com.numericalmethod.suanshu.vector.doubles.dense.operation
Class Basis

java.lang.Object
  com.numericalmethod.suanshu.vector.doubles.dense.DenseVector
      com.numericalmethod.suanshu.vector.doubles.dense.operation.Basis

All Implemented Interfaces:

DeepCopyable, AbelianGroup<Vector>, BanachSpace<Vector,Real>, HilbertSpace<Vector,Real>, VectorSpace<Vector,Real>, Vector

public class Basis
extends DenseVector

A basis is a set of linearly independent vectors spanning a vector space. Every element in this space can be uniquely represented by a linear combination of elements in the basis.

This class implements the standard basis in the Euclidean Rⁿ space.

See Also:

Wikipedia: Basis (linear algebra)

Field Summary

Fields inherited from class com.numericalmethod.suanshu.vector.doubles.dense.DenseVector

length

Constructor Summary

Basis(int dim, int i)
Construct a vector which corresponds to the i-th dimension in Rⁿ.

Method Summary

double	angle(Vector that) Measure the angle between this and that.
static VectorList	basis(int dim) Get the full set of standard basis vectors.
static VectorList	basis(int dim, int ncols) Get a subset of standard basis vectors.
double	norm() Compute the length or magnitude or Euclidean norm of a vector, namely, ||v||.
Vector	opposite() Get the opposite of this vector.
Vector	scaled(Real scalar) scalar * that If scalar is 1, it simply returns itself.

Methods inherited from class com.numericalmethod.suanshu.vector.doubles.dense.DenseVector

add, add, add, as, deepCopy, divide, divide, equals, get, hashCode, innerProduct, minus, minus, minus, multiply, multiply, norm, pow, scaled, set, set, size, toArray, toString, ZERO

Methods inherited from class java.lang.Object

clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Constructor Detail

Basis

public Basis(int dim,
             int i)

Construct a vector which corresponds to the i-th dimension in Rⁿ. That is,

 |  0  |
 | ... |
 |  1  | <-- the i-th entry
 | ... |
 |  0  |
 

Parameters:

dim - the dimension

i - the i-th dimension in Rⁿ

Method Detail

basis

public static VectorList basis(int dim)

Get the full set of standard basis vectors.

Parameters:

dim - the dimension

Returns:

the basis vectors

basis

public static VectorList basis(int dim,
                               int ncols)

Get a subset of standard basis vectors.

Parameters:

dim - the dimension

ncols - number of basis vectors requested; it must be smaller than dim

Returns:

the basis vectors

Throws:

java.lang.IllegalArgumentException - if there are more columns requested than the dimension

scaled

public Vector scaled(Real scalar)

Description copied from interface: Vector

scalar * that

If scalar is 1, it simply returns itself. So, here is a way to get a unit version of the vector:

vector.scaled(1. / vector.norm())

Specified by:

scaled in interface VectorSpace<Vector,Real>

Specified by:

scaled in interface Vector

Parameters:

scalar - a Real number scalar

Returns:

a vector which is scaled by scalar

See Also:

Wikipedia: Scalar multiplication

opposite

public Vector opposite()

Description copied from interface: Vector

Get the opposite of this vector.

Specified by:

opposite in interface AbelianGroup<Vector>

Specified by:

opposite in interface Vector

Returns:

-v

See Also:

Wikipedia: Additive inverse

angle

public double angle(Vector that)

Description copied from interface: Vector

Measure the angle between this and that.

That is,

this ∙ that = ||this|| * ||that|| cos(angle)

Specified by:

angle in interface HilbertSpace<Vector,Real>

Specified by:

angle in interface Vector

Parameters:

that - a vector

Returns:

the angle between this and that

norm

public double norm()

Description copied from interface: Vector

Compute the length or magnitude or Euclidean norm of a vector, namely, ||v||.

Specified by:

norm in interface BanachSpace<Vector,Real>

Specified by:

norm in interface Vector

Returns:

the Euclidean norm

See Also:

Wikipedia: Norm (mathematics)