9.37 Kernel

A kernel is a symbolic function application (such as sin(x+ y)) or a symbol (such as x). More precisely, a non-symbol kernel over a set S is an operator applied to a given list of arguments from S. The operator has type BasicOperator (see BasicOperatorXmpPage ) and the kernel object is usually part of an expression object (see ExpressionXmpPage ).

Kernels are created implicitly for you when you create expressions.

x :: Expression Integer

\[\]

x

_{Type: Expression Integer}

You can directly create a symbol kernel by using the kernelkernelKernel operation.

kernel x

\[\]

x

_{Type: Kernel Expression Integer}

This expression has two different kernels.

sin(x) + cos(x)

\[\]

sin(x)+cos(x)

_{Type: Expression Integer}

The operator kernelskernelsExpression returns a list of the kernels in an object of type Expression.

kernels %

\[\]

[sin(x),cos(x)]

_{Type: List Kernel Expression Integer}

This expression also has two different kernels.

sin(x)^2 + sin(x) + cos(x)

\[\]

sin(x)2+sin(x)+cos(x)

_{Type: Expression Integer}

The sin(x) kernel is used twice.

kernels %

\[\]

[sin(x),cos(x)]

_{Type: List Kernel Expression Integer}

An expression need not contain any kernels.

kernels(1 :: Expression Integer)

\[\]

[]

_{Type: List Kernel Expression Integer}

If one or more kernels are present, one of them is designated the main kernel.

mainKernel(cos(x) + tan(x))

\[\]

tan(x)

_{Type: Union(Kernel Expression Integer,...)}

Kernels can be nested. Use heightheightKernel to determine the nesting depth.

height kernel x

\[\]

1

_{Type: PositiveInteger}

This has height 2 because the x has height 1 and then we apply an operator to that.

height mainKernel(sin x)

\[\]

2

_{Type: PositiveInteger}

height mainKernel(sin cos x)

\[\]

3

_{Type: PositiveInteger}

height mainKernel(sin cos (tan x + sin x))

\[\]

4

_{Type: PositiveInteger}

Use the operatoroperatorKernel operation to extract the operator component of the kernel. The operator has type BasicOperator.

operator mainKernel(sin cos (tan x + sin x))

\[\]

sin

_{Type: BasicOperator}

Use the namenameKernel operation to extract the name of the operator component of the kernel. The name has type Symbol. This is really just a shortcut for a two-step process of extracting the operator and then calling namenameBasicOperator on the operator.

name mainKernel(sin cos (tan x + sin x))

\[\]

sin

_{Type: Symbol}

FriCAS knows about functions such as sin, cos and so on and can make kernels and then expressions using them. To create a kernel and expression using an arbitrary operator, use operatoroperatorBasicOperator.

Now f can be used to create symbolic function applications.

f := operator 'f

\[\]

f

_{Type: BasicOperator}

e := f(x, y, 10)

\[\]

f(x,y,10)

_{Type: Expression Integer}

Use the is?is?Kernel operation to learn if the operator component of a kernel is equal to a given operator.

is?(e, f)

\[\]

true

_{Type: Boolean}

You can also use a symbol or a string as the second argument to is?is?Kernel.

is?(e, 'f)

\[\]

true

_{Type: Boolean}

Use the argumentargumentKernel operation to get a list containing the argument component of a kernel.

argument mainKernel e

\[\]

[x,y,10]

_{Type: List Expression Integer}

Conceptually, an object of type Expression can be thought of a quotient of multivariate polynomials, where the variables are kernels. The arguments of the kernels are again expressions and so the structure recurses. See ExpressionXmpPage for examples of using kernels to take apart expression objects.