Jama
Class Matrix

java.lang.Object
  Jama.Matrix

All Implemented Interfaces:

Serializable, Cloneable

public class Matrix
extends Object
implements Cloneable, Serializable

Jama = Java Matrix class.

The Java Matrix Class provides the fundamental operations of numerical linear algebra. Various constructors create Matrices from two dimensional arrays of double precision floating point numbers. Various "gets" and "sets" provide access to submatrices and matrix elements. Several methods implement basic matrix arithmetic, including matrix addition and multiplication, matrix norms, and element-by-element array operations. Methods for reading and printing matrices are also included. All the operations in this version of the Matrix Class involve real matrices. Complex matrices may be handled in a future version.

Five fundamental matrix decompositions, which consist of pairs or triples of matrices, permutation vectors, and the like, produce results in five decomposition classes. These decompositions are accessed by the Matrix class to compute solutions of simultaneous linear equations, determinants, inverses and other matrix functions. The five decompositions are:

• Cholesky Decomposition of symmetric, positive definite matrices.
• LU Decomposition of rectangular matrices.
• QR Decomposition of rectangular matrices.
• Singular Value Decomposition of rectangular matrices.
• Eigenvalue Decomposition of both symmetric and nonsymmetric square matrices.

Example of use:

Solve a linear system A x = b and compute the residual norm, ||b - A x||.

      double[][] vals = {{1.,2.,3},{4.,5.,6.},{7.,8.,10.}};
      Matrix A = new Matrix(vals);
      Matrix b = Matrix.random(3,1);
      Matrix x = A.solve(b);
      Matrix r = A.times(x).minus(b);
      double rnorm = r.normInf();

See Also:

Serialized Form

Constructor Summary

Matrix(double[][] A)
Construct a matrix from a 2-D array.

Matrix(double[][] A, int m, int n)
Construct a matrix quickly without checking arguments.

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

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

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

Method Summary

Matrix	arrayLeftDivide(Matrix B) Element-by-element left division, C = A.\B
Matrix	arrayLeftDivideEquals(Matrix B) Element-by-element left division in place, A = A.\B
Matrix	arrayRightDivide(Matrix B) Element-by-element right division, C = A./B
Matrix	arrayRightDivideEquals(Matrix B) Element-by-element right division in place, A = A./B
Matrix	arrayTimes(Matrix B) Element-by-element multiplication, C = A.*B
Matrix	arrayTimesEquals(Matrix B) Element-by-element multiplication in place, A = A.*B
CholeskyDecomposition	chol() Cholesky Decomposition
Object	clone() Clone the Matrix object.
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
EigenvalueDecomposition	eig() Eigenvalue Decomposition
double	get(int i, int j) Get a single element.
double[][]	getArray() Access the internal two-dimensional array.
double[][]	getArrayCopy() Copy the internal two-dimensional array.
int	getColumnDimension() Get column dimension.
double[]	getColumnPackedCopy() Make a one-dimensional column packed copy of the internal array.
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.
int	getRowDimension() Get row dimension.
double[]	getRowPackedCopy() Make a one-dimensional row packed copy of the internal array.
static Matrix	identity(int m, int n) Generate identity matrix
Matrix	inverse() Matrix inverse or pseudoinverse
LUDecomposition	lu() LU Decomposition
Matrix	minus(Matrix B) C = A - B
Matrix	minusEquals(Matrix B) A = A - B
double	norm1() One norm
double	norm2() Two norm
double	normF() Frobenius norm
double	normInf() Infinity norm
Matrix	plus(Matrix B) C = A + B
Matrix	plusEquals(Matrix B) A = A + B
void	print(int w, int d) Print the matrix to stdout.
void	print(NumberFormat format, int width) Print the matrix to stdout.
void	print(PrintWriter output, int w, int d) Print the matrix to the output stream.
void	print(PrintWriter output, NumberFormat format, int width) Print the matrix to the output stream.
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.
void	set(int i, int j, double s) Set a single element.
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.
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
Matrix	timesEquals(double s) Multiply a matrix by a scalar in place, A = s*A
double	trace() Matrix trace.
Matrix	transpose() Matrix transpose.
Matrix	uminus() Unary minus

Methods inherited from class java.lang.Object

equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

Matrix

public Matrix(int m,
              int n)

Construct an m-by-n matrix of zeros.

Parameters:

m - Number of rows.

n - Number of colums.

Matrix

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

Construct an m-by-n constant matrix.

Parameters:

m - Number of rows.

n - Number of colums.

s - Fill the matrix with this scalar value.

Matrix

public Matrix(double[][] A)

Construct a matrix from a 2-D array.

Parameters:

A - Two-dimensional array of doubles.

Throws:

IllegalArgumentException - All rows must have the same length

See Also:

constructWithCopy(double[][])

Matrix

public Matrix(double[][] A,
              int m,
              int n)

Construct a matrix quickly without checking arguments.

Parameters:

A - Two-dimensional array of doubles.

m - Number of rows.

n - Number of colums.

Matrix

public Matrix(double[] vals,
              int m)

Construct a matrix from a one-dimensional packed array

Parameters:

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

m - Number of rows.

Throws:

IllegalArgumentException - Array length must be a multiple of m.

Method Detail

constructWithCopy

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

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

Parameters:

A - Two-dimensional array of doubles.

Throws:

IllegalArgumentException - All rows must have the same length

copy

public Matrix copy()

Make a deep copy of a matrix

clone

public Object clone()

Clone the Matrix object.

Overrides:

clone in class Object

getArray

public double[][] getArray()

Access the internal two-dimensional array.

Returns:

Pointer to the two-dimensional array of matrix elements.

getArrayCopy

public double[][] getArrayCopy()

Copy the internal two-dimensional array.

Returns:

Two-dimensional array copy of matrix elements.

getColumnPackedCopy

public double[] getColumnPackedCopy()

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

Returns:

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

getRowPackedCopy

public double[] getRowPackedCopy()

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

Returns:

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

getRowDimension

public int getRowDimension()

Get row dimension.

Returns:

m, the number of rows.

getColumnDimension

public int getColumnDimension()

Get column dimension.

Returns:

n, the number of columns.

get

public double get(int i,
                  int j)

Get a single element.

Parameters:

i - Row index.

j - Column index.

Returns:

A(i,j)

Throws:

ArrayIndexOutOfBoundsException

getMatrix

public 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 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 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

getMatrix

public 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

set

public void set(int i,
                int j,
                double s)

Set a single element.

Parameters:

i - Row index.

j - Column index.

s - A(i,j).

Throws:

ArrayIndexOutOfBoundsException

setMatrix

public 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 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 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 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

transpose

public Matrix transpose()

Matrix transpose.

Returns:

A'

norm1

public double norm1()

One norm

Returns:

maximum column sum.

norm2

public 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.

uminus

public Matrix uminus()

Unary minus

Returns:

-A

plus

public Matrix plus(Matrix B)

C = A + B

Parameters:

B - another matrix

Returns:

A + B

plusEquals

public Matrix plusEquals(Matrix B)

A = A + B

Parameters:

B - another matrix

Returns:

A + B

minus

public Matrix minus(Matrix B)

C = A - B

Parameters:

B - another matrix

Returns:

A - B

minusEquals

public Matrix minusEquals(Matrix B)

A = A - B

Parameters:

B - another matrix

Returns:

A - B

arrayTimes

public Matrix arrayTimes(Matrix B)

Element-by-element multiplication, C = A.*B

Parameters:

B - another matrix

Returns:

A.*B

arrayTimesEquals

public Matrix arrayTimesEquals(Matrix B)

Element-by-element multiplication in place, A = A.*B

Parameters:

B - another matrix

Returns:

A.*B

arrayRightDivide

public Matrix arrayRightDivide(Matrix B)

Element-by-element right division, C = A./B

Parameters:

B - another matrix

Returns:

A./B

arrayRightDivideEquals

public Matrix arrayRightDivideEquals(Matrix B)

Element-by-element right division in place, A = A./B

Parameters:

B - another matrix

Returns:

A./B

arrayLeftDivide

public Matrix arrayLeftDivide(Matrix B)

Element-by-element left division, C = A.\B

Parameters:

B - another matrix

Returns:

A.\B

arrayLeftDivideEquals

public Matrix arrayLeftDivideEquals(Matrix B)

Element-by-element left division in place, A = A.\B

Parameters:

B - another matrix

Returns:

A.\B

times

public Matrix times(double s)

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

Parameters:

s - scalar

Returns:

s*A

timesEquals

public Matrix timesEquals(double s)

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

Parameters:

s - scalar

Returns:

replace A by s*A

times

public 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.

lu

public LUDecomposition lu()

LU Decomposition

Returns:

LUDecomposition

See Also:

LUDecomposition

qr

public QRDecomposition qr()

QR Decomposition

Returns:

QRDecomposition

See Also:

QRDecomposition

chol

public CholeskyDecomposition chol()

Cholesky Decomposition

Returns:

CholeskyDecomposition

See Also:

CholeskyDecomposition

svd

public SingularValueDecomposition svd()

Singular Value Decomposition

Returns:

SingularValueDecomposition

See Also:

SingularValueDecomposition

eig

public EigenvalueDecomposition eig()

Eigenvalue Decomposition

Returns:

EigenvalueDecomposition

See Also:

EigenvalueDecomposition

solve

public 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 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 Matrix inverse()

Matrix inverse or pseudoinverse

Returns:

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

det

public double det()

Matrix determinant

Returns:

determinant

rank

public int rank()

Matrix rank

Returns:

effective numerical rank, obtained from SVD.

cond

public double cond()

Matrix condition (2 norm)

Returns:

ratio of largest to smallest singular value.

trace

public double trace()

Matrix trace.

Returns:

sum of the diagonal elements.

random

public static Matrix random(int m,
                            int n)

Generate matrix with random elements

Parameters:

m - Number of rows.

n - Number of colums.

Returns:

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

identity

public static Matrix identity(int m,
                              int n)

Generate identity matrix

Parameters:

m - Number of rows.

n - Number of colums.

Returns:

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

print

public void print(int w,
                  int d)

Print the matrix to stdout. Line the elements up in columns with a Fortran-like 'Fw.d' style format.

Parameters:

w - Column width.

d - Number of digits after the decimal.

print

public void print(PrintWriter output,
                  int w,
                  int d)

Print the matrix to the output stream. Line the elements up in columns with a Fortran-like 'Fw.d' style format.

Parameters:

output - Output stream.

w - Column width.

d - Number of digits after the decimal.

print

public void print(NumberFormat format,
                  int width)

Print the matrix to stdout. Line the elements up in columns. Use the format object, and right justify within columns of width characters. Note that is the matrix is to be read back in, you probably will want to use a NumberFormat that is set to US Locale.

Parameters:

format - A Formatting object for individual elements.

width - Field width for each column.

See Also:

DecimalFormat.setDecimalFormatSymbols(java.text.DecimalFormatSymbols)

print

public void print(PrintWriter output,
                  NumberFormat format,
                  int width)

Print the matrix to the output stream. Line the elements up in columns. Use the format object, and right justify within columns of width characters. Note that is the matrix is to be read back in, you probably will want to use a NumberFormat that is set to US Locale.

Parameters:

output - the output stream.

format - A formatting object to format the matrix elements

width - Column width.

See Also:

DecimalFormat.setDecimalFormatSymbols(java.text.DecimalFormatSymbols)

read

public static 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.

Throws:

IOException