de.lmu.ifi.dbs.elki.math.linearalgebra
Class Matrix

java.lang.Object
  de.lmu.ifi.dbs.elki.math.linearalgebra.Matrix

All Implemented Interfaces:

MatrixLike<Matrix>, Serializable, Cloneable

public class Matrix
extends Object
implements MatrixLike<Matrix>, Serializable

A two-dimensional matrix class, where the data is stored as two-dimensional array. Implementation note: this class contains various optimizations that theoretically the java hotspot compiler should optimize on its own. However, they do show up a hotspots in the profiler (in cpu=times mode), so it does make a difference at least when optimizing other parts of ELKI.

See Also:

Serialized Form

Field Summary

(package private) int	columndimension Column dimension.
static double	DELTA A small number to handle numbers near 0 as 0.
protected double[][]	elements Array for internal storage of elements.
private static long	serialVersionUID Serial version

Constructor Summary

Matrix(double[][] elements)
Constructs a matrix from a 2-D array.

Matrix(double[] values, int m)
Construct a matrix from a one-dimensional packed array

Matrix(int m, int n)
Constructs an m-by-n matrix of zeros.

Matrix(int m, int n, double s)
Constructs an m-by-n constant matrix.

Matrix(Matrix mat)
Constructor, cloning an existing matrix.

Matrix(RationalNumber[][] q)
Constructs a Matrix for a given array of arrays of RationalNumbers.

Method Summary

boolean	almostEquals(Object obj) Compare two matrices with a delta parameter to take numerical errors into account.
boolean	almostEquals(Object obj, double maxdelta) Compare two matrices with a delta parameter to take numerical errors into account.
Matrix	appendColumns(Matrix columns) Returns a matrix which consists of this matrix and the specified columns.
Matrix	cheatToAvoidSingularity(double constant) Adds a given value to the diagonal entries if the entry is smaller than the constant.
protected void	checkMatrixDimensions(MatrixLike<?> B) Check if size(A) == size(B)
CholeskyDecomposition	chol() Cholesky Decomposition
Matrix	clone() Clone the Matrix object.
Matrix	completeBasis() Completes this d x c basis of a subspace of R^d to a d x d basis of R^d, i.e. appends c-d columns to this basis.
Matrix	completeToOrthonormalBasis() Completes this d x c basis of a subspace of R^d to a d x d basis of R^d, i.e. appends c-d columns to this basis.
double	cond() Matrix condition (2 norm)
static Matrix	constructWithCopy(double[][] A) Construct a matrix from a copy of a 2-D array.
Matrix	copy() Make a deep copy of a matrix.
double	det() Matrix determinant
static Matrix	diagonal(double[] diagonal) Returns a quadratic Matrix consisting of zeros and of the given values on the diagonal.
static Matrix	diagonal(Vector diagonal) Returns a quadratic Matrix consisting of zeros and of the given values on the diagonal.
String	dimensionInfo() Returns the dimensionality of this matrix as a string.
double	distanceCov(Matrix B) distanceCov returns distance of two Matrices A and B, i.e. the root of the sum of the squared distances Aij-Bij.
EigenvalueDecomposition	eig() Eigenvalue Decomposition
boolean	equals(Object obj)
private RationalNumber[][]	exactGaussElimination() Perform an exact Gauss-elimination of this Matrix using RationalNumbers to yield highest possible accuracy.
private static RationalNumber[][]	exactGaussElimination(RationalNumber[][] gauss) Perform recursive Gauss-elimination on the given matrix of RationalNumbers.
Matrix	exactGaussJordanElimination() Returns a matrix derived by Gauss-Jordan-elimination using RationalNumbers for the transformations.
double	get(int i, int j) Get a single element.
double[][]	getArrayCopy() Copy the internal two-dimensional array.
double[][]	getArrayRef() Access the internal two-dimensional array.
Matrix	getColumn(int j) Returns the jth column of this matrix.
int	getColumnDimensionality() Returns the dimensionality of the columns of this matrix.
double[]	getColumnPackedCopy() Make a one-dimensional column packed copy of the internal array.
Vector	getColumnVector(int j) Returns the jth column of this matrix as vector.
double[]	getDiagonal() getDiagonal returns array of diagonal-elements.
Matrix	getMatrix(int[] r, int[] c) Get a submatrix.
Matrix	getMatrix(int[] r, int j0, int j1) Get a submatrix.
Matrix	getMatrix(int i0, int i1, int[] c) Get a submatrix.
Matrix	getMatrix(int i0, int i1, int j0, int j1) Get a submatrix.
Matrix	getRow(int i) Returns the ith row of this matrix.
int	getRowDimensionality() Returns the dimensionality of the rows of this matrix.
double[]	getRowPackedCopy() Make a one-dimensional row packed copy of the internal array.
Vector	getRowVector(int i) Returns the ith row of this matrix as vector.
int	hashCode()
static Matrix	identity(int m, int n) Generate identity matrix
Matrix	increment(int i, int j, double s) Increments a single element.
Matrix	inverse() Matrix inverse or pseudoinverse
boolean	isSymmetric() Returns true, if this matrix is symmetric, false otherwise.
boolean	linearlyIndependent(Matrix columnMatrix) Returns true if the specified column matrix a is linearly independent to the columns of this matrix.
LUDecomposition	lu() LU Decomposition
Matrix	minus(Matrix B) C = A - B
Matrix	minusEquals(Matrix B) A = A - B
Matrix	minusTimes(Matrix B, double s) C = A - s * B
Matrix	minusTimesEquals(Matrix B, double s) A = A - s * B
double	norm1() One norm
double	norm2() Two norm
void	normalizeColumns() Normalizes the columns of this matrix to length of 1.0.
double	normF() Frobenius norm
double	normInf() Infinity norm
Matrix	orthonormalize() Returns an orthonormalization of this matrix.
Matrix	plus(Matrix B) C = A + B
Matrix	plusEquals(Matrix B) A = A + B
Matrix	plusTimes(Matrix B, double s) C = A + s * B
Matrix	plusTimesEquals(Matrix B, double s) A = A + s * B
QRDecomposition	qr() QR Decomposition
static Matrix	random(int m, int n) Generate matrix with random elements
int	rank() Matrix rank
static Matrix	read(BufferedReader input) Read a matrix from a stream.
double	scalarProduct(int colA, Matrix B, int colB) Returns the scalar product of the colA column of this and the colB column of B.
double	scalarProduct(int colA, Vector B) Returns the scalar product of the colA column of this and the colB column of B.
Matrix	set(int i, int j, double s) Set a single element.
void	setColumn(int j, Matrix column) Sets the jth column of this matrix to the specified column.
void	setColumnVector(int j, Vector column) Sets the jth column of this matrix to the specified column.
void	setMatrix(int[] r, int[] c, Matrix X) Set a submatrix.
void	setMatrix(int[] r, int j0, int j1, Matrix X) Set a submatrix.
void	setMatrix(int i0, int i1, int[] c, Matrix X) Set a submatrix.
void	setMatrix(int i0, int i1, int j0, int j1, Matrix X) Set a submatrix.
void	setRow(int j, Matrix row) Sets the jth row of this matrix to the specified vector.
void	setRowVector(int j, Vector row) Sets the jth row of this matrix to the specified vector.
Matrix	solve(Matrix B) Solve A*X = B
Matrix	solveTranspose(Matrix B) Solve X*A = B, which is also A'*X' = B'
SingularValueDecomposition	svd() Singular Value Decomposition
Matrix	times(double s) Multiply a matrix by a scalar, C = s*A
Matrix	times(Matrix B) Linear algebraic matrix multiplication, A * B
Vector	times(Vector B) Linear algebraic matrix multiplication, A * B
Matrix	timesEquals(double s) Multiply a matrix by a scalar in place, A = s*A
Matrix	timesTranspose(Matrix B) Linear algebraic matrix multiplication, A * B^T
String	toString() toString returns String-representation of Matrix.
double	trace() Matrix trace.
Matrix	transpose() Matrix transpose.
Matrix	transposeTimes(Matrix B) Linear algebraic matrix multiplication, AT * B
Vector	transposeTimes(Vector B) Linear algebraic matrix multiplication, AT * B
Matrix	transposeTimesTranspose(Matrix B) Linear algebraic matrix multiplication, A^T * B^T.
static Matrix	unitMatrix(int dim) Returns the unit matrix of the specified dimension.
static Matrix	zeroMatrix(int dim) Returns the zero matrix of the specified dimension.

Methods inherited from class java.lang.Object

finalize, getClass, notify, notifyAll, wait, wait, wait

Field Detail

serialVersionUID

private static final long serialVersionUID

Serial version

See Also:

Constant Field Values

DELTA

public static final double DELTA

A small number to handle numbers near 0 as 0.

See Also:

Constant Field Values

elements

protected final double[][] elements

Array for internal storage of elements.

columndimension

final int columndimension

Column dimension.

Constructor Detail

Matrix

public Matrix(int m,
              int n)

Constructs an m-by-n matrix of zeros.

Parameters:

m - number of rows

n - number of columns

Matrix

public Matrix(int m,
              int n,
              double s)

Constructs an m-by-n constant matrix.

Parameters:

m - number of rows

n - number of columns

s - A scalar value defining the constant value in the matrix

Matrix

public Matrix(double[][] elements)

Constructs a matrix from a 2-D array.

Parameters:

elements - an array of arrays of doubles defining the values of the matrix

Throws:

IllegalArgumentException - if not all rows conform in the same length

Matrix

public Matrix(RationalNumber[][] q)

Constructs a Matrix for a given array of arrays of RationalNumbers.

Parameters:

q - an array of arrays of RationalNumbers. q is not checked for consistency (i.e. whether all rows are of equal length)

Matrix

public Matrix(double[] values,
              int m)

Construct a matrix from a one-dimensional packed array

Parameters:

values - One-dimensional array of doubles, packed by columns (ala Fortran).

m - Number of rows.

Throws:

IllegalArgumentException - Array length must be a multiple of m.

Matrix

public Matrix(Matrix mat)

Constructor, cloning an existing matrix.

Parameters:

mat - Matrix to clone

Method Detail

constructWithCopy

public static final Matrix constructWithCopy(double[][] A)

Construct a matrix from a copy of a 2-D array.

Parameters:

A - Two-dimensional array of doubles.

Returns:

new matrix

Throws:

IllegalArgumentException - All rows must have the same length

unitMatrix

public static final Matrix unitMatrix(int dim)

Returns the unit matrix of the specified dimension.

Parameters:

dim - the dimensionality of the unit matrix

Returns:

the unit matrix of the specified dimension

zeroMatrix

public static final Matrix zeroMatrix(int dim)

Returns the zero matrix of the specified dimension.

Parameters:

dim - the dimensionality of the unit matrix

Returns:

the zero matrix of the specified dimension

random

public static final Matrix random(int m,
                                  int n)

Generate matrix with random elements

Parameters:

m - Number of rows.

n - Number of columns.

Returns:

An m-by-n matrix with uniformly distributed random elements.

identity

public static final Matrix identity(int m,
                                    int n)

Generate identity matrix

Parameters:

m - Number of rows.

n - Number of columns.

Returns:

An m-by-n matrix with ones on the diagonal and zeros elsewhere.

diagonal

public static final Matrix diagonal(double[] diagonal)

Returns a quadratic Matrix consisting of zeros and of the given values on the diagonal.

Parameters:

diagonal - the values on the diagonal

Returns:

the resulting matrix

diagonal

public static final Matrix diagonal(Vector diagonal)

Returns a quadratic Matrix consisting of zeros and of the given values on the diagonal.

Parameters:

diagonal - the values on the diagonal

Returns:

the resulting matrix

copy

public final Matrix copy()

Make a deep copy of a matrix.

Specified by:

copy in interface MatrixLike<Matrix>

Returns:

a new matrix containing the same values as this matrix

clone

public Matrix clone()

Clone the Matrix object.

Specified by:

clone in interface MatrixLike<Matrix>

Overrides:

clone in class Object

getArrayRef

public final double[][] getArrayRef()

Access the internal two-dimensional array.

Returns:

Pointer to the two-dimensional array of matrix elements.

getArrayCopy

public final double[][] getArrayCopy()

Copy the internal two-dimensional array.

Returns:

Two-dimensional array copy of matrix elements.

getRowDimensionality

public final int getRowDimensionality()

Returns the dimensionality of the rows of this matrix.

Specified by:

getRowDimensionality in interface MatrixLike<Matrix>

Returns:

m, the number of rows.

getColumnDimensionality

public final int getColumnDimensionality()

Returns the dimensionality of the columns of this matrix.

Specified by:

getColumnDimensionality in interface MatrixLike<Matrix>

Returns:

n, the number of columns.

get

public final double get(int i,
                        int j)

Get a single element.

Specified by:

get in interface MatrixLike<Matrix>

Parameters:

i - Row index.

j - Column index.

Returns:

A(i,j)

Throws:

ArrayIndexOutOfBoundsException - on bounds error

set

public final Matrix set(int i,
                        int j,
                        double s)

Set a single element.

Specified by:

set in interface MatrixLike<Matrix>

Parameters:

i - Row index.

j - Column index.

s - A(i,j).

Returns:

modified matrix

Throws:

ArrayIndexOutOfBoundsException - on bounds error

increment

public final Matrix increment(int i,
                              int j,
                              double s)

Increments a single element.

Specified by:

increment in interface MatrixLike<Matrix>

Parameters:

i - the row index

j - the column index

s - the increment value: A(i,j) = A(i.j) + s.

Returns:

modified matrix

Throws:

ArrayIndexOutOfBoundsException - on bounds error

getRowPackedCopy

public final double[] getRowPackedCopy()

Make a one-dimensional row packed copy of the internal array.

Returns:

Matrix elements packed in a one-dimensional array by rows.

getColumnPackedCopy

public final double[] getColumnPackedCopy()

Make a one-dimensional column packed copy of the internal array.

Returns:

Matrix elements packed in a one-dimensional array by columns.

getMatrix

public final Matrix getMatrix(int i0,
                              int i1,
                              int j0,
                              int j1)

Get a submatrix.

Parameters:

i0 - Initial row index

i1 - Final row index

j0 - Initial column index

j1 - Final column index

Returns:

A(i0:i1,j0:j1)

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

getMatrix

public final Matrix getMatrix(int[] r,
                              int[] c)

Get a submatrix.

Parameters:

r - Array of row indices.

c - Array of column indices.

Returns:

A(r(:),c(:))

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

getMatrix

public final Matrix getMatrix(int[] r,
                              int j0,
                              int j1)

Get a submatrix.

Parameters:

r - Array of row indices.

j0 - Initial column index

j1 - Final column index

Returns:

A(r(:),j0:j1)

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

getMatrix

public final Matrix getMatrix(int i0,
                              int i1,
                              int[] c)

Get a submatrix.

Parameters:

i0 - Initial row index

i1 - Final row index

c - Array of column indices.

Returns:

A(i0:i1,c(:))

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

setMatrix

public final void setMatrix(int i0,
                            int i1,
                            int j0,
                            int j1,
                            Matrix X)

Set a submatrix.

Parameters:

i0 - Initial row index

i1 - Final row index

j0 - Initial column index

j1 - Final column index

X - A(i0:i1,j0:j1)

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

setMatrix

public final void setMatrix(int[] r,
                            int[] c,
                            Matrix X)

Set a submatrix.

Parameters:

r - Array of row indices.

c - Array of column indices.

X - A(r(:),c(:))

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

setMatrix

public final void setMatrix(int[] r,
                            int j0,
                            int j1,
                            Matrix X)

Set a submatrix.

Parameters:

r - Array of row indices.

j0 - Initial column index

j1 - Final column index

X - A(r(:),j0:j1)

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

setMatrix

public final void setMatrix(int i0,
                            int i1,
                            int[] c,
                            Matrix X)

Set a submatrix.

Parameters:

i0 - Initial row index

i1 - Final row index

c - Array of column indices.

X - A(i0:i1,c(:))

Throws:

ArrayIndexOutOfBoundsException - Submatrix indices

getRow

public final Matrix getRow(int i)

Returns the ith row of this matrix.

Parameters:

i - the index of the row to be returned

Returns:

the ith row of this matrix

getRowVector

public final Vector getRowVector(int i)

Returns the ith row of this matrix as vector.

Parameters:

i - the index of the row to be returned

Returns:

the ith row of this matrix

setRow

public final void setRow(int j,
                         Matrix row)

Sets the jth row of this matrix to the specified vector.

Parameters:

j - the index of the row to be set

row - the value of the row to be set

setRowVector

public final void setRowVector(int j,
                               Vector row)

Sets the jth row of this matrix to the specified vector.

Parameters:

j - the index of the column to be set

row - the value of the column to be set

getColumn

public final Matrix getColumn(int j)

Returns the jth column of this matrix.

Parameters:

j - the index of the column to be returned

Returns:

the jth column of this matrix

getColumnVector

public final Vector getColumnVector(int j)

Returns the jth column of this matrix as vector.

Specified by:

getColumnVector in interface MatrixLike<Matrix>

Parameters:

j - the index of the column to be returned

Returns:

the jth column of this matrix

setColumn

public final void setColumn(int j,
                            Matrix column)

Sets the jth column of this matrix to the specified column.

Parameters:

j - the index of the column to be set

column - the value of the column to be set

setColumnVector

public final void setColumnVector(int j,
                                  Vector column)

Sets the jth column of this matrix to the specified column.

Parameters:

j - the index of the column to be set

column - the value of the column to be set

transpose

public final Matrix transpose()

Matrix transpose.

Specified by:

transpose in interface MatrixLike<Matrix>

Returns:

A^T

plus

public final Matrix plus(Matrix B)

C = A + B

Specified by:

plus in interface MatrixLike<Matrix>

Parameters:

B - another matrix

Returns:

A + B in a new Matrix

plusTimes

public final Matrix plusTimes(Matrix B,
                              double s)

C = A + s * B

Specified by:

plusTimes in interface MatrixLike<Matrix>

Parameters:

B - another matrix

s - scalar

Returns:

A + s * B in a new Matrix

plusEquals

public final Matrix plusEquals(Matrix B)

A = A + B

Specified by:

plusEquals in interface MatrixLike<Matrix>

Parameters:

B - another matrix

Returns:

A + B in this Matrix

plusTimesEquals

public final Matrix plusTimesEquals(Matrix B,
                                    double s)

A = A + s * B

Specified by:

plusTimesEquals in interface MatrixLike<Matrix>

Parameters:

B - another matrix

s - Scalar

Returns:

A + s * B in this Matrix

minus

public final Matrix minus(Matrix B)

C = A - B

Specified by:

minus in interface MatrixLike<Matrix>

Parameters:

B - another matrix

Returns:

A - B in a new Matrix

minusTimes

public final Matrix minusTimes(Matrix B,
                               double s)

C = A - s * B

Specified by:

minusTimes in interface MatrixLike<Matrix>

Parameters:

B - another matrix

s - Scalar

Returns:

A - s * B in a new Matrix

minusEquals

public final Matrix minusEquals(Matrix B)

A = A - B

Specified by:

minusEquals in interface MatrixLike<Matrix>

Parameters:

B - another matrix

Returns:

A - B in this Matrix

minusTimesEquals

public final Matrix minusTimesEquals(Matrix B,
                                     double s)

A = A - s * B

Specified by:

minusTimesEquals in interface MatrixLike<Matrix>

Parameters:

B - another matrix

s - Scalar

Returns:

A - s * B in this Matrix

times

public final Matrix times(double s)

Multiply a matrix by a scalar, C = s*A

Specified by:

times in interface MatrixLike<Matrix>

Parameters:

s - scalar

Returns:

s*A

timesEquals

public final Matrix timesEquals(double s)

Multiply a matrix by a scalar in place, A = s*A

Specified by:

timesEquals in interface MatrixLike<Matrix>

Parameters:

s - scalar

Returns:

replace A by s*A

times

public final Matrix times(Matrix B)

Linear algebraic matrix multiplication, A * B

Parameters:

B - another matrix

Returns:

Matrix product, A * B

Throws:

IllegalArgumentException - Matrix inner dimensions must agree.

times

public final Vector times(Vector B)

Linear algebraic matrix multiplication, A * B

Parameters:

B - a vector

Returns:

Matrix product, A * B

Throws:

IllegalArgumentException - Matrix inner dimensions must agree.

transposeTimes

public final Vector transposeTimes(Vector B)

Linear algebraic matrix multiplication, A^T * B

Parameters:

B - another matrix

Returns:

Matrix product, A^T * B

Throws:

IllegalArgumentException - Matrix inner dimensions must agree.

transposeTimes

public final Matrix transposeTimes(Matrix B)

Linear algebraic matrix multiplication, A^T * B

Parameters:

B - another matrix

Returns:

Matrix product, A^T * B

Throws:

IllegalArgumentException - Matrix inner dimensions must agree.

timesTranspose

public final Matrix timesTranspose(Matrix B)

Linear algebraic matrix multiplication, A * B^T

Parameters:

B - another matrix

Returns:

Matrix product, A * B^T

Throws:

IllegalArgumentException - Matrix inner dimensions must agree.

transposeTimesTranspose

public final Matrix transposeTimesTranspose(Matrix B)

Linear algebraic matrix multiplication, A^T * B^T. Computed as (B*A)^T

Parameters:

B - another matrix

Returns:

Matrix product, A^T * B^T

Throws:

IllegalArgumentException - Matrix inner dimensions must agree.

scalarProduct

public double scalarProduct(int colA,
                            Matrix B,
                            int colB)

Returns the scalar product of the colA column of this and the colB column of B.

Parameters:

colA - The column of A to compute scalar product for

B - second Matrix

colB - The column of B to compute scalar product for

Returns:

double The scalar product of the first column of this and B

scalarProduct

public double scalarProduct(int colA,
                            Vector B)

Returns the scalar product of the colA column of this and the colB column of B.

Parameters:

colA - The column of A to compute scalar product for

B - Vector

Returns:

double The scalar product of the first column of this and B

lu

public final LUDecomposition lu()

LU Decomposition

Returns:

LUDecomposition

See Also:

LUDecomposition

qr

public final QRDecomposition qr()

QR Decomposition

Returns:

QRDecomposition

See Also:

QRDecomposition

chol

public final CholeskyDecomposition chol()

Cholesky Decomposition

Returns:

CholeskyDecomposition

See Also:

CholeskyDecomposition

svd

public final SingularValueDecomposition svd()

Singular Value Decomposition

Returns:

SingularValueDecomposition

See Also:

SingularValueDecomposition

eig

public final EigenvalueDecomposition eig()

Eigenvalue Decomposition

Returns:

EigenvalueDecomposition

See Also:

EigenvalueDecomposition

solve

public final Matrix solve(Matrix B)

Solve A*X = B

Parameters:

B - right hand side

Returns:

solution if A is square, least squares solution otherwise

solveTranspose

public final Matrix solveTranspose(Matrix B)

Solve X*A = B, which is also A'*X' = B'

Parameters:

B - right hand side

Returns:

solution if A is square, least squares solution otherwise.

inverse

public final Matrix inverse()

Matrix inverse or pseudoinverse

Returns:

inverse(A) if A is square, pseudoinverse otherwise.

det

public final double det()

Matrix determinant

Returns:

determinant

rank

public final int rank()

Matrix rank

Returns:

effective numerical rank, obtained from SVD.

cond

public final double cond()

Matrix condition (2 norm)

Returns:

ratio of largest to smallest singular value.

trace

public final double trace()

Matrix trace.

Returns:

sum of the diagonal elements.

norm1

public double norm1()

One norm

Returns:

maximum column sum.

norm2

public final double norm2()

Two norm

Returns:

maximum singular value.

normInf

public double normInf()

Infinity norm

Returns:

maximum row sum.

normF

public double normF()

Frobenius norm

Returns:

sqrt of sum of squares of all elements.

distanceCov

public final double distanceCov(Matrix B)

distanceCov returns distance of two Matrices A and B, i.e. the root of the sum of the squared distances A_ij-B_ij.

Parameters:

B - Matrix to compute distance from this (A)

Returns:

distance of Matrices

getDiagonal

public final double[] getDiagonal()

getDiagonal returns array of diagonal-elements.

Returns:

double[] the values on the diagonal of the Matrix

normalizeColumns

public void normalizeColumns()

Normalizes the columns of this matrix to length of 1.0.

linearlyIndependent

public final boolean linearlyIndependent(Matrix columnMatrix)

Returns true if the specified column matrix a is linearly independent to the columns of this matrix. Linearly independence is given, if the matrix resulting from appending a to this matrix has full rank.

Parameters:

columnMatrix - the column matrix to be tested for linear independence

Returns:

true if the specified column matrix is linearly independent to the columns of this matrix

exactGaussJordanElimination

public final Matrix exactGaussJordanElimination()

Returns a matrix derived by Gauss-Jordan-elimination using RationalNumbers for the transformations.

Returns:

a matrix derived by Gauss-Jordan-elimination using RationalNumbers for the transformations

exactGaussElimination

private final RationalNumber[][] exactGaussElimination()

Perform an exact Gauss-elimination of this Matrix using RationalNumbers to yield highest possible accuracy.

Returns:

an array of arrays of RationalNumbers representing the Gauss-eliminated form of this Matrix

exactGaussElimination

private static final RationalNumber[][] exactGaussElimination(RationalNumber[][] gauss)

Perform recursive Gauss-elimination on the given matrix of RationalNumbers.

Parameters:

gauss - an array of arrays of RationalNumber

Returns:

recursive derived Gauss-elimination-form of the given matrix of RationalNumbers

isSymmetric

public final boolean isSymmetric()

Returns true, if this matrix is symmetric, false otherwise.

Returns:

true, if this matrix is symmetric, false otherwise

completeBasis

public final Matrix completeBasis()

Completes this d x c basis of a subspace of R^d to a d x d basis of R^d, i.e. appends c-d columns to this basis.

Returns:

the appended columns

completeToOrthonormalBasis

public final Matrix completeToOrthonormalBasis()

Completes this d x c basis of a subspace of R^d to a d x d basis of R^d, i.e. appends c-d columns to this basis.

Returns:

the appended columns

appendColumns

public final Matrix appendColumns(Matrix columns)

Returns a matrix which consists of this matrix and the specified columns.

Parameters:

columns - the columns to be appended

Returns:

the new matrix with the appended columns

orthonormalize

public final Matrix orthonormalize()

Returns an orthonormalization of this matrix.

Returns:

the orthonormalized matrix

cheatToAvoidSingularity

public final Matrix cheatToAvoidSingularity(double constant)

Adds a given value to the diagonal entries if the entry is smaller than the constant.

Parameters:

constant - value to add to the diagonal entries

Returns:

a new Matrix differing from this Matrix by the given value added to the diagonal entries

read

public static final Matrix read(BufferedReader input)
                         throws IOException

Read a matrix from a stream. The format is the same the print method, so printed matrices can be read back in (provided they were printed using US Locale). Elements are separated by whitespace, all the elements for each row appear on a single line, the last row is followed by a blank line.

Parameters:

input - the input stream.

Returns:

New matrix

Throws:

IOException - on input error

checkMatrixDimensions

protected void checkMatrixDimensions(MatrixLike<?> B)

Check if size(A) == size(B)

hashCode

public int hashCode()

Overrides:

hashCode in class Object

equals

public boolean equals(Object obj)

Overrides:

equals in class Object

almostEquals

public boolean almostEquals(Object obj,
                            double maxdelta)

Compare two matrices with a delta parameter to take numerical errors into account.

Parameters:

obj - other object to compare with

maxdelta - maximum delta allowed

Returns:

true if delta smaller than maximum

almostEquals

public boolean almostEquals(Object obj)

Compare two matrices with a delta parameter to take numerical errors into account.

Parameters:

obj - other object to compare with

Returns:

almost equals with delta DELTA

dimensionInfo

public String dimensionInfo()

Returns the dimensionality of this matrix as a string.

Returns:

the dimensionality of this matrix as a string

toString

public String toString()

toString returns String-representation of Matrix.

Overrides:

toString in class Object