# Linear Algebra Symbol API¶

## Overview¶

This document lists the linear algebra routines of the symbolic expression package:

 mxnet.symbol.linalg Linear Algebra Symbol API of MXNet.

The Linear Algebra Symbol API, defined in the symbol.linalg package, provides symbolic expressions for linear algebra routines.

In the rest of this document, we list routines provided by the symbol.linalg package.

## Linear Algebra¶

 gemm Performs general matrix multiplication and accumulation. gemm2 Performs general matrix multiplication. potrf Performs Cholesky factorization of a symmetric positive-definite matrix. potri Performs matrix inversion from a Cholesky factorization. trmm Performs multiplication with a lower triangular matrix. trsm Solves matrix equation involving a lower triangular matrix. sumlogdiag Computes the sum of the logarithms of the diagonal elements of a square matrix. syrk Multiplication of matrix with its transpose. gelqf LQ factorization for general matrix. syevd Eigendecomposition for symmetric matrix.

## API Reference¶

Linear Algebra Symbol API of MXNet.

mxnet.symbol.linalg.gelqf(A=None, name=None, attr=None, out=None, **kwargs)

LQ factorization for general matrix. Input is a tensor A of dimension n >= 2.

If n=2, we compute the LQ factorization (LAPACK gelqf, followed by orglq). A must have shape (x, y) with x <= y, and must have full rank =x. The LQ factorization consists of L with shape (x, x) and Q with shape (x, y), so that:

A = L * Q

Here, L is lower triangular (upper triangle equal to zero) with nonzero diagonal, and Q is row-orthonormal, meaning that

Q * QT

is equal to the identity matrix of shape (x, x).

If n>2, gelqf is performed separately on the trailing two dimensions for all inputs (batch mode).

Note

The operator supports float32 and float64 data types only.

Examples:

// Single LQ factorization
A = [[1., 2., 3.], [4., 5., 6.]]
Q, L = gelqf(A)
Q = [[-0.26726124, -0.53452248, -0.80178373],
[0.87287156, 0.21821789, -0.43643578]]
L = [[-3.74165739, 0.],
[-8.55235974, 1.96396101]]

// Batch LQ factorization
A = [[[1., 2., 3.], [4., 5., 6.]],
[[7., 8., 9.], [10., 11., 12.]]]
Q, L = gelqf(A)
Q = [[[-0.26726124, -0.53452248, -0.80178373],
[0.87287156, 0.21821789, -0.43643578]],
[[-0.50257071, -0.57436653, -0.64616234],
[0.7620735, 0.05862104, -0.64483142]]]
L = [[[-3.74165739, 0.],
[-8.55235974, 1.96396101]],
[[-13.92838828, 0.],
[-19.09768702, 0.52758934]]]


Defined in src/operator/tensor/la_op.cc:L552

Parameters: A (Symbol) – Tensor of input matrices to be factorized name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.gemm(A=None, B=None, C=None, transpose_a=_Null, transpose_b=_Null, alpha=_Null, beta=_Null, axis=_Null, name=None, attr=None, out=None, **kwargs)

Performs general matrix multiplication and accumulation. Input are tensors A, B, C, each of dimension n >= 2 and having the same shape on the leading n-2 dimensions.

If n=2, the BLAS3 function gemm is performed:

out = alpha * op(A) * op(B) + beta * C

Here, alpha and beta are scalar parameters, and op() is either the identity or matrix transposition (depending on transpose_a, transpose_b).

If n>2, gemm is performed separately for a batch of matrices. The column indices of the matrices are given by the last dimensions of the tensors, the row indices by the axis specified with the axis parameter. By default, the trailing two dimensions will be used for matrix encoding.

For a non-default axis parameter, the operation performed is equivalent to a series of swapaxes/gemm/swapaxes calls. For example let A, B, C be 5 dimensional tensors. Then gemm(A, B, C, axis=1) is equivalent to

A1 = swapaxes(A, dim1=1, dim2=3) B1 = swapaxes(B, dim1=1, dim2=3) C = swapaxes(C, dim1=1, dim2=3) C = gemm(A1, B1, C) C = swapaxis(C, dim1=1, dim2=3)

Note

The operator supports float32 and float64 data types only.

Examples:

// Single matrix multiply-add
A = [[1.0, 1.0], [1.0, 1.0]]
B = [[1.0, 1.0], [1.0, 1.0], [1.0, 1.0]]
C = [[1.0, 1.0, 1.0], [1.0, 1.0, 1.0]]
gemm(A, B, C, transpose_b=True, alpha=2.0, beta=10.0)
= [[14.0, 14.0, 14.0], [14.0, 14.0, 14.0]]

A = [[[1.0, 1.0]], [[0.1, 0.1]]]
B = [[[1.0, 1.0]], [[0.1, 0.1]]]
C = [[[10.0]], [[0.01]]]
gemm(A, B, C, transpose_b=True, alpha=2.0 , beta=10.0)
= [[[104.0]], [[0.14]]]


Defined in src/operator/tensor/la_op.cc:L81

Parameters: A (Symbol) – Tensor of input matrices B (Symbol) – Tensor of input matrices C (Symbol) – Tensor of input matrices transpose_a (boolean, optional, default=0) – Multiply with transposed of first input (A). transpose_b (boolean, optional, default=0) – Multiply with transposed of second input (B). alpha (double, optional, default=1) – Scalar factor multiplied with A*B. beta (double, optional, default=1) – Scalar factor multiplied with C. axis (int, optional, default='-2') – Axis corresponding to the matrix rows. name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.gemm2(A=None, B=None, transpose_a=_Null, transpose_b=_Null, alpha=_Null, axis=_Null, name=None, attr=None, out=None, **kwargs)

Performs general matrix multiplication. Input are tensors A, B, each of dimension n >= 2 and having the same shape on the leading n-2 dimensions.

If n=2, the BLAS3 function gemm is performed:

out = alpha * op(A) * op(B)

Here alpha is a scalar parameter and op() is either the identity or the matrix transposition (depending on transpose_a, transpose_b).

If n>2, gemm is performed separately for a batch of matrices. The column indices of the matrices are given by the last dimensions of the tensors, the row indices by the axis specified with the axis parameter. By default, the trailing two dimensions will be used for matrix encoding.

For a non-default axis parameter, the operation performed is equivalent to a series of swapaxes/gemm/swapaxes calls. For example let A, B be 5 dimensional tensors. Then gemm(A, B, axis=1) is equivalent to

A1 = swapaxes(A, dim1=1, dim2=3) B1 = swapaxes(B, dim1=1, dim2=3) C = gemm2(A1, B1) C = swapaxis(C, dim1=1, dim2=3)

Note

The operator supports float32 and float64 data types only.

Examples:

// Single matrix multiply
A = [[1.0, 1.0], [1.0, 1.0]]
B = [[1.0, 1.0], [1.0, 1.0], [1.0, 1.0]]
gemm2(A, B, transpose_b=True, alpha=2.0)
= [[4.0, 4.0, 4.0], [4.0, 4.0, 4.0]]

// Batch matrix multiply
A = [[[1.0, 1.0]], [[0.1, 0.1]]]
B = [[[1.0, 1.0]], [[0.1, 0.1]]]
gemm2(A, B, transpose_b=True, alpha=2.0)
= [[[4.0]], [[0.04 ]]]


Defined in src/operator/tensor/la_op.cc:L151

Parameters: A (Symbol) – Tensor of input matrices B (Symbol) – Tensor of input matrices transpose_a (boolean, optional, default=0) – Multiply with transposed of first input (A). transpose_b (boolean, optional, default=0) – Multiply with transposed of second input (B). alpha (double, optional, default=1) – Scalar factor multiplied with A*B. axis (int, optional, default='-2') – Axis corresponding to the matrix row indices. name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.potrf(A=None, name=None, attr=None, out=None, **kwargs)

Performs Cholesky factorization of a symmetric positive-definite matrix. Input is a tensor A of dimension n >= 2.

If n=2, the Cholesky factor L of the symmetric, positive definite matrix A is computed. L is lower triangular (entries of upper triangle are all zero), has positive diagonal entries, and:

A = L * LT

If n>2, potrf is performed separately on the trailing two dimensions for all inputs (batch mode).

Note

The operator supports float32 and float64 data types only.

Examples:

// Single matrix factorization
A = [[4.0, 1.0], [1.0, 4.25]]
potrf(A) = [[2.0, 0], [0.5, 2.0]]

// Batch matrix factorization
A = [[[4.0, 1.0], [1.0, 4.25]], [[16.0, 4.0], [4.0, 17.0]]]
potrf(A) = [[[2.0, 0], [0.5, 2.0]], [[4.0, 0], [1.0, 4.0]]]


Defined in src/operator/tensor/la_op.cc:L201

Parameters: A (Symbol) – Tensor of input matrices to be decomposed name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.potri(A=None, name=None, attr=None, out=None, **kwargs)

Performs matrix inversion from a Cholesky factorization. Input is a tensor A of dimension n >= 2.

If n=2, A is a lower triangular matrix (entries of upper triangle are all zero) with positive diagonal. We compute:

out = A-T * A-1

In other words, if A is the Cholesky factor of a symmetric positive definite matrix B (obtained by potrf), then

out = B-1

If n>2, potri is performed separately on the trailing two dimensions for all inputs (batch mode).

Note

The operator supports float32 and float64 data types only.

Note

Use this operator only if you are certain you need the inverse of B, and cannot use the Cholesky factor A (potrf), together with backsubstitution (trsm). The latter is numerically much safer, and also cheaper.

Examples:

// Single matrix inverse
A = [[2.0, 0], [0.5, 2.0]]
potri(A) = [[0.26563, -0.0625], [-0.0625, 0.25]]

// Batch matrix inverse
A = [[[2.0, 0], [0.5, 2.0]], [[4.0, 0], [1.0, 4.0]]]
potri(A) = [[[0.26563, -0.0625], [-0.0625, 0.25]],
[[0.06641, -0.01562], [-0.01562, 0,0625]]]


Defined in src/operator/tensor/la_op.cc:L259

Parameters: A (Symbol) – Tensor of lower triangular matrices name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.sumlogdiag(A=None, name=None, attr=None, out=None, **kwargs)

Computes the sum of the logarithms of the diagonal elements of a square matrix. Input is a tensor A of dimension n >= 2.

If n=2, A must be square with positive diagonal entries. We sum the natural logarithms of the diagonal elements, the result has shape (1,).

If n>2, sumlogdiag is performed separately on the trailing two dimensions for all inputs (batch mode).

Note

The operator supports float32 and float64 data types only.

Examples:

// Single matrix reduction
A = [[1.0, 1.0], [1.0, 7.0]]
sumlogdiag(A) = [1.9459]

// Batch matrix reduction
A = [[[1.0, 1.0], [1.0, 7.0]], [[3.0, 0], [0, 17.0]]]
sumlogdiag(A) = [1.9459, 3.9318]


Defined in src/operator/tensor/la_op.cc:L428

Parameters: A (Symbol) – Tensor of square matrices name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.syevd(A=None, name=None, attr=None, out=None, **kwargs)

Eigendecomposition for symmetric matrix. Input is a tensor A of dimension n >= 2.

If n=2, A must be symmetric, of shape (x, x). We compute the eigendecomposition, resulting in the orthonormal matrix U of eigenvectors, shape (x, x), and the vector L of eigenvalues, shape (x,), so that:

U * A = diag(L) * U

Here:

U * UT = UT * U = I

where I is the identity matrix. Also, L(0) <= L(1) <= L(2) <= ... (ascending order).

If n>2, syevd is performed separately on the trailing two dimensions of A (batch mode). In this case, U has n dimensions like A, and L has n-1 dimensions.

Note

The operator supports float32 and float64 data types only.

Note

Derivatives for this operator are defined only if A is such that all its eigenvalues are distinct, and the eigengaps are not too small. If you need gradients, do not apply this operator to matrices with multiple eigenvalues.

Examples:

// Single symmetric eigendecomposition
A = [[1., 2.], [2., 4.]]
U, L = syevd(A)
U = [[0.89442719, -0.4472136],
[0.4472136, 0.89442719]]
L = [0., 5.]

// Batch symmetric eigendecomposition
A = [[[1., 2.], [2., 4.]],
[[1., 2.], [2., 5.]]]
U, L = syevd(A)
U = [[[0.89442719, -0.4472136],
[0.4472136, 0.89442719]],
[[0.92387953, -0.38268343],
[0.38268343, 0.92387953]]]
L = [[0., 5.],
[0.17157288, 5.82842712]]


Defined in src/operator/tensor/la_op.cc:L621

Parameters: A (Symbol) – Tensor of input matrices to be factorized name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.syrk(A=None, transpose=_Null, alpha=_Null, name=None, attr=None, out=None, **kwargs)

Multiplication of matrix with its transpose. Input is a tensor A of dimension n >= 2.

If n=2, the operator performs the BLAS3 function syrk:

out = alpha * A * AT

if transpose=False, or

out = alpha * AT * A

if transpose=True.

If n>2, syrk is performed separately on the trailing two dimensions for all inputs (batch mode).

Note

The operator supports float32 and float64 data types only.

Examples:

// Single matrix multiply
A = [[1., 2., 3.], [4., 5., 6.]]
syrk(A, alpha=1., transpose=False)
= [[14., 32.],
[32., 77.]]
syrk(A, alpha=1., transpose=True)
= [[17., 22., 27.],
[22., 29., 36.],
[27., 36., 45.]]

// Batch matrix multiply
A = [[[1., 1.]], [[0.1, 0.1]]]
syrk(A, alpha=2., transpose=False) = [[[4.]], [[0.04]]]


Defined in src/operator/tensor/la_op.cc:L484

Parameters: A (Symbol) – Tensor of input matrices transpose (boolean, optional, default=0) – Use transpose of input matrix. alpha (double, optional, default=1) – Scalar factor to be applied to the result. name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.trmm(A=None, B=None, transpose=_Null, rightside=_Null, alpha=_Null, name=None, attr=None, out=None, **kwargs)

Performs multiplication with a lower triangular matrix. Input are tensors A, B, each of dimension n >= 2 and having the same shape on the leading n-2 dimensions.

If n=2, A must be lower triangular. The operator performs the BLAS3 function trmm:

out = alpha * op(A) * B

if rightside=False, or

out = alpha * B * op(A)

if rightside=True. Here, alpha is a scalar parameter, and op() is either the identity or the matrix transposition (depending on transpose).

If n>2, trmm is performed separately on the trailing two dimensions for all inputs (batch mode).

Note

The operator supports float32 and float64 data types only.

Examples:

// Single triangular matrix multiply
A = [[1.0, 0], [1.0, 1.0]]
B = [[1.0, 1.0, 1.0], [1.0, 1.0, 1.0]]
trmm(A, B, alpha=2.0) = [[2.0, 2.0, 2.0], [4.0, 4.0, 4.0]]

// Batch triangular matrix multiply
A = [[[1.0, 0], [1.0, 1.0]], [[1.0, 0], [1.0, 1.0]]]
B = [[[1.0, 1.0, 1.0], [1.0, 1.0, 1.0]], [[0.5, 0.5, 0.5], [0.5, 0.5, 0.5]]]
trmm(A, B, alpha=2.0) = [[[2.0, 2.0, 2.0], [4.0, 4.0, 4.0]],
[[1.0, 1.0, 1.0], [2.0, 2.0, 2.0]]]


Defined in src/operator/tensor/la_op.cc:L316

Parameters: A (Symbol) – Tensor of lower triangular matrices B (Symbol) – Tensor of matrices transpose (boolean, optional, default=0) – Use transposed of the triangular matrix rightside (boolean, optional, default=0) – Multiply triangular matrix from the right to non-triangular one. alpha (double, optional, default=1) – Scalar factor to be applied to the result. name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol
mxnet.symbol.linalg.trsm(A=None, B=None, transpose=_Null, rightside=_Null, alpha=_Null, name=None, attr=None, out=None, **kwargs)

Solves matrix equation involving a lower triangular matrix. Input are tensors A, B, each of dimension n >= 2 and having the same shape on the leading n-2 dimensions.

If n=2, A must be lower triangular. The operator performs the BLAS3 function trsm, solving for out in:

op(A) * out = alpha * B

if rightside=False, or

out * op(A) = alpha * B

if rightside=True. Here, alpha is a scalar parameter, and op() is either the identity or the matrix transposition (depending on transpose).

If n>2, trsm is performed separately on the trailing two dimensions for all inputs (batch mode).

Note

The operator supports float32 and float64 data types only.

Examples:

// Single matrix solve
A = [[1.0, 0], [1.0, 1.0]]
B = [[2.0, 2.0, 2.0], [4.0, 4.0, 4.0]]
trsm(A, B, alpha=0.5) = [[1.0, 1.0, 1.0], [1.0, 1.0, 1.0]]

// Batch matrix solve
A = [[[1.0, 0], [1.0, 1.0]], [[1.0, 0], [1.0, 1.0]]]
B = [[[2.0, 2.0, 2.0], [4.0, 4.0, 4.0]],
[[4.0, 4.0, 4.0], [8.0, 8.0, 8.0]]]
trsm(A, B, alpha=0.5) = [[[1.0, 1.0, 1.0], [1.0, 1.0, 1.0]],
[[2.0, 2.0, 2.0], [2.0, 2.0, 2.0]]]


Defined in src/operator/tensor/la_op.cc:L379

Parameters: A (Symbol) – Tensor of lower triangular matrices B (Symbol) – Tensor of matrices transpose (boolean, optional, default=0) – Use transposed of the triangular matrix rightside (boolean, optional, default=0) – Multiply triangular matrix from the right to non-triangular one. alpha (double, optional, default=1) – Scalar factor to be applied to the result. name (string, optional.) – Name of the resulting symbol. The result symbol. Symbol