orbital.math
Interface Vector

All Superinterfaces:

Arithmetic, Normed, Tensor

public interface Vector
extends Tensor

Represents a mathematical vector of any dimension n.

v∈V=Rn, v = (v0,v1,v2,…,vn-1)T =	(	v0	)
		v1
		⋮
		vn-1

The components v_i∈R are any arithmetic objects forming a ring. (Rⁿ,+,·) form a (free) R-module with the law of composition + and the law of action ·. If R is a field the vectors even form a vector space over R, which justifies the name vector. If R is an integrity domain, then Rⁿ≅R^m ⇔ n=m.

A vector space over a field R is a set V with an algebraic structure of

(1)	(V,+) is an Abelian group with the law of composition
	+:V×V→V; (v,w)↦v+w
(2)	·:R×V→V; (λ,v)↦λ·v is a law of action (the scalar multiplication · or sometimes ·)
("a")	λ·(μ·v) = (λ⋅μ)·v
("d")	(λ+μ)·v = λ·v + μ·v
("d")	λ·(v+w) = λ·v + λ·w
("n")	1·v = v

∀λ,μ∈R, v,w∈V.

If you intend to use mutable arithmetic elements, note the discussion of mutations per reference vs. explicit cloning in Tensor.set(int[],Arithmetic) which generally holds for all operations that set component values.

Also note that some few methods will change its instance and explicitly return this to allow chaining of structural changes, whilst arithmetic methods will leave a vector unchanged but return a modified version. Refer to the documentation of the individual methods for details.

Author:

André Platzer

See Also:

ValueFactory.tensor(Arithmetic[]), ValueFactory.valueOf(Arithmetic[]), ValueFactory.valueOf(double[])

Invariants:

super ∧ rank()==1

Structure:

extends Tensor

Field Summary

Fields inherited from interface orbital.math.Arithmetic

numerical

Method Summary

Vector	add(Vector b) Adds two vectors returning a vector.
Vector	cross(Vector b) Vector-cross-product of two vectors.
int	dimension() Returns the dimension of the vector.
Arithmetic	get(int i) Returns the value at the component specified by index.
Vector	insert(Arithmetic v) Append a value to this vector.
Vector	insert(int index, Arithmetic v) Insert a value into this vector at the specified index.
Vector	insertAll(int index, Vector v) Insert all components of a vector into this vector at the specified index.
Vector	insertAll(Vector v) Append all components of a vector to this vector.
java.util.ListIterator	iterator() Returns an iterator over all components.
Vector	multiply(Matrix B) Multiplies a vector with a matrix returning a vector.
Vector	multiply(Scalar s) Multiplies a vector with a scalar returning a vector.
Arithmetic	multiply(Vector b) Scalar-dot-product ⟨·,·⟩:V×V→F of two vectors.
Real	norm(double p) Returns the norm || ||p of this vector.
Vector	remove(int index) Remove the component at an index from this vector.
Vector	scale(Scalar s) Multiplies a vector with a scalar returning a vector.
void	set(int i, Arithmetic vi) Sets a value at the component specified by index.
Vector	subtract(Vector b) Subtracts two vectors returning a vector.
Vector	subVector(int i1, int i2) Get a sub-vector view ranging (i1:i2) inclusive.
Arithmetic[]	toArray() Returns an array containing all the elements in this vector.
Matrix	transpose() Returns this vector transposed.

Methods inherited from interface orbital.math.Tensor

add, clone, dimensions, entries, get, indices, multiply, rank, set, setSubTensor, subTensor, subTensor, subTensorTransposed, subtract, tensor

Methods inherited from interface orbital.math.Arithmetic

add, divide, equals, inverse, isOne, isZero, minus, multiply, one, power, scale, subtract, toString, valueFactory, zero

Methods inherited from interface orbital.math.Normed

norm

Method Detail

dimension

int dimension()

Returns the dimension of the vector. The dimension is the number n of elements contained.

get

Arithmetic get(int i)

Returns the value at the component specified by index.

Of course, this method only has a meaning for free modules like vector spaces.

set

void set(int i,
         Arithmetic vi)
         throws java.lang.UnsupportedOperationException

Sets a value at the component specified by index.

Of course, this method only has a meaning for free modules like vector spaces.

Throws:

java.lang.UnsupportedOperationException - if this vector is constant and does not allow modifications.

iterator

java.util.ListIterator iterator()

Returns an iterator over all components.

Specified by:

iterator in interface Tensor

Returns:

an iterator that iterates over v₀,…,v_n-1.

Postconditions:

RES.supports(#nextIndex()} ∧ RES.supports(#previousIndex()} ∧ RES.supports(#add(Object)} ∧ RES.supports(#remove()}

subVector

Vector subVector(int i1,
                 int i2)

Get a sub-vector view ranging (i1:i2) inclusive.

The returned vector is a structurally unmodifiable view.

Returns:

a matrix view of the specified part of this matrix.

Preconditions:

i1<=i2 && valid(i1) && valid(i2)

norm

Real norm(double p)

Returns the norm || ||_p of this vector.

This method implements p-norms, where
||x||_p = (|x₁|^p + … + |x_n|^p)^1/p.
||x||_∞ = max {|x₁|,…,|x_n|}.

Preconditions:

p>=1

add

Vector add(Vector b)

Adds two vectors returning a vector.

Preconditions:

dimension() == b.dimension()

Postconditions:

RES.dimension() == dimension() && RES.get(i) == get(i) + b.get(i)

Attributes:

associative, neutral (0), inverse (-v), commutative

subtract

Vector subtract(Vector b)

Subtracts two vectors returning a vector.

Preconditions:

dimension() == b.dimension()

Postconditions:

RES.dimension() == dimension() && RES.get(i) == get(i) - b.get(i)

Attributes:

associative

multiply

Arithmetic multiply(Vector b)

Scalar-dot-product ⟨·,·⟩:V×V→F of two vectors.

⟨·,·⟩:V×V→R is a real scalar product and V a euclidian vector space, if ∀x,y∈V:

(bl)	⟨.,y⟩ and ⟨x,.⟩ are linear	"bilinear"
(s)	⟨x,y⟩ = ⟨y,x⟩	"symmetric" (this method is commutative)
(pdef)	⟨x,x⟩≥0 and ⟨x,x⟩=0 ⇔ x=0	"positive definite"

⟨·,·⟩:V×V→C is a complex scalar product and V a unitarian vector space, if ∀x,y∈V:

(ll)	⟨·,y⟩ is linear	"left-linear"
(h)	⟨x,y⟩ = ⟨y,x⟩	"hermite"
(pdef)	⟨x,x⟩≥0 and ⟨x,x⟩=0 ⇔ x=0	"positive definite"

A scalar product ⟨·,·⟩ induces a norm ||.||:V→[0,∞); x ↦ ||x|| := √⟨x,x⟩

The standard scalar-product which will often be implemented, is
(x,y) ↦ ⟨x,y⟩ = x^T·y = ∑_i=0^n-1 x_i⋅y_i. It belongs to the euclidian 2-norm and is the inner product of vectors.

See Also:

Normed

Preconditions:

dimension() == b.dimension()

Postconditions:

RES.dimension() == dimension() && RES == ⟨this, b⟩

scale

Vector scale(Scalar s)

Multiplies a vector with a scalar returning a vector.

Returns:

s·v

Preconditions:

true

Postconditions:

RES.dimension().equals(dimension()) && RES.get(i) == s⋅get(i)

Attributes:

associative, neutral

multiply

Vector multiply(Scalar s)

Multiplies a vector with a scalar returning a vector.

See Also:

scale(Scalar)

multiply

Vector multiply(Matrix B)

Multiplies a vector with a matrix returning a vector. If row-vector v is sized n and the matrix A is sized n×m, the resulting row-vector v∙A is sized m. This is an inner product.

Preconditions:

dimension() == B.dimension().height

cross

Vector cross(Vector b)

Vector-cross-product of two vectors.
×:R³×R³→R³; (x,y) ↦ x×y = (x₁y₂-x₂y₁, x₂y₀-x₀y₂, x₀y₁-x₁y₀)

cross is antisymmetric: y×x = -(x×y)

Returns:

the cross-product vector which will be orthogonal on this and b. x×y ⊥ x ∧ x×y ⊥ y.

Preconditions:

dimension() == 3 && dimension() == b.dimension(), (dimension() == 3 || dimension() == 2) && dimension() == b.dimension()

Postconditions:

RES.multiply(this) == 0 && RES.multiply(b) == 0.

Attributes:

antisymmetric

transpose

Matrix transpose()

Returns this vector transposed. Also distinguished via vector.multiply(matrix) or matrix.multiply(vector) instead.

See Also:

Matrix.transpose()

insert

Vector insert(int index,
              Arithmetic v)

Insert a value into this vector at the specified index.

Returns:

this.

Preconditions:

0<=index && index<=dimension()

Postconditions:

RES == this && RES.dimension() == OLD(dimension()) + 1

insertAll

Vector insertAll(int index,
                 Vector v)

Insert all components of a vector into this vector at the specified index.

Returns:

this.

Preconditions:

0<=index && index<=dimension()

Postconditions:

RES == this && RES.dimension() == OLD(dimension()) + v.dimension()

insert

Vector insert(Arithmetic v)

Append a value to this vector.

Returns:

this.

Postconditions:

RES == this && RES.dimension() == OLD(dimension()) + 1

insertAll

Vector insertAll(Vector v)

Append all components of a vector to this vector.

Returns:

this.

Postconditions:

RES == this && RES.dimension() == OLD(dimension()) + v.dimension()

remove

Vector remove(int index)

Remove the component at an index from this vector.

Returns:

this.

Postconditions:

RES == this && RES.dimension() == OLD(dimension()) - 1

toArray

Arithmetic[] toArray()

Returns an array containing all the elements in this vector.

See Also:

Object.clone()

Postconditions:

RES[i] == get(i) && RES != RES