it.

Examples

iex> Nx.LinAlg.adjoint(Nx.tensor([[1, 2], [3, 4]]))
#Nx.Tensor<
  s64[2][2]
  [
    [1, 3],
    [2, 4]
  ]
>

iex> Nx.LinAlg.adjoint(Nx.tensor([[1, Complex.new(0, 2)], [3, Complex.new(0, -4)]]))
#Nx.Tensor<
  c64[2][2]
  [
    [1.0-0.0i, 3.0-0.0i],
    [0.0-2.0i, 0.0+4.0i]
  ]
>

cholesky(tensor)

Performs a Cholesky decomposition of a batch of square matrices.

The matrices must be positive-definite and either Hermitian if complex or symmetric if real. An error is raised by the default backend if those conditions are not met. Other backends may emit undefined behaviour.

Examples

iex> Nx.LinAlg.cholesky(Nx.tensor([[20.0, 17.6], [17.6, 16.0]]))
#Nx.Tensor<
  f32[2][2]
  [
    [4.4721360206604, 0.0],
    [3.9354796409606934, 0.7155418395996094]
  ]
>

iex> Nx.LinAlg.cholesky(Nx.tensor([[[2.0, 3.0], [3.0, 5.0]], [[1.0, 0.0], [0.0, 1.0]]]))
#Nx.Tensor<
  f32[2][2][2]
  [
    [
      [1.4142135381698608, 0.0],
      [2.1213204860687256, 0.7071064710617065]
    ],
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ]
  ]
>

iex> t = Nx.tensor([
...>   [6.0, 3.0, 4.0, 8.0],
...>   [3.0, 6.0, 5.0, 1.0],
...>   [4.0, 5.0, 10.0, 7.0],
...>   [8.0, 1.0, 7.0, 25.0]
...> ])
iex> Nx.LinAlg.cholesky(t)
#Nx.Tensor<
  f32[4][4]
  [
    [2.4494898319244385, 0.0, 0.0, 0.0],
    [1.2247447967529297, 2.1213202476501465, 0.0, 0.0],
    [1.6329931020736694, 1.41421377658844, 2.309401035308838, 0.0],
    [3.265986204147339, -1.4142134189605713, 1.5877134799957275, 3.132491111755371]
  ]
>

iex> Nx.LinAlg.cholesky(Nx.tensor([[1.0, Complex.new(0, -2)], [Complex.new(0, 2), 5.0]]))
#Nx.Tensor<
  c64[2][2]
  [
    [1.0+0.0i, 0.0+0.0i],
    [0.0+2.0i, 1.0+0.0i]
  ]
>

iex> t = Nx.tensor([[[2.0, 3.0], [3.0, 5.0]], [[1.0, 0.0], [0.0, 1.0]]]) |> Nx.vectorize(x: 2)
iex> Nx.LinAlg.cholesky(t)
#Nx.Tensor<
  vectorized[x: 2]
  f32[2][2]
  [
    [
      [1.4142135381698608, 0.0],
      [2.1213204860687256, 0.7071064710617065]
    ],
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ]
  ]
>

determinant(tensor)

Calculates the determinant of batched square matrices.

Examples

For 2x2 and 3x3, the results are given by the closed formulas:

iex> Nx.LinAlg.determinant(Nx.tensor([[1, 2], [3, 4]]))
#Nx.Tensor<
  f32
  -2.0
>

iex> Nx.LinAlg.determinant(Nx.tensor([[1.0, 2.0, 3.0], [1.0, -2.0, 3.0], [7.0, 8.0, 9.0]]))
#Nx.Tensor<
  f32
  48.0
>

When there are linearly dependent rows or columns, the determinant is 0:

iex> Nx.LinAlg.determinant(Nx.tensor([[1.0, 0.0], [3.0, 0.0]]))
#Nx.Tensor<
  f32
  0.0
>

iex> Nx.LinAlg.determinant(Nx.tensor([[1.0, 2.0, 3.0], [-1.0, -2.0, -3.0], [4.0, 5.0, 6.0]]))
#Nx.Tensor<
  f32
  0.0
>

The determinant can also be calculated when the axes are bigger than 3:

iex> Nx.LinAlg.determinant(Nx.tensor([
...> [1, 0, 0, 0],
...> [0, 1, 2, 3],
...> [0, 1, -2, 3],
...> [0, 7, 8, 9.0]
...> ]))
#Nx.Tensor<
  f32
  48.0
>

iex> Nx.LinAlg.determinant(Nx.tensor([
...> [0, 0, 0, 0, -1],
...> [0, 1, 2, 3, 0],
...> [0, 1, -2, 3, 0],
...> [0, 7, 8, 9, 0],
...> [1, 0, 0, 0, 0]
...> ]))
#Nx.Tensor<
  f32
  48.0
>

iex> Nx.LinAlg.determinant(Nx.tensor([
...> [[2, 4, 6, 7], [5, 1, 8, 8], [1, 7, 3, 1], [3, 9, 2, 4]],
...> [[2, 5, 1, 3], [4, 1, 7, 9], [6, 8, 3, 2], [7, 8, 1, 4]]
...> ]))
#Nx.Tensor<
  f32[2]
  [630.0, 630.0]
>

iex> t = Nx.tensor([[[1, 0], [0, 2]], [[3, 0], [0, 4]]]) |> Nx.vectorize(x: 2)
iex> Nx.LinAlg.determinant(t)
#Nx.Tensor<
  vectorized[x: 2]
  f32
  [2.0, 12.0]
>

If the axes are named, their names are not preserved in the output:

iex> two_by_two = Nx.tensor([[1, 2], [3, 4]], names: [:x, :y])
iex> Nx.LinAlg.determinant(two_by_two)
#Nx.Tensor<
  f32
  -2.0
>

iex> three_by_three = Nx.tensor([[1.0, 2.0, 3.0], [1.0, -2.0, 3.0], [7.0, 8.0, 9.0]], names: [:x, :y])
iex> Nx.LinAlg.determinant(three_by_three)
#Nx.Tensor<
  f32
  48.0
>

Also supports complex inputs:

iex> t = Nx.tensor([[1, 0, 0], [0, Complex.new(0, 2), 0], [0, 0, 3]])
iex> Nx.LinAlg.determinant(t)
#Nx.Tensor<
  c64
  0.0+6.0i
>

iex> t = Nx.tensor([[0, 0, 0, 1], [0, Complex.new(0, 2), 0, 0], [0, 0, 3, 0], [1, 0, 0, 0]])
iex> Nx.LinAlg.determinant(t)
#Nx.Tensor<
  c64
  -0.0-6.0i
>

eigh(tensor, opts \\ [])

Calculates the Eigenvalues and Eigenvectors of batched Hermitian 2-D matrices.

It returns {eigenvals, eigenvecs}.

Options

:max_iter - integer. Defaults to 1_000 Number of maximum iterations before stopping the decomposition
:eps - float. Defaults to 1.0e-4 Tolerance applied during the decomposition

Note not all options apply to all backends, as backends may have specific optimizations that render these mechanisms unnecessary.

Examples

iex> {eigenvals, eigenvecs} = Nx.LinAlg.eigh(Nx.tensor([[1, 0], [0, 2]]))
iex> Nx.round(eigenvals)
#Nx.Tensor<
  f32[2]
  [1.0, 2.0]
>
iex> eigenvecs
#Nx.Tensor<
  f32[2][2]
  [
    [1.0, 0.0],
    [0.0, 1.0]
  ]
>

iex> {eigenvals, eigenvecs} = Nx.LinAlg.eigh(Nx.tensor([[0, 1, 2], [1, 0, 2], [2, 2, 3]]))
iex> Nx.round(eigenvals)
#Nx.Tensor<
  f32[3]
  [5.0, -1.0, -1.0]
>
iex> eigenvecs
#Nx.Tensor<
  f32[3][3]
  [
    [0.4075949788093567, 0.9131628274917603, 0.0],
    [0.40837883949279785, -0.18228201568126678, 0.8944271802902222],
    [0.8167576789855957, -0.36456403136253357, -0.4472135901451111]
  ]
>

iex> {eigenvals, eigenvecs} = Nx.LinAlg.eigh(Nx.tensor([[[2, 5],[5, 6]], [[1, 0], [0, 4]]]))
iex> Nx.round(eigenvals)
#Nx.Tensor<
  f32[2][2]
  [
    [9.0, -1.0],
    [1.0, 4.0]
  ]
>
iex> eigenvecs
#Nx.Tensor<
  f32[2][2][2]
  [
    [
      [0.5612090229988098, -0.8276740908622742],
      [0.8276740908622742, 0.5612090229988098]
    ],
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ]
  ]
>

iex> t = Nx.tensor([[[2, 5],[5, 6]], [[1, 0], [0, 4]]]) |> Nx.vectorize(x: 2)
iex> {eigenvals, eigenvecs} = Nx.LinAlg.eigh(t)
iex> Nx.round(eigenvals)
#Nx.Tensor<
  vectorized[x: 2]
  f32[2]
  [
    [9.0, -1.0],
    [1.0, 4.0]
  ]
>
iex> eigenvecs
#Nx.Tensor<
  vectorized[x: 2]
  f32[2][2]
  [
    [
      [0.5612090229988098, -0.8276740908622742],
      [0.8276740908622742, 0.5612090229988098]
    ],
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ]
  ]
>

Error cases

iex> Nx.LinAlg.eigh(Nx.tensor([[1, 2, 3], [4, 5, 6]]))
** (ArgumentError) tensor must be a square matrix or a batch of square matrices, got shape: {2, 3}

invert(tensor)

Inverts a batch of square matrices.

For non-square matrices, use pinv/2 for pseudo-inverse calculations.

Examples

iex> a = Nx.tensor([[1, 2, 1, 1], [0, 1, 0, 1], [0, 0, 1, 1], [0 , 0, 0, 1]])
iex> a_inv = Nx.LinAlg.invert(a)
#Nx.Tensor<
  f32[4][4]
  [
    [1.0, -2.0, -1.0, 2.0],
    [0.0, 1.0, 0.0, -1.0],
    [0.0, 0.0, 1.0, -1.0],
    [0.0, 0.0, 0.0, 1.0]
  ]
>
iex> Nx.dot(a, a_inv)
#Nx.Tensor<
  f32[4][4]
  [
    [1.0, 0.0, 0.0, 0.0],
    [0.0, 1.0, 0.0, 0.0],
    [0.0, 0.0, 1.0, 0.0],
    [0.0, 0.0, 0.0, 1.0]
  ]
>
iex> Nx.dot(a_inv, a)
#Nx.Tensor<
  f32[4][4]
  [
    [1.0, 0.0, 0.0, 0.0],
    [0.0, 1.0, 0.0, 0.0],
    [0.0, 0.0, 1.0, 0.0],
    [0.0, 0.0, 0.0, 1.0]
  ]
>

iex> a = Nx.tensor([[[1, 2], [0, 1]], [[1, 1], [0, 1]]])
iex> a_inv = Nx.LinAlg.invert(a)
#Nx.Tensor<
  f32[2][2][2]
  [
    [
      [1.0, -2.0],
      [0.0, 1.0]
    ],
    [
      [1.0, -1.0],
      [0.0, 1.0]
    ]
  ]
>
iex> Nx.dot(a, [2], [0], a_inv, [1], [0])
#Nx.Tensor<
  f32[2][2][2]
  [
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ],
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ]
  ]
>
iex> Nx.dot(a_inv, [2], [0], a, [1], [0])
#Nx.Tensor<
  f32[2][2][2]
  [
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ],
    [
      [1.0, 0.0],
      [0.0, 1.0]
    ]
  ]
>

If a singular matrix is passed, the result will silently fail.

iex> Nx.LinAlg.invert(Nx.tensor([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1, 1, 1, 1]]))
#Nx.Tensor<
  f32[4][4]
  [
    [NaN, NaN, NaN, NaN],
    [NaN, NaN, NaN, NaN],
    [NaN, NaN, NaN, NaN],
    [NaN, NaN, NaN, NaN]
  ]
>

Error cases

iex> Nx.LinAlg.invert(Nx.tensor([[3, 0, 0, 0], [2, 1, 0, 0]]))
** (ArgumentError) invert/1 expects a square matrix or a batch of square matrices, got tensor with shape: {2, 4}

least_squares(a, b)

Return the least-squares solution to a linear matrix equation Ax = b.

Examples

iex> Nx.LinAlg.least_squares(Nx.tensor([[1, 2], [2, 3]]), Nx.tensor([1, 2]))
#Nx.Tensor<
  f32[2]
  [1.0000004768371582, -2.665601925855299e-7]
>

iex> Nx.LinAlg.least_squares(Nx.tensor([[0, 1], [1, 1], [2, 1], [3, 1]]), Nx.tensor([-1, 0.2, 0.9, 2.1]))
#Nx.Tensor<
  f32[2]
  [0.9966151118278503, -0.947966456413269]
>

iex> Nx.LinAlg.least_squares(Nx.tensor([[1, 2, 3], [4, 5, 6]]), Nx.tensor([1, 2]))
#Nx.Tensor<
  f32[3]
  [-0.05534052848815918, 0.1111316829919815, 0.27760395407676697]
>

Error cases

iex> Nx.LinAlg.least_squares(Nx.tensor([1, 2, 3]), Nx.tensor([1, 2]))
** (ArgumentError) tensor of 1st argument must have rank 2, got rank 1 with shape {3}

iex> Nx.LinAlg.least_squares(Nx.tensor([[1, 2], [2, 3]]), Nx.tensor([[1, 2], [3, 4]]))
** (ArgumentError) tensor of 2nd argument must have rank 1, got rank 2 with shape {2, 2}

iex> Nx.LinAlg.least_squares(Nx.tensor([[1, Complex.new(0, 2)], [3, Complex.new(0, -4)]]),  Nx.tensor([1, 2]))
** (ArgumentError) Nx.LinAlg.least_squares/2 is not yet implemented for complex inputs

iex> Nx.LinAlg.least_squares(Nx.tensor([[1, 2], [2, 3]]), Nx.tensor([1, 2, 3]))
** (ArgumentError) the number of rows of the matrix as the 1st argument and the number of columns of the vector as the 2nd argument must be the same, got 1st argument shape {2, 2} and 2nd argument shape {3}

lu(tensor, opts \\ [])

Calculates the A = PLU decomposition of batched square 2-D matrices A.

Options

:eps - Rounding error threshold that can be applied during the factorization

Examples

iex> {p, l, u} = Nx.LinAlg.lu(Nx.tensor([[1, 2, 3], [4, 5, 6], [7, 8, 9]]))
iex> p
#Nx.Tensor<
  s64[3][3]
  [
    [0, 0, 1],
    [0, 1, 0],
    [1, 0, 0]
  ]
>
iex> l
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 0.0],
    [0.5714285969734192, 1.0, 0.0],
    [0.1428571492433548, 2.0, 1.0]
  ]
>
iex> u
#Nx.Tensor<
  f32[3][3]
  [
    [7.0, 8.0, 9.0],
    [0.0, 0.4285714328289032, 0.8571428656578064],
    [0.0, 0.0, 0.0]
  ]
>
iex> p |> Nx.dot(l) |> Nx.dot(u)
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 2.0, 3.0],
    [4.0, 5.0, 6.0],
    [7.0, 8.0, 9.0]
  ]
>

iex> {p, l, u} = Nx.LinAlg.lu(Nx.tensor([[1, 0, 1], [-1, 0, -1], [1, 1, 1]]))
iex> p
#Nx.Tensor<
  s64[3][3]
  [
    [1, 0, 0],
    [0, 0, 1],
    [0, 1, 0]
  ]
>
iex> l
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 0.0],
    [1.0, 1.0, 0.0],
    [-1.0, 0.0, 1.0]
  ]
>
iex> u
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 1.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 0.0]
  ]
>
iex> p |> Nx.dot(l) |> Nx.dot(u)
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 1.0],
    [-1.0, 0.0, -1.0],
    [1.0, 1.0, 1.0]
  ]
>

iex> {p, l, u} = Nx.LinAlg.lu(Nx.tensor([[[9, 8, 7], [6, 5, 4], [3, 2, 1]], [[-1, 0, -1], [1, 0, 1], [1, 1, 1]]]))
iex> p
#Nx.Tensor<
  s64[2][3][3]
  [
    [
      [1, 0, 0],
      [0, 1, 0],
      [0, 0, 1]
    ],
    [
      [1, 0, 0],
      [0, 0, 1],
      [0, 1, 0]
    ]
  ]
>
iex> l
#Nx.Tensor<
  f32[2][3][3]
  [
    [
      [1.0, 0.0, 0.0],
      [0.6666666865348816, 1.0, 0.0],
      [0.3333333432674408, 2.0, 1.0]
    ],
    [
      [1.0, 0.0, 0.0],
      [-1.0, 1.0, 0.0],
      [-1.0, 0.0, 1.0]
    ]
  ]
>
iex> u
#Nx.Tensor<
  f32[2][3][3]
  [
    [
      [9.0, 8.0, 7.0],
      [0.0, -0.3333333432674408, -0.6666666865348816],
      [0.0, 0.0, 0.0]
    ],
    [
      [-1.0, 0.0, -1.0],
      [0.0, 1.0, 0.0],
      [0.0, 0.0, 0.0]
    ]
  ]
>
iex> p |> Nx.dot([2], [0], l, [1], [0]) |> Nx.dot([2], [0], u, [1], [0])
#Nx.Tensor<
  f32[2][3][3]
  [
    [
      [9.0, 8.0, 7.0],
      [6.0, 5.0, 4.0],
      [3.0, 2.0, 1.0]
    ],
    [
      [-1.0, 0.0, -1.0],
      [1.0, 0.0, 1.0],
      [1.0, 1.0, 1.0]
    ]
  ]
>

iex> t = Nx.tensor([[[9, 8, 7], [6, 5, 4], [3, 2, 1]], [[-1, 0, -1], [1, 0, 1], [1, 1, 1]]]) |> Nx.vectorize(x: 2)
iex> {p, l, u} = Nx.LinAlg.lu(t)
iex> p
#Nx.Tensor<
  vectorized[x: 2]
  s64[3][3]
  [
    [
      [1, 0, 0],
      [0, 1, 0],
      [0, 0, 1]
    ],
    [
      [1, 0, 0],
      [0, 0, 1],
      [0, 1, 0]
    ]
  ]
>
iex> l
#Nx.Tensor<
  vectorized[x: 2]
  f32[3][3]
  [
    [
      [1.0, 0.0, 0.0],
      [0.6666666865348816, 1.0, 0.0],
      [0.3333333432674408, 2.0, 1.0]
    ],
    [
      [1.0, 0.0, 0.0],
      [-1.0, 1.0, 0.0],
      [-1.0, 0.0, 1.0]
    ]
  ]
>
iex> u
#Nx.Tensor<
  vectorized[x: 2]
  f32[3][3]
  [
    [
      [9.0, 8.0, 7.0],
      [0.0, -0.3333333432674408, -0.6666666865348816],
      [0.0, 0.0, 0.0]
    ],
    [
      [-1.0, 0.0, -1.0],
      [0.0, 1.0, 0.0],
      [0.0, 0.0, 0.0]
    ]
  ]
>

Error cases

iex> Nx.LinAlg.lu(Nx.tensor([[1, 1, 1, 1], [-1, 4, 4, -1], [4, -2, 2, 0]]))
** (ArgumentError) tensor must be a square matrix or a batch of square matrices, got shape: {3, 4}

matrix_power(tensor, power)

Produces the tensor taken to the given power by matrix dot-product.

The input is always a tensor of batched square matrices and an integer, and the output is a tensor of the same dimensions as the input tensor.

The dot-products are unrolled inside defn.

Examples

iex> Nx.LinAlg.matrix_power(Nx.tensor([[1, 2], [3, 4]]), 0)
#Nx.Tensor<
  s64[2][2]
  [
    [1, 0],
    [0, 1]
  ]
>

iex> Nx.LinAlg.matrix_power(Nx.tensor([[1, 2], [3, 4]]), 6)
#Nx.Tensor<
  s64[2][2]
  [
    [5743, 8370],
    [12555, 18298]
  ]
>

iex> Nx.LinAlg.matrix_power(Nx.eye(3), 65535)
#Nx.Tensor<
  s64[3][3]
  [
    [1, 0, 0],
    [0, 1, 0],
    [0, 0, 1]
  ]
>

iex> Nx.LinAlg.matrix_power(Nx.tensor([[1, 2], [3, 4]]), -1)
#Nx.Tensor<
  f32[2][2]
  [
    [-2.000000476837158, 1.0000003576278687],
    [1.5000004768371582, -0.5000002384185791]
  ]
>

iex> Nx.LinAlg.matrix_power(Nx.iota({2, 2, 2}), 3)
#Nx.Tensor<
  s64[2][2][2]
  [
    [
      [6, 11],
      [22, 39]
    ],
    [
      [514, 615],
      [738, 883]
    ]
  ]
>

iex> Nx.LinAlg.matrix_power(Nx.iota({2, 2, 2}), -3)
#Nx.Tensor<
  f32[2][2][2]
  [
    [
      [-4.875, 1.375],
      [2.75, -0.75]
    ],
    [
      [-110.37397766113281, 76.8742904663086],
      [92.24915313720703, -64.2494125366211]
    ]
  ]
>

iex> Nx.LinAlg.matrix_power(Nx.tensor([[1, 2], [3, 4], [5, 6]]), 1)
** (ArgumentError) matrix_power/2 expects a square matrix or a batch of square matrices, got tensor with shape: {3, 2}

matrix_rank(a, opts \\ [])

Return matrix rank of input M × N matrix using Singular Value Decomposition method.

Approximate the number of linearly independent rows by calculating the number of singular values greater than eps * max(singular values) * max(M, N).

This also appears in Numerical recipes in the discussion of SVD solutions for linear least squares [1].

[1] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes (3rd edition)”, Cambridge University Press, 2007, page 795.

Options

:eps - Rounding error threshold used to assume values as 0. Defaults to 1.0e-7

Examples

iex> Nx.LinAlg.matrix_rank(Nx.tensor([[1, 2], [3, 4]]))
#Nx.Tensor<
  u64
  2
>

iex> Nx.LinAlg.matrix_rank(Nx.tensor([[1, 1, 1, 1], [1, 1, 1, 1], [1, 2, 3, 4]]))
#Nx.Tensor<
  u64
  2
>

iex> Nx.LinAlg.matrix_rank(Nx.tensor([[1, 1, 1], [2, 2, 2], [8, 9, 10], [-2, 1, 5]]))
#Nx.Tensor<
  u64
  3
>

Error cases

iex> Nx.LinAlg.matrix_rank(Nx.tensor([1, 2, 3]))
** (ArgumentError) tensor must have rank 2, got rank 1 with shape {3}

iex> Nx.LinAlg.matrix_rank(Nx.tensor([[1, Complex.new(0, 2)], [3, Complex.new(0, -4)]]))
** (ArgumentError) Nx.LinAlg.matrix_rank/2 is not yet implemented for complex inputs

norm(tensor, opts \\ [])

Calculates the p-norm of a tensor.

For the 0-norm, the norm is the number of non-zero elements in the tensor.

Options

:axes - defines the axes upon which the norm will be calculated. Applies only on 2-norm for 2-D tensors. Default: nil.
:keep_axes - whether the calculation axes should be kept with length 1. Defaults to false
:ord - defines which norm will be calculated according to the table below. Default: nil.

ord	2-D	1-D
`nil`	Frobenius norm	2-norm
`:nuclear`	Nuclear norm	-
`:frobenius`	Frobenius norm	-
`:inf`	`max(sum(abs(x), axes: [1]))`	`max(abs(x))`
`:neg_inf`	`min(sum(abs(x), axes: [1]))`	`min(abs(x))`
0	-	Number of non-zero elements
1	`max(sum(abs(x), axes: [0]))`	as below
-1	`min(sum(abs(x), axes: [0]))`	as below
2	2-norm	as below
-2	smallest singular value	as below
other	-	pow(sum(pow(abs(x), p)), 1/p)

Examples

Vector norms

iex> Nx.LinAlg.norm(Nx.tensor([3, 4]))
#Nx.Tensor<
  f32
  5.0
>

iex> Nx.LinAlg.norm(Nx.tensor([3, 4]), ord: 1)
#Nx.Tensor<
  f32
  7.0
>

iex> Nx.LinAlg.norm(Nx.tensor([3, -4]), ord: :inf)
#Nx.Tensor<
  f32
  4.0
>

iex> Nx.LinAlg.norm(Nx.tensor([3, -4]), ord: :neg_inf)
#Nx.Tensor<
  f32
  3.0
>

iex> Nx.LinAlg.norm(Nx.tensor([3, -4, 0, 0]), ord: 0)
#Nx.Tensor<
  f32
  2.0
>

Matrix norms

iex> Nx.LinAlg.norm(Nx.tensor([[3, -1], [2, -4]]), ord: -1)
#Nx.Tensor<
  f32
  5.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[3, -2], [2, -4]]), ord: 1)
#Nx.Tensor<
  f32
  6.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[3, -2], [2, -4]]), ord: :neg_inf)
#Nx.Tensor<
  f32
  5.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[3, -2], [2, -4]]), ord: :inf)
#Nx.Tensor<
  f32
  6.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[3, 0], [0, -4]]), ord: :frobenius)
#Nx.Tensor<
  f32
  5.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[1, 0, 0], [0, -4, 0], [0, 0, 9]]), ord: :nuclear)
#Nx.Tensor<
  f32
  14.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[1, 0, 0], [0, -4, 0], [0, 0, 9]]), ord: -2)
#Nx.Tensor<
  f32
  1.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[3, 0], [0, -4]]))
#Nx.Tensor<
  f32
  5.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[3, 4], [0, -4]]), axes: [1])
#Nx.Tensor<
  f32[2]
  [5.0, 4.0]
>

iex> Nx.LinAlg.norm(Nx.tensor([[Complex.new(0, 3), 4], [4, 0]]), axes: [0])
#Nx.Tensor<
  f32[2]
  [5.0, 4.0]
>

iex> Nx.LinAlg.norm(Nx.tensor([[Complex.new(0, 3), 0], [4, 0]]), ord: :neg_inf)
#Nx.Tensor<
  f32
  3.0
>

iex> Nx.LinAlg.norm(Nx.tensor([[0, 0], [0, 0]]))
#Nx.Tensor<
  f32
  0.0
>

Error cases

iex> Nx.LinAlg.norm(Nx.tensor([3, 4]), ord: :frobenius)
** (ArgumentError) expected a 2-D tensor for ord: :frobenius, got a 1-D tensor

pinv(tensor, opts \\ [])

Calculates the Moore-Penrose inverse, or the pseudoinverse, of a matrix.

Options

:eps - Rounding error threshold used to assume values as 0. Defaults to 1.0e-10

Examples

Scalar case:

iex> Nx.LinAlg.pinv(2)
#Nx.Tensor<
  f32
  0.5
>

iex> Nx.LinAlg.pinv(0)
#Nx.Tensor<
  f32
  0.0
>

Vector case:

iex> Nx.LinAlg.pinv(Nx.tensor([0, 1, 2]))
#Nx.Tensor<
  f32[3]
  [0.0, 0.20000000298023224, 0.4000000059604645]
>

iex> Nx.LinAlg.pinv(Nx.tensor([0, 0, 0]))
#Nx.Tensor<
  f32[3]
  [0.0, 0.0, 0.0]
>

Matrix case:

iex> Nx.LinAlg.pinv(Nx.tensor([[1, 1], [3, 4]]))
#Nx.Tensor<
  f32[2][2]
  [
    [3.9924824237823486, -1.0052783489227295],
    [-3.0051186084747314, 1.0071179866790771]
  ]
>

iex> Nx.LinAlg.pinv(Nx.tensor([[0.5, 0], [0, 1], [0.5, 0]]))
#Nx.Tensor<
  f32[2][3]
  [
    [0.9999999403953552, 0.0, 0.9999998807907104],
    [0.0, 1.0, 0.0]
  ]
>

qr(tensor, opts \\ [])

Calculates the QR decomposition of a tensor with shape {..., M, N}.

Options

:mode - Can be one of :reduced, :complete. Defaults to :reduced For the following, K = min(M, N)
- :reduced - returns q and r with shapes {..., M, K} and {..., K, N}
- :complete - returns q and r with shapes {..., M, M} and {..., M, N}
:eps - Rounding error threshold that can be applied during the triangularization. Defaults to 1.0e-10

Examples

iex> {q, r} = Nx.LinAlg.qr(Nx.tensor([[-3, 2, 1], [0, 1, 1], [0, 0, -1]]))
iex> q
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 1.0]
  ]
>
iex> r
#Nx.Tensor<
  f32[3][3]
  [
    [-3.0, 2.0, 1.0],
    [0.0, 1.0, 1.0],
    [0.0, 0.0, -1.0]
  ]
>

iex> t = Nx.tensor([[3, 2, 1], [0, 1, 1], [0, 0, 1]])
iex> {q, r} = Nx.LinAlg.qr(t)
iex> q
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 1.0]
  ]
>
iex> r
#Nx.Tensor<
  f32[3][3]
  [
    [3.0, 2.0, 1.0],
    [0.0, 1.0, 1.0],
    [0.0, 0.0, 1.0]
  ]
>

iex> {qs, rs} = Nx.LinAlg.qr(Nx.tensor([[[-3, 2, 1], [0, 1, 1], [0, 0, -1]],[[3, 2, 1], [0, 1, 1], [0, 0, 1]]]))
iex> qs
#Nx.Tensor<
  f32[2][3][3]
  [
    [
      [1.0, 0.0, 0.0],
      [0.0, 1.0, 0.0],
      [0.0, 0.0, 1.0]
    ],
    [
      [1.0, 0.0, 0.0],
      [0.0, 1.0, 0.0],
      [0.0, 0.0, 1.0]
    ]
  ]
>
iex> rs
#Nx.Tensor<
  f32[2][3][3]
  [
    [
      [-3.0, 2.0, 1.0],
      [0.0, 1.0, 1.0],
      [0.0, 0.0, -1.0]
    ],
    [
      [3.0, 2.0, 1.0],
      [0.0, 1.0, 1.0],
      [0.0, 0.0, 1.0]
    ]
  ]
>

iex> t = Nx.tensor([[3, 2, 1], [0, 1, 1], [0, 0, 1], [0, 0, 1]], type: :f32)
iex> {q, r} = Nx.LinAlg.qr(t, mode: :reduced)
iex> q
#Nx.Tensor<
  f32[4][3]
  [
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 0.7071068286895752],
    [0.0, 0.0, 0.7071067690849304]
  ]
>
iex> r
#Nx.Tensor<
  f32[3][3]
  [
    [3.0, 2.0, 1.0],
    [0.0, 1.0, 1.0],
    [0.0, 0.0, 1.4142136573791504]
  ]
>

iex> t = Nx.tensor([[3, 2, 1], [0, 1, 1], [0, 0, 1], [0, 0, 0]], type: :f32)
iex> {q, r} = Nx.LinAlg.qr(t, mode: :complete)
iex> q
#Nx.Tensor<
  f32[4][4]
  [
    [1.0, 0.0, 0.0, 0.0],
    [0.0, 1.0, 0.0, 0.0],
    [0.0, 0.0, 1.0, 0.0],
    [0.0, 0.0, 0.0, 1.0]
  ]
>
iex> r
#Nx.Tensor<
  f32[4][3]
  [
    [3.0, 2.0, 1.0],
    [0.0, 1.0, 1.0],
    [0.0, 0.0, 1.0],
    [0.0, 0.0, 0.0]
  ]
>

iex> t = Nx.tensor([[[-3, 2, 1], [0, 1, 1], [0, 0, -1]],[[3, 2, 1], [0, 1, 1], [0, 0, 1]]]) |> Nx.vectorize(x: 2)
iex> {qs, rs} = Nx.LinAlg.qr(t)
iex> qs
#Nx.Tensor<
  vectorized[x: 2]
  f32[3][3]
  [
    [
      [1.0, 0.0, 0.0],
      [0.0, 1.0, 0.0],
      [0.0, 0.0, 1.0]
    ],
    [
      [1.0, 0.0, 0.0],
      [0.0, 1.0, 0.0],
      [0.0, 0.0, 1.0]
    ]
  ]
>
iex> rs
#Nx.Tensor<
  vectorized[x: 2]
  f32[3][3]
  [
    [
      [-3.0, 2.0, 1.0],
      [0.0, 1.0, 1.0],
      [0.0, 0.0, -1.0]
    ],
    [
      [3.0, 2.0, 1.0],
      [0.0, 1.0, 1.0],
      [0.0, 0.0, 1.0]
    ]
  ]
>

Error cases

iex> Nx.LinAlg.qr(Nx.tensor([1, 2, 3, 4, 5]))
** (ArgumentError) tensor must have at least rank 2, got rank 1 with shape {5}

iex> t = Nx.tensor([[-3, 2, 1], [0, 1, 1], [0, 0, -1]])
iex> Nx.LinAlg.qr(t, mode: :error_test)
** (ArgumentError) invalid :mode received. Expected one of [:reduced, :complete], received: :error_test

solve(a, b)

Solves the system AX = B.

A must have shape {..., n, n} and B must have shape {..., n, m} or {..., n}. X has the same shape as B.

Examples

iex> a = Nx.tensor([[1, 3, 2, 1], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]])
iex> Nx.LinAlg.solve(a, Nx.tensor([-3, 0, 4, -2])) |> Nx.round()
#Nx.Tensor<
  f32[4]
  [1.0, -2.0, 3.0, -4.0]
>

iex> a = Nx.tensor([[1, 0, 1], [1, 1, 0], [1, 1, 1]], type: :f64)
iex> Nx.LinAlg.solve(a, Nx.tensor([0, 2, 1])) |> Nx.round()
#Nx.Tensor<
  f64[3]
  [1.0, 1.0, -1.0]
>

iex> a = Nx.tensor([[1, 0, 1], [1, 1, 0], [0, 1, 1]])
iex> b = Nx.tensor([[2, 2, 3], [2, 2, 4], [2, 0, 1]])
iex> Nx.LinAlg.solve(a, b) |> Nx.round()
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 2.0, 3.0],
    [1.0, 0.0, 1.0],
    [1.0, 0.0, 0.0]
  ]
>

iex> a = Nx.tensor([[[14, 10], [9, 9]], [[4, 11], [2, 3]]])
iex> b = Nx.tensor([[[2, 4], [3, 2]], [[1, 5], [-3, -1]]])
iex> Nx.LinAlg.solve(a, b) |> Nx.round()
#Nx.Tensor<
  f32[2][2][2]
  [
    [
      [0.0, 0.0],
      [1.0, 0.0]
    ],
    [
      [-4.0, -3.0],
      [1.0, 1.0]
    ]
  ]
>

iex> a = Nx.tensor([[[1, 1], [0, 1]], [[2, 0], [0, 2]]]) |> Nx.vectorize(x: 2)
iex> b = Nx.tensor([[[2, 1], [5, -1]]]) |> Nx.vectorize(x: 1, y: 2)
iex> Nx.LinAlg.solve(a, b)
#Nx.Tensor<
  vectorized[x: 2][y: 2]
  f32[2]
  [
    [
      [1.0, 1.0],
      [6.0, -1.0]
    ],
    [
      [1.0, 0.5],
      [2.5, -0.5]
    ]
  ]
>

If the axes are named, their names are not preserved in the output:

iex> a = Nx.tensor([[1, 0, 1], [1, 1, 0], [1, 1, 1]], names: [:x, :y])
iex> Nx.LinAlg.solve(a, Nx.tensor([0, 2, 1], names: [:z])) |> Nx.round()
#Nx.Tensor<
  f32[3]
  [1.0, 1.0, -1.0]
>

Error cases

iex> Nx.LinAlg.solve(Nx.tensor([[1, 0], [0, 1]]), Nx.tensor([4, 2, 4, 2]))
** (ArgumentError) `b` tensor has incompatible dimensions, expected {2, 2} or {2}, got: {4}

iex> Nx.LinAlg.solve(Nx.tensor([[3, 0, 0, 0], [2, 1, 0, 0], [1, 1, 1, 1]]), Nx.tensor([4]))
** (ArgumentError) `a` tensor has incompatible dimensions, expected a square matrix or a batch of square matrices, got: {3, 4}

svd(tensor, opts \\ [])

Calculates the Singular Value Decomposition of batched 2-D matrices.

It returns {u, s, vt} where the elements of s are sorted from highest to lowest.

Options

:max_iter - integer. Defaults to 100 Number of maximum iterations before stopping the decomposition
:full_matrices? - boolean. Defaults to true If true, u and vt are of shape (M, M), (N, N). Otherwise, the shapes are (M, K) and (K, N), where K = min(M, N).

Note not all options apply to all backends, as backends may have specific optimizations that render these mechanisms unnecessary.

Examples

iex> {u, s, vt} = Nx.LinAlg.svd(Nx.tensor([[1, 0, 0], [0, 1, 0], [0, 0, -1]]))
iex> u
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, -1.0]
  ]
>
iex> s
#Nx.Tensor<
  f32[3]
  [1.0, 1.0, 1.0]
>
iex> vt
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 1.0]
  ]
>

iex> {u, s, vt} = Nx.LinAlg.svd(Nx.tensor([[2, 0, 0], [0, 3, 0], [0, 0, -1], [0, 0, 0]]))
iex> u
#Nx.Tensor<
  f32[4][4]
  [
    [0.0, 0.9999999403953552, 0.0, 0.0],
    [1.0, 0.0, 0.0, 0.0],
    [0.0, 0.0, -1.0, 0.0],
    [0.0, 0.0, 0.0, 1.0]
  ]
>
iex> s
#Nx.Tensor<
  f32[3]
  [3.0, 1.9999998807907104, 1.0]
>
iex> vt
#Nx.Tensor<
  f32[3][3]
  [
    [0.0, 1.0, 0.0],
    [1.0, 0.0, 0.0],
    [0.0, 0.0, 1.0]
  ]
>

iex> {u, s, vt} = Nx.LinAlg.svd(Nx.tensor([[2, 0, 0], [0, 3, 0], [0, 0, -1], [0, 0, 0]]), full_matrices?: false)
iex> u
#Nx.Tensor<
  f32[4][3]
  [
    [0.0, 0.9999999403953552, 0.0],
    [1.0, 0.0, 0.0],
    [0.0, 0.0, -1.0],
    [0.0, 0.0, 0.0]
  ]
>
iex> s
#Nx.Tensor<
  f32[3]
  [3.0, 1.9999998807907104, 1.0]
>
iex> vt
#Nx.Tensor<
  f32[3][3]
  [
    [0.0, 1.0, 0.0],
    [1.0, 0.0, 0.0],
    [0.0, 0.0, 1.0]
  ]
>

triangular_solve(a, b, opts \\ [])

Solve the equation a x = b for x, assuming a is a batch of triangular matrices. Can also solve x a = b for x. See the :left_side option below.

b must either be a batch of square matrices with the same dimensions as a or a batch of 1-D tensors with as many rows as a. Batch dimensions of a and b must be the same.

Options

The following options are defined in order of precedence

:transform_a - Defines op(a), depending on its value. Can be one of:
- :none -> op(a) = a
- :transpose -> op(a) = transpose(a) Defaults to :none
:lower - When true, defines the a matrix as lower triangular. If false, a is upper triangular. Defaults to true
:left_side - When true, solves the system as op(A).X = B. Otherwise, solves X.op(A) = B. Defaults to true.

Examples

iex> a = Nx.tensor([[3, 0, 0, 0], [2, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 1]])
iex> Nx.LinAlg.triangular_solve(a, Nx.tensor([4, 2, 4, 2]))
#Nx.Tensor<
  f32[4]
  [1.3333333730697632, -0.6666666865348816, 2.6666667461395264, -1.3333333730697632]
>

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [1, 1, 1]], type: :f64)
iex> Nx.LinAlg.triangular_solve(a, Nx.tensor([1, 2, 1]))
#Nx.Tensor<
  f64[3]
  [1.0, 1.0, -1.0]
>

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [0, 1, 1]])
iex> b = Nx.tensor([[1, 2, 3], [2, 2, 4], [2, 0, 1]])
iex> Nx.LinAlg.triangular_solve(a, b)
#Nx.Tensor<
  f32[3][3]
  [
    [1.0, 2.0, 3.0],
    [1.0, 0.0, 1.0],
    [1.0, 0.0, 0.0]
  ]
>

iex> a = Nx.tensor([[1, 1, 1, 1], [0, 1, 0, 1], [0, 0, 1, 2], [0, 0, 0, 3]])
iex> Nx.LinAlg.triangular_solve(a, Nx.tensor([2, 4, 2, 4]), lower: false)
#Nx.Tensor<
  f32[4]
  [-1.3333333730697632, 2.6666667461395264, -0.6666666865348816, 1.3333333730697632]
>

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [1, 2, 1]])
iex> b = Nx.tensor([[0, 2, 1], [1, 1, 0], [3, 3, 1]])
iex> Nx.LinAlg.triangular_solve(a, b, left_side: false)
#Nx.Tensor<
  f32[3][3]
  [
    [-1.0, 0.0, 1.0],
    [0.0, 1.0, 0.0],
    [1.0, 1.0, 1.0]
  ]
>

iex> a = Nx.tensor([[1, 1, 1], [0, 1, 1], [0, 0, 1]], type: :f64)
iex> Nx.LinAlg.triangular_solve(a, Nx.tensor([1, 2, 1]), transform_a: :transpose, lower: false)
#Nx.Tensor<
  f64[3]
  [1.0, 1.0, -1.0]
>

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [1, 1, 1]], type: :f64)
iex> Nx.LinAlg.triangular_solve(a, Nx.tensor([1, 2, 1]), transform_a: :none)
#Nx.Tensor<
  f64[3]
  [1.0, 1.0, -1.0]
>

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [1, 2, 1]])
iex> b = Nx.tensor([[0, 1, 3], [2, 1, 3]])
iex> Nx.LinAlg.triangular_solve(a, b, left_side: false)
#Nx.Tensor<
  f32[2][3]
  [
    [2.0, -5.0, 3.0],
    [4.0, -5.0, 3.0]
  ]
>

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [1, 2, 1]])
iex> b = Nx.tensor([[0, 2], [3, 0], [0, 0]])
iex> Nx.LinAlg.triangular_solve(a, b, left_side: true)
#Nx.Tensor<
  f32[3][2]
  [
    [0.0, 2.0],
    [3.0, -2.0],
    [-6.0, 2.0]
  ]
>

iex> a = Nx.tensor([
...> [1, 0, 0],
...> [1, Complex.new(0, 1), 0],
...> [Complex.new(0, 1), 1, 1]
...>])
iex> b = Nx.tensor([1, -1, Complex.new(3, 3)])
iex> Nx.LinAlg.triangular_solve(a, b)
#Nx.Tensor<
  c64[3]
  [1.0+0.0i, 0.0+2.0i, 3.0+0.0i]
>

iex> a = Nx.tensor([[[1, 0], [2, 3]], [[4, 0], [5, 6]]])
iex> b = Nx.tensor([[2, -1], [3, 7]])
iex> Nx.LinAlg.triangular_solve(a, b)
#Nx.Tensor<
  f32[2][2]
  [
    [2.0, -1.6666666269302368],
    [0.75, 0.5416666865348816]
  ]
>

iex> a = Nx.tensor([[[1, 1], [0, 1]], [[2, 0], [0, 2]]]) |> Nx.vectorize(x: 2)
iex> b = Nx.tensor([[[2, 1], [5, -1]]]) |> Nx.vectorize(x: 1, y: 2)
iex> Nx.LinAlg.triangular_solve(a, b, lower: false)
#Nx.Tensor<
  vectorized[x: 2][y: 2]
  f32[2]
  [
    [
      [1.0, 1.0],
      [6.0, -1.0]
    ],
    [
      [1.0, 0.5],
      [2.5, -0.5]
    ]
  ]
>

Error cases

iex> Nx.LinAlg.triangular_solve(Nx.tensor([[3, 0, 0, 0], [2, 1, 0, 0]]), Nx.tensor([4, 2, 4, 2]))
** (ArgumentError) triangular_solve/3 expected a square matrix or a batch of square matrices, got tensor with shape: {2, 4}

iex> Nx.LinAlg.triangular_solve(Nx.tensor([[3, 0, 0, 0], [2, 1, 0, 0], [1, 1, 1, 1], [1, 1, 1, 1]]), Nx.tensor([4]))
** (ArgumentError) incompatible dimensions for a and b on triangular solve

iex> Nx.LinAlg.triangular_solve(Nx.tensor([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1, 1, 1, 1]]), Nx.tensor([4, 2, 4, 2]))
** (ArgumentError) can't solve for singular matrix

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [1, 1, 1]], type: :f64)
iex> Nx.LinAlg.triangular_solve(a, Nx.tensor([1, 2, 1]), transform_a: :conjugate)
** (ArgumentError) complex numbers not supported yet

iex> a = Nx.tensor([[1, 0, 0], [1, 1, 0], [1, 1, 1]], type: :f64)
iex> Nx.LinAlg.triangular_solve(a, Nx.tensor([1, 2, 1]), transform_a: :other)
** (ArgumentError) invalid value for :transform_a option, expected :none, :transpose, or :conjugate, got: :other