SparseMatrix (Smile Math 1.0.4 API)

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

All Implemented Interfaces:

IMatrix
```
public class SparseMatrix
extends Object
implements IMatrix
```
A sparse matrix is a matrix populated primarily with zeros. Conceptually, sparsity corresponds to systems which are loosely coupled. Huge sparse matrices often appear when solving partial differential equations.
Operations using standard dense matrix structures and algorithms are slow and consume large amounts of memory when applied to large sparse matrices. Indeed, some very large sparse matrices are infeasible to manipulate with the standard dense algorithms. Sparse data is by nature easily compressed, and this compression almost always results in significantly less computer data storage usage.
This class employs Harwell-Boeing column-compressed sparse matrix format. Nonzero values are stored in an array (top-to-bottom, then left-to-right-bottom). The row indices corresponding to the values are also stored. Besides, a list of pointers are indexes where each column starts. This format is efficient for arithmetic operations, column slicing, and matrix-vector products. One typically uses SparseDataset for construction of SparseMatrix.

Author:

Haifeng Li

Constructor Summary

Constructors
Constructor and Description
`SparseMatrix(double[][] D)` Constructor.
`SparseMatrix(double[][] D, double tol)` Constructor.
`SparseMatrix(int nrows, int ncols, double[] x, int[] rowIndex, int[] colIndex)` Constructor.

Method Summary

All Methods Static Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`static SparseMatrix`	`AAT(SparseMatrix A, SparseMatrix AT)` Returns A * A^T.
`void`	`asolve(double[] b, double[] x)` Solve A_d * x = b for the preconditioner matrix A_d.
`void`	`atx(double[] x, double[] y)` y = A' * x
`void`	`atxpy(double[] x, double[] y)` y = A' * x + y
`void`	`atxpy(double[] x, double[] y, double b)` y = A' * x + b * y
`void`	`ax(double[] x, double[] y)` y = A * x
`void`	`axpy(double[] x, double[] y)` y = A * x + y
`void`	`axpy(double[] x, double[] y, double b)` y = A * x + b * y
`double`	`eigen(double[] v)` Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other its other eigenvalues and the starting vectorhas a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
`EigenValueDecomposition`	`eigen(int k)` Returns the k largest eigen pairs.
`double`	`get(int i, int j)` Returns the entry value at row i and column j.
`int`	`ncols()` Returns the number of columns.
`int`	`nrows()` Returns the number of rows.
`SparseMatrix`	`set(int i, int j, double x)` Set the entry value at row i and column j.
`int`	`size()` Returns the number of nonzero values.
`SparseMatrix`	`times(SparseMatrix B)` Returns the matrix multiplication C = A * B.
`SparseMatrix`	`transpose()` Returns the matrix transpose.
`double[]`	`values()` Returns all nonzero values.

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

- Constructor Detail
  - SparseMatrix
```
public SparseMatrix(int nrows,
                    int ncols,
                    double[] x,
                    int[] rowIndex,
                    int[] colIndex)
```
    Constructor.
    
    Parameters:
    
    nrows - the number of rows in the matrix.
    
    ncols - the number of columns in the matrix.
    
    rowIndex - the row indices of nonzero values.
    
    colIndex - the index of the start of columns.
    
    x - the array of nonzero values stored column by column.
  - SparseMatrix
```
public SparseMatrix(double[][] D)
```
    Constructor.
    
    Parameters:
    
    D - a dense matrix to converted into sparse matrix format.
  - SparseMatrix
```
public SparseMatrix(double[][] D,
                    double tol)
```
    Constructor.
    
    Parameters:
    
    D - a dense matrix to converted into sparse matrix format.
    
    tol - the tolerance to regard a value as zero if |x| < tol.
- Method Detail
  - nrows
```
public int nrows()
```
    Description copied from interface: IMatrix
    
    Returns the number of rows.
    
    Specified by:
    
    nrows in interface IMatrix
  - ncols
```
public int ncols()
```
    Description copied from interface: IMatrix
    
    Returns the number of columns.
    
    Specified by:
    
    ncols in interface IMatrix
  - size
```
public int size()
```
    Returns the number of nonzero values.
  - values
```
public double[] values()
```
    Returns all nonzero values.
    
    Returns:
    
    all nonzero values
  - get
```
public double get(int i,
                  int j)
```
    Description copied from interface: IMatrix
    
    Returns the entry value at row i and column j.
    
    Specified by:
    
    get in interface IMatrix
  - set
```
public SparseMatrix set(int i,
                        int j,
                        double x)
```
    Description copied from interface: IMatrix
    
    Set the entry value at row i and column j.
    
    Specified by:
    
    set in interface IMatrix
  - ax
```
public void ax(double[] x,
               double[] y)
```
    Description copied from interface: IMatrix
    
    y = A * x
    
    Specified by:
    
    ax in interface IMatrix
  - axpy
```
public void axpy(double[] x,
                 double[] y)
```
    Description copied from interface: IMatrix
    
    y = A * x + y
    
    Specified by:
    
    axpy in interface IMatrix
  - axpy
```
public void axpy(double[] x,
                 double[] y,
                 double b)
```
    Description copied from interface: IMatrix
    
    y = A * x + b * y
    
    Specified by:
    
    axpy in interface IMatrix
  - atx
```
public void atx(double[] x,
                double[] y)
```
    Description copied from interface: IMatrix
    
    y = A' * x
    
    Specified by:
    
    atx in interface IMatrix
  - atxpy
```
public void atxpy(double[] x,
                  double[] y)
```
    Description copied from interface: IMatrix
    
    y = A' * x + y
    
    Specified by:
    
    atxpy in interface IMatrix
  - atxpy
```
public void atxpy(double[] x,
                  double[] y,
                  double b)
```
    Description copied from interface: IMatrix
    
    y = A' * x + b * y
    
    Specified by:
    
    atxpy in interface IMatrix
  - transpose
```
public SparseMatrix transpose()
```
    Returns the matrix transpose.
  - times
```
public SparseMatrix times(SparseMatrix B)
```
    Returns the matrix multiplication C = A * B.
  - AAT
```
public static SparseMatrix AAT(SparseMatrix A,
                               SparseMatrix AT)
```
    Returns A * A^T.
  - eigen
```
public double eigen(double[] v)
```
    Returns the largest eigen pair of matrix with the power iteration under the assumptions A has an eigenvalue that is strictly greater in magnitude than its other its other eigenvalues and the starting vectorhas a nonzero component in the direction of an eigenvector associated with the dominant eigenvalue.
    
    Parameters:
    
    v - on input, it is the non-zero initial guess of the eigen vector. On output, it is the eigen vector corresponding largest eigen value.
    
    Returns:
    
    the largest eigen value.
  - eigen
```
public EigenValueDecomposition eigen(int k)
```
    Returns the k largest eigen pairs. Only works for symmetric matrix.
  - asolve
```
public void asolve(double[] b,
                   double[] x)
```
    Description copied from interface: IMatrix
    
    Solve A_d * x = b for the preconditioner matrix A_d. The preconditioner matrix A_d is close to A and should be easy to solve for linear systems. This method is useful for preconditioned conjugate gradient method. The preconditioner matrix could be as simple as the trivial diagonal part of A in some cases.
    
    Specified by:
    
    asolve in interface IMatrix

Class SparseMatrix

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

SparseMatrix

SparseMatrix

SparseMatrix

Method Detail

nrows

ncols

size

values

get

set

ax

axpy

axpy

atx

atxpy

atxpy

transpose

times

AAT

eigen

eigen

asolve