VectorSymbol

VectorSymbol[v]

represents a vector with name v.

VectorSymbol[v,d]

represents a vector of length d.

VectorSymbol[v,d,dom]

represents a vector with elements in the domain dom.

Details

The name v in VectorSymbol[v,d,dom] can be any expression.
A valid dimension specification d in VectorSymbol[v,d,dom] is any positive integer. It is also possible to work with symbolic dimension specifications.
Element domain specifications dom in VectorSymbol[v,d,dom] include:

	Complexes	complex numbers
	Integers	integers
	Reals	real numbers
	NonNegativeReals	real numbers x with x≥0
	PositiveReals	real numbers x with x>0

In arithmetic and many other functions that work with lists, VectorSymbol objects do not automatically combine with other list arguments.
Optimization functions, equation solvers and D recognize that VectorSymbol objects represent vector variables.

Examples

open allclose all

Basic Examples (1)

Assign the value of the variable v to represent a length n vector with name "v":

Arithmetic operations recognize that v is not a scalar:

D recognizes that v is a vector variable:

Scope (5)

Compute derivatives with respect to a vector variable:

Compute derivatives involving vector-valued functions:

This requires a real-valued vector:

Use a vector variable in optimization:

Solve equations and inequalities involving a vector variable:

Applications (7)

Use a symbolic vector as a variable for optimization:

Find the vector that has the smallest sum of elements with a constraint:

Using a variable y treated by default as a scalar does not work since Total[y + {1,2,3}] expands:

Find the radius and center of a minimal enclosing ball that encompasses a set of k points in n dimensions:

The ball encloses point if Norm[p_i-c]<=r:

Visualize the sphere and points:

The optimization works in any dimension:

Solve a differential equation for a vector-valued function:

Plot the solution:

Check that the residual is small:

Approximate the variance for a perturbed vector:

Order zero approximation:

Compare with the exact value:

Order-one approximation:

Compare with the exact value:

Order-two approximation:

Since the second derivative does not depend on , the order-two approximation equals the exact value:

Find GammaDistribution parameters that best fit the given data using the maximum likelihood method:

Maximize the log-likelihood function :

Compute the gradient of :

Find a zero of the gradient, with replaced by :

Visualize the result:

Compare with the result computed using EstimatedDistribution:

Find an optimality condition for a portfolio optimization problem with the expected return and standard deviation :

The goal is to maximize when the vector of asset weights satisfies . The constraint can be used to represent , where the unconstrained vector variable consists of the first coordinates of :

The maximum occurs at a critical point of :

Express the condition in terms of :

Compute the gradient of the log-likelihood function of the linear regression model represented by the equation , where are normally distributed random variables with mean zero and variance :