# 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.