dev.nm.algebra.linear.matrix.doubles.operation.householder

Class HouseholderContext

java.lang.Object

dev.nm.algebra.linear.matrix.doubles.operation.householder.HouseholderContext

public class HouseholderContext
extends Object

This is the context information about a Householder transformation. It tells

1. v, the defining vector which is perpendicular to the Householder hyperplane;
2. generator, the vector used to generate the Householder defining vector;
3. λ, the norm of the generator with the sign chosen to be the opposite of the first coordinate of generator.

Field Summary

Fields

Modifier and Type	Field and Description
double	beta β = 2 / v'v.
Vector	generator The vector which is used to generate the Householder vector.
double	lambda The norm of the generator with the sign chosen to be the opposite of the first coordinate of the generator.
Vector	v The defining vector which is perpendicular to the Householder hyperplane.

Constructor Summary

Constructors

Constructor and Description
HouseholderContext(Vector v, double beta, Vector generator, double lambda) Constructs a Householder context information.

Method Summary

Modifier and Type	Method and Description
static HouseholderContext	getContext(Vector x) Generates the context information from a generating vector x.
String	toString()

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Field Detail

v

public final Vector v

The defining vector which is perpendicular to the Householder hyperplane.

beta

public final double beta

β = 2 / v'v.

generator

public final Vector generator

The vector which is used to generate the Householder vector.

lambda

public final double lambda

The norm of the generator with the sign chosen to be the opposite of the first coordinate of the generator.

Constructor Detail

HouseholderContext

public HouseholderContext(Vector v,
                          double beta,
                          Vector generator,
                          double lambda)

Constructs a Householder context information.

Parameters:

v - the defining vector which is perpendicular to the Householder hyperplane

beta - β

generator - the vector that generates the Householder reflection

lambda - λ, the negative norm of the generator

Method Detail

getContext

public static HouseholderContext getContext(Vector x)

Generates the context information from a generating vector x. Given a vector x, return a vector v, such that \[ Hx = \pm ||x|| \cdot e_{1} \] That is,

 H.reflect(x) == new DenseVector(new double[]{±x.norm(), 0, ...})
 

Parameters:

x - a vector

Returns:

the context information for a Householder transformation

See Also:

"G. W. Steward, "Algorithm 1.1," Matrix Algorithms, Volume 1"

toString

public String toString()

Overrides:

toString in class Object