smile.cas

abs(x)

x + y

A + B

x + y

acos(x)

acot(x)

asin(x)

atan(x)

ceil(x)

Constant value.

Constant matrix.

Constant vector. Different from VectorVal that has concrete values, this is of constant yet abstract value.

cos(x)

cot(x)

Diagonal matrix

x / y

exp(x)

floor(x)

The derivative of a scalar y with respect to a vector x.

Identity matrix

Inner product (x * y)

Explicit conversion of int to float.

x + y

Integer constant value.

x / y

x * y

-x

x ** y

Scalar: rank-0 tensor.

x - y

Integer scalar value.

Integer scalar variable

log(x)

Matrix: rank-2 tensor.

inv(A)

Matrix multiplication (A * B)

A'

Abstract matrix variable

Matrix vector multiplication (A * x)

x % y

x * y

-x

-A

-x

Matrix of all 1's

Vector of all 1's

Outer product (x ** y)

x ** y

round(x)

Row-wise matrix

Scalar: rank-0 tensor.

a * A

a * x

sin(x)

x - y

tan(x)

The derivative of a matrix y with respect to a scalar x.

The derivative of a vector y with respect to a scalar x.

A tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors (which are often, but not always, understood as arrows with length that point in a direction) and scalars (which are often familiar numbers such as the real numbers), and, recursively, even other tensors. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.

The shape of tensor (the number of dimensions and the size of each dimension) might be only partially known.

Scalar value.

Scalar variable

Vector variable

Vector: rank-1 tensor.

Vector value.

Abstract vector variable

Matrix of all 0's

Vector of all 0's