SingularValueDecomposition (Smile Math 1.0.3 API)

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

```
public class SingularValueDecomposition
extends 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.

Author:

Haifeng Li

Method Summary

Methods
Modifier and Type	Method and Description
`double`	`condition()` Returns the L₂ norm condition number, which is max(S) / min(S).
`static SingularValueDecomposition`	`decompose(double[][] A)` Returns the singular value decomposition.
`static SingularValueDecomposition`	`decompose(IMatrix A, int k)` Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
`static SingularValueDecomposition`	`decompose(IMatrix A, int k, double kappa)` Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
`double[][]`	`getS()` Returns the diagonal matrix of singular values
`double[]`	`getSingularValues()` Returns the one-dimensional array of singular values, ordered by from largest to smallest.
`double[][]`	`getU()` Returns the left singular vectors
`double[][]`	`getV()` Returns the right singular vectors
`double`	`norm()` Returns the L2 matrix norm.
`int`	`nullity()` Returns the dimension of null space.
`double[][]`	`nullspace()` Returns a matrix of which columns give an orthonormal basis for the null space.
`double[][]`	`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(double[][] B, double[][] X)` Solve A * X = B using the pseudoinverse of A as obtained by SVD.
`void`	`solve(double[] b, double[] x)` Solve A * x = b using the pseudoinverse of A as obtained by SVD.

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

- Method Detail
  - getU
```
public double[][] getU()
```
    Returns the left singular vectors
  - getV
```
public double[][] 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 double[][] 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 double[][] range()
```
    Returns a matrix of which columns give an orthonormal basis for the range space.
  - nullspace
```
public double[][] nullspace()
```
    Returns a matrix of which columns give an orthonormal basis for the null space.
  - solve
```
public void solve(double[] b,
         double[] x)
```
    Solve A * x = b using the pseudoinverse of A as obtained by SVD.
  - solve
```
public void solve(double[][] B,
         double[][] X)
```
    Solve A * X = B using the pseudoinverse of A as obtained by SVD.
  - decompose
```
public static SingularValueDecomposition decompose(IMatrix A,
                                   int k)
```
    Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
    
    Parameters:
    A - the matrix supporting matrix vector multiplication operation.
    k - the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.
  - decompose
```
public static SingularValueDecomposition decompose(IMatrix A,
                                   int k,
                                   double kappa)
```
    Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
    
    Parameters:
    A - the matrix supporting matrix vector multiplication operation.
    k - the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.
    kappa - relative accuracy of ritz values acceptable as singular values.
  - decompose
```
public static SingularValueDecomposition decompose(double[][] A)
```
    Returns the singular value decomposition. Note that the input matrix A will hold U on output.
    
    Parameters:
    A - rectangular matrix. Row number should be equal to or larger than column number for current implementation.

Class SingularValueDecomposition

Method Summary

Methods inherited from class java.lang.Object

Method Detail

getU

getV

getSingularValues

getS

norm

rank

nullity

condition

range

nullspace

solve

solve

decompose

decompose

decompose