smile.math.matrix

Class Matrix

java.lang.Object

smile.math.matrix.Matrix

All Implemented Interfaces:

java.io.Serializable

Direct Known Subclasses:

BandMatrix, DenseMatrix, SparseMatrix

public abstract class Matrix
extends java.lang.Object
implements java.io.Serializable

An abstract interface of matrix. The most important method is the matrix vector multiplication, which is the only operation needed in many iterative matrix algorithms, e.g. biconjugate gradient method for solving linear equations and power iteration and Lanczos algorithm for eigen decomposition, which are usually very efficient for very large and sparse matrices.

A matrix is a rectangular array of numbers. An item in a matrix is called an entry or an element. Entries are often denoted by a variable with two subscripts. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general.

Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations and matrix multiplication corresponds to composition of linear transformations. Matrices can also keep track of the coefficients in a system of linear equations. For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.

There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix transformation or matrix decomposition techniques. The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.

The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The QR decomposition factors matrices as a product of an orthogonal (Q) and a right triangular matrix (R). QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm. Singular value decomposition expresses any matrix A as a product UDV', where U and V are unitary matrices and D is a diagonal matrix. The eigendecomposition or diagonalization expresses A as a product VDV^-1, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable.

See Also:

Serialized Form

Constructor Summary

Constructors

Constructor and Description
Matrix()

Method Summary

Modifier and Type	Method and Description
abstract Matrix	aat() Returns A * A'
double	apply(int i, int j) Returns the entry value at row i and column j.
abstract Matrix	ata() Returns A' * A
abstract double[]	atx(double[] x, double[] y) y = A' * x
abstract double[]	atxpy(double[] x, double[] y) y = A' * x + y
abstract double[]	atxpy(double[] x, double[] y, double b) y = A' * x + b * y
abstract double[]	ax(double[] x, double[] y) y = A * x
abstract double[]	axpy(double[] x, double[] y) y = A * x + y
abstract double[]	axpy(double[] x, double[] y, double b) y = A * x + b * y
double[]	diag() Returns the diagonal elements.
static DenseMatrix	diag(double[] A) Returns a square diagonal matrix with the elements of vector diag on the main diagonal.
EVD	eigen(int k) Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
EVD	eigen(int k, double kappa, int maxIter) Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.
static DenseMatrix	eye(int n) Returns an n-by-n identity matrix.
static DenseMatrix	eye(int m, int n) Returns an m-by-n identity matrix.
abstract double	get(int i, int j) Returns the entry value at row i and column j.
boolean	isSymmetric() Returns true if the matrix is symmetric.
abstract int	ncols() Returns the number of columns.
static DenseMatrix	newInstance(double[] A) Returns a column vector/matrix initialized by given one-dimensional array.
static DenseMatrix	newInstance(double[][] A) Returns an matrix initialized by given two-dimensional array.
static DenseMatrix	newInstance(int rows, int cols, double value) Creates a matrix filled with given value.
abstract int	nrows() Returns the number of rows.
static DenseMatrix	ones(int rows, int cols) Return an all-one matrix.
static DenseMatrix	randn(int rows, int cols) Returns a random matrix of standard normal distributed values with given mean and standard dev.
static DenseMatrix	randn(int rows, int cols, double mu, double sigma) Returns a random matrix of normal distributed values with given mean and standard dev.
void	setSymmetric(boolean symmetric) Sets if the matrix is symmetric.
SVD	svd(int k) Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
SVD	svd(int k, double kappa, int maxIter) Find k largest approximate singular triples of a matrix by the Lanczos algorithm.
java.lang.String	toString()
java.lang.String	toString(boolean full) Returns the string representation of matrix.
double	trace() Returns the matrix trace.
abstract Matrix	transpose() Returns the matrix transpose.
static DenseMatrix	zeros(int rows, int cols) Returns all-zero matrix.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

Matrix

public Matrix()

Method Detail

newInstance

public static DenseMatrix newInstance(double[][] A)

Returns an matrix initialized by given two-dimensional array.

newInstance

public static DenseMatrix newInstance(double[] A)

Returns a column vector/matrix initialized by given one-dimensional array.

newInstance

public static DenseMatrix newInstance(int rows,
                                      int cols,
                                      double value)

Creates a matrix filled with given value.

zeros

public static DenseMatrix zeros(int rows,
                                int cols)

Returns all-zero matrix.

ones

public static DenseMatrix ones(int rows,
                               int cols)

Return an all-one matrix.

eye

public static DenseMatrix eye(int n)

Returns an n-by-n identity matrix.

eye

public static DenseMatrix eye(int m,
                              int n)

Returns an m-by-n identity matrix.

diag

public static DenseMatrix diag(double[] A)

Returns a square diagonal matrix with the elements of vector diag on the main diagonal.

Parameters:

A - the array of diagonal elements.

randn

public static DenseMatrix randn(int rows,
                                int cols)

Returns a random matrix of standard normal distributed values with given mean and standard dev.

randn

public static DenseMatrix randn(int rows,
                                int cols,
                                double mu,
                                double sigma)

Returns a random matrix of normal distributed values with given mean and standard dev.

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object

toString

public java.lang.String toString(boolean full)

Returns the string representation of matrix.

Parameters:

full - Print the full matrix if true. Otherwise only print top left 7 x 7 submatrix.

isSymmetric

public boolean isSymmetric()

Returns true if the matrix is symmetric.

setSymmetric

public void setSymmetric(boolean symmetric)

Sets if the matrix is symmetric. It is the caller's responability to make sure if the matrix symmetric. Also the matrix won't update this property if the matrix values are changed.

nrows

public abstract int nrows()

Returns the number of rows.

ncols

public abstract int ncols()

Returns the number of columns.

transpose

public abstract Matrix transpose()

Returns the matrix transpose.

get

public abstract double get(int i,
                           int j)

Returns the entry value at row i and column j.

apply

public double apply(int i,
                    int j)

Returns the entry value at row i and column j. For Scala users.

diag

public double[] diag()

Returns the diagonal elements.

trace

public double trace()

Returns the matrix trace. The sum of the diagonal elements.

ata

public abstract Matrix ata()

Returns A' * A

aat

public abstract Matrix aat()

Returns A * A'

ax

public abstract double[] ax(double[] x,
                            double[] y)

y = A * x

Returns:

y

axpy

public abstract double[] axpy(double[] x,
                              double[] y)

y = A * x + y

Returns:

y

axpy

public abstract double[] axpy(double[] x,
                              double[] y,
                              double b)

y = A * x + b * y

Returns:

y

atx

public abstract double[] atx(double[] x,
                             double[] y)

y = A' * x

Returns:

y

atxpy

public abstract double[] atxpy(double[] x,
                               double[] y)

y = A' * x + y

Returns:

y

atxpy

public abstract double[] atxpy(double[] x,
                               double[] y,
                               double b)

y = A' * x + b * y

Returns:

y

eigen

public EVD eigen(int k)

Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.

Parameters:

k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.

eigen

public EVD eigen(int k,
                 double kappa,
                 int maxIter)

Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.

Parameters:

k - the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.

kappa - relative accuracy of ritz values acceptable as eigenvalues.

maxIter - Maximum number of iterations.

svd

public SVD svd(int k)

Find k largest approximate singular triples of a matrix by the Lanczos algorithm.

Parameters:

k - the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.

svd

public SVD svd(int k,
               double kappa,
               int maxIter)

Find k largest approximate singular triples of a matrix by the Lanczos algorithm.

Parameters:

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.

maxIter - Maximum number of iterations.