SVD

java.lang.Object
- smile.math.matrix.SVD

```
public class SVD
extends java.lang.Object
```
Singular Value Decomposition.
For an m-by-n matrix A with m ≥ n, the singular value decomposition is an m-by-n orthogonal matrix U, an n-by-n diagonal matrix Σ, and an n-by-n orthogonal matrix V so that A = U*Σ*V'.
For m < n, only the first m columns of V are computed and Σ is m-by-m.
The singular values, σ_k = Σ_kk, are ordered so that σ₀ ≥ σ₁ ≥ ... ≥ σ_n-1.
The singular value decompostion always exists. The matrix condition number and the effective numerical rank can be computed from this decomposition.
SVD is a very powerful technique for dealing with sets of equations or matrices that are either singular or else numerically very close to singular. In many cases where Gaussian elimination and LU decomposition fail to give satisfactory results, SVD will diagnose precisely what the problem is. SVD is also the method of choice for solving most linear least squares problems.
Applications which employ the SVD include computing the pseudoinverse, least squares fitting of data, matrix approximation, and determining the rank, range and null space of a matrix. The SVD is also applied extensively to the study of linear inverse problems, and is useful in the analysis of regularization methods such as that of Tikhonov. It is widely used in statistics where it is related to principal component analysis. Yet another usage is latent semantic indexing in natural language text processing.

Field Summary

Fields
Modifier and Type	Field and Description
`protected boolean`	`full` Is this a full decomposition?
`protected int`	`m` The number of rows.
`protected int`	`n` The number of columns.
`protected double[]`	`s` Array for internal storage of singular values.
`protected double`	`tol` Threshold of estimated roundoff.
`protected DenseMatrix`	`U` Arrays for internal storage of left singular vectors U.
`protected DenseMatrix`	`V` Arrays for internal storage of right singular vectors V.

Constructor Summary

Constructors
Constructor and Description

SVD(DenseMatrix U, DenseMatrix V, double[] s)
Constructor.

Constructors
Constructor and Description
`SVD(DenseMatrix U, DenseMatrix V, double[] s)` Constructor.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`Cholesky`	`CholeskyOfAtA()` Returns the Cholesky decomposition of A'A.
`double`	`condition()` Returns the L₂ norm condition number, which is max(S) / min(S).
`DenseMatrix`	`getS()` Returns the diagonal matrix of singular values
`double[]`	`getSingularValues()` Returns the one-dimensional array of singular values, ordered by from largest to smallest.
`DenseMatrix`	`getU()` Returns the left singular vectors
`DenseMatrix`	`getV()` Returns the right singular vectors
`double`	`norm()` Returns the L2 matrix norm.
`int`	`nullity()` Returns the dimension of null space.
`DenseMatrix`	`nullspace()` Returns a matrix of which columns give an orthonormal basis for the null space.
`DenseMatrix`	`range()` Returns a matrix of which columns give an orthonormal basis for the range space.
`int`	`rank()` Returns the effective numerical matrix rank.
`void`	`solve(DenseMatrix B)` Solve the least squares A * X = B.
`void`	`solve(double[] b, double[] x)` Solve the least squares A*x = b.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Field Detail
  - U
```
protected DenseMatrix U
```
    Arrays for internal storage of left singular vectors U.
  - V
```
protected DenseMatrix V
```
    Arrays for internal storage of right singular vectors V.
  - s
```
protected double[] s
```
    Array for internal storage of singular values.
  - full
```
protected boolean full
```
    Is this a full decomposition?
  - m
```
protected int m
```
    The number of rows.
  - n
```
protected int n
```
    The number of columns.
  - tol
```
protected double tol
```
    Threshold of estimated roundoff.
- Constructor Detail
  - SVD
```
public SVD(DenseMatrix U,
           DenseMatrix V,
           double[] s)
```
    Constructor.
- Method Detail
  - getU
```
public DenseMatrix getU()
```
    Returns the left singular vectors
  - getV
```
public DenseMatrix getV()
```
    Returns the right singular vectors
  - getSingularValues
```
public double[] getSingularValues()
```
    Returns the one-dimensional array of singular values, ordered by from largest to smallest.
  - getS
```
public DenseMatrix getS()
```
    Returns the diagonal matrix of singular values
  - norm
```
public double norm()
```
    Returns the L2 matrix norm. The largest singular value.
  - rank
```
public int rank()
```
    Returns the effective numerical matrix rank. The number of nonnegligible singular values.
  - nullity
```
public int nullity()
```
    Returns the dimension of null space. The number of negligible singular values.
  - condition
```
public double condition()
```
    Returns the L₂ norm condition number, which is max(S) / min(S). A system of equations is considered to be well-conditioned if a small change in the coefficient matrix or a small change in the right hand side results in a small change in the solution vector. Otherwise, it is called ill-conditioned. Condition number is defined as the product of the norm of A and the norm of A^-1. If we use the usual L₂ norm on vectors and the associated matrix norm, then the condition number is the ratio of the largest singular value of matrix A to the smallest. Condition number depends on the underlying norm. However, regardless of the norm, it is always greater or equal to 1. If it is close to one, the matrix is well conditioned. If the condition number is large, then the matrix is said to be ill-conditioned. A matrix that is not invertible has the condition number equal to infinity.
  - range
```
public DenseMatrix range()
```
    Returns a matrix of which columns give an orthonormal basis for the range space.
  - nullspace
```
public DenseMatrix nullspace()
```
    Returns a matrix of which columns give an orthonormal basis for the null space.
  - CholeskyOfAtA
```
public Cholesky CholeskyOfAtA()
```
    Returns the Cholesky decomposition of A'A.
  - solve
```
public void solve(double[] b,
                  double[] x)
```
    Solve the least squares A*x = b.
    
    Parameters:
    
    b - right hand side of linear system.
    
    x - the output solution vector that minimizes the L2 norm of Q*R*x - b.
    
    Throws:
    
    java.lang.RuntimeException - if matrix is rank deficient.
  - solve
```
public void solve(DenseMatrix B)
```
    Solve the least squares A * X = B. B will be overwritten with the solution matrix on output.
    
    Parameters:
    
    B - right hand side of linear system. B will be overwritten with the solution matrix on output.
    
    Throws:
    
    java.lang.RuntimeException - Matrix is rank deficient.

Class SVD

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

U

V

s

full

m

n

tol

Constructor Detail

SVD

Method Detail

getU

getV

getSingularValues

getS

norm

rank

nullity

condition

range

nullspace

CholeskyOfAtA

solve

solve