cern.colt.matrix.linalg
Class Algebra

java.lang.Object
  extended by cern.colt.PersistentObject
      extended by cern.colt.matrix.linalg.Algebra
All Implemented Interfaces:
java.io.Serializable, java.lang.Cloneable

public class Algebra
extends PersistentObject

Linear algebraic matrix operations operating on DoubleMatrix2D; concentrates most functionality of this package.

Author:
wolfgang.hoschek@cern.ch
See Also:
Serialized Form

Field Summary
static Algebra DEFAULT
          A default Algebra object; has Property.DEFAULT attached for tolerance.
protected  Property property
          The property object attached to this instance.
static Algebra ZERO
          A default Algebra object; has Property.ZERO attached for tolerance.
 
Fields inherited from class cern.colt.PersistentObject
serialVersionUID
 
Constructor Summary
Algebra()
          Constructs a new instance with an equality tolerance given by Property.DEFAULT.tolerance().
Algebra(double tolerance)
          Constructs a new instance with the given equality tolerance.
 
Method Summary
 java.lang.Object clone()
          Returns a copy of the receiver.
 double cond(DoubleMatrix2D A)
          Returns the condition of matrix A, which is the ratio of largest to smallest singular value.
 double det(DoubleMatrix2D A)
          Returns the determinant of matrix A.
protected static double hypot(double a, double b)
          Returns sqrt(a^2 + b^2) without under/overflow.
protected static DoubleDoubleFunction hypotFunction()
          Returns sqrt(a^2 + b^2) without under/overflow.
 DoubleMatrix2D inverse(DoubleMatrix2D A)
          Returns the inverse or pseudo-inverse of matrix A.
 double mult(DoubleMatrix1D x, DoubleMatrix1D y)
          Inner product of two vectors; Sum(x[i] * y[i]).
 DoubleMatrix1D mult(DoubleMatrix2D A, DoubleMatrix1D y)
          Linear algebraic matrix-vector multiplication; z = A * y.
 DoubleMatrix2D mult(DoubleMatrix2D A, DoubleMatrix2D B)
          Linear algebraic matrix-matrix multiplication; C = A x B.
 DoubleMatrix2D multOuter(DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix2D A)
          Outer product of two vectors; Sets A[i,j] = x[i] * y[j].
 double norm1(DoubleMatrix1D x)
          Returns the one-norm of vector x, which is Sum(abs(x[i])).
 double norm1(DoubleMatrix2D A)
          Returns the one-norm of matrix A, which is the maximum absolute column sum.
 double norm2(DoubleMatrix1D x)
          Returns the two-norm (aka euclidean norm) of vector x; equivalent to mult(x,x).
 double norm2(DoubleMatrix2D A)
          Returns the two-norm of matrix A, which is the maximum singular value; obtained from SVD.
 double normF(DoubleMatrix2D A)
          Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]2)).
 double normInfinity(DoubleMatrix1D x)
          Returns the infinity norm of vector x, which is Max(abs(x[i])).
 double normInfinity(DoubleMatrix2D A)
          Returns the infinity norm of matrix A, which is the maximum absolute row sum.
 DoubleMatrix1D permute(DoubleMatrix1D A, int[] indexes, double[] work)
          Modifies the given vector A such that it is permuted as specified; Useful for pivoting.
 DoubleMatrix2D permute(DoubleMatrix2D A, int[] rowIndexes, int[] columnIndexes)
          Constructs and returns a new row and column permuted selection view of matrix A; equivalent to DoubleMatrix2D.viewSelection(int[],int[]).
 DoubleMatrix2D permuteColumns(DoubleMatrix2D A, int[] indexes, int[] work)
          Modifies the given matrix A such that it's columns are permuted as specified; Useful for pivoting.
 DoubleMatrix2D permuteRows(DoubleMatrix2D A, int[] indexes, int[] work)
          Modifies the given matrix A such that it's rows are permuted as specified; Useful for pivoting.
 DoubleMatrix2D pow(DoubleMatrix2D A, int p)
          Linear algebraic matrix power; B = Ak <==> B = A*A*...*A.
 Property property()
          Returns the property object attached to this Algebra, defining tolerance.
 int rank(DoubleMatrix2D A)
          Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.
 void setProperty(Property property)
          Attaches the given property object to this Algebra, defining tolerance.
 DoubleMatrix2D solve(DoubleMatrix2D A, DoubleMatrix2D B)
          Solves A*X = B.
 DoubleMatrix2D solveTranspose(DoubleMatrix2D A, DoubleMatrix2D B)
          Solves X*A = B, which is also A'*X' = B'.
 DoubleMatrix2D subMatrix(DoubleMatrix2D A, int fromRow, int toRow, int fromColumn, int toColumn)
          Constructs and returns a new sub-range view which is the sub matrix A[fromRow..toRow,fromColumn..toColumn].
 java.lang.String toString(DoubleMatrix2D matrix)
          Returns a String with (propertyName, propertyValue) pairs.
 java.lang.String toVerboseString(DoubleMatrix2D matrix)
          Returns the results of toString(A) and additionally the results of all sorts of decompositions applied to the given matrix.
 double trace(DoubleMatrix2D A)
          Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).
 DoubleMatrix2D transpose(DoubleMatrix2D A)
          Constructs and returns a new view which is the transposition of the given matrix A.
protected  DoubleMatrix2D trapezoidalLower(DoubleMatrix2D A)
          Modifies the matrix to be a lower trapezoidal matrix.
 
Methods inherited from class java.lang.Object
equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Field Detail

DEFAULT

public static final Algebra DEFAULT
A default Algebra object; has Property.DEFAULT attached for tolerance. Allows ommiting to construct an Algebra object time and again. Note that this Algebra object is immutable. Any attempt to assign a new Property object to it (via method setProperty), or to alter the tolerance of its property object (via property().setTolerance(...)) will throw an exception.


ZERO

public static final Algebra ZERO
A default Algebra object; has Property.ZERO attached for tolerance. Allows ommiting to construct an Algebra object time and again. Note that this Algebra object is immutable. Any attempt to assign a new Property object to it (via method setProperty), or to alter the tolerance of its property object (via property().setTolerance(...)) will throw an exception.


property

protected Property property
The property object attached to this instance.

Constructor Detail

Algebra

public Algebra()
Constructs a new instance with an equality tolerance given by Property.DEFAULT.tolerance().


Algebra

public Algebra(double tolerance)
Constructs a new instance with the given equality tolerance.

Parameters:
tolerance - the tolerance to be used for equality operations.
Method Detail

clone

public java.lang.Object clone()
Returns a copy of the receiver. The attached property object is also copied. Hence, the property object of the copy is mutable.

Overrides:
clone in class PersistentObject
Returns:
a copy of the receiver.

cond

public double cond(DoubleMatrix2D A)
Returns the condition of matrix A, which is the ratio of largest to smallest singular value.


det

public double det(DoubleMatrix2D A)
Returns the determinant of matrix A.

Returns:
the determinant.

hypot

protected static double hypot(double a,
                              double b)
Returns sqrt(a^2 + b^2) without under/overflow.


hypotFunction

protected static DoubleDoubleFunction hypotFunction()
Returns sqrt(a^2 + b^2) without under/overflow.


inverse

public DoubleMatrix2D inverse(DoubleMatrix2D A)
Returns the inverse or pseudo-inverse of matrix A.

Returns:
a new independent matrix; inverse(matrix) if the matrix is square, pseudoinverse otherwise.

mult

public double mult(DoubleMatrix1D x,
                   DoubleMatrix1D y)
Inner product of two vectors; Sum(x[i] * y[i]). Also known as dot product.
Equivalent to x.zDotProduct(y).

Parameters:
x - the first source vector.
y - the second source matrix.
Returns:
the inner product.
Throws:
java.lang.IllegalArgumentException - if x.size() != y.size().

mult

public DoubleMatrix1D mult(DoubleMatrix2D A,
                           DoubleMatrix1D y)
Linear algebraic matrix-vector multiplication; z = A * y. z[i] = Sum(A[i,j] * y[j]), i=0..A.rows()-1, j=0..y.size()-1.

Parameters:
A - the source matrix.
y - the source vector.
Returns:
z; a new vector with z.size()==A.rows().
Throws:
java.lang.IllegalArgumentException - if A.columns() != y.size().

mult

public DoubleMatrix2D mult(DoubleMatrix2D A,
                           DoubleMatrix2D B)
Linear algebraic matrix-matrix multiplication; C = A x B. C[i,j] = Sum(A[i,k] * B[k,j]), k=0..n-1.
Matrix shapes: A(m x n), B(n x p), C(m x p).

Parameters:
A - the first source matrix.
B - the second source matrix.
Returns:
C; a new matrix holding the results, with C.rows()=A.rows(), C.columns()==B.columns().
Throws:
java.lang.IllegalArgumentException - if B.rows() != A.columns().

multOuter

public DoubleMatrix2D multOuter(DoubleMatrix1D x,
                                DoubleMatrix1D y,
                                DoubleMatrix2D A)
Outer product of two vectors; Sets A[i,j] = x[i] * y[j].

Parameters:
x - the first source vector.
y - the second source vector.
A - the matrix to hold the results. Set this parameter to null to indicate that a new result matrix shall be constructed.
Returns:
A (for convenience only).
Throws:
java.lang.IllegalArgumentException - if A.rows() != x.size() || A.columns() != y.size().

norm1

public double norm1(DoubleMatrix1D x)
Returns the one-norm of vector x, which is Sum(abs(x[i])).


norm1

public double norm1(DoubleMatrix2D A)
Returns the one-norm of matrix A, which is the maximum absolute column sum.


norm2

public double norm2(DoubleMatrix1D x)
Returns the two-norm (aka euclidean norm) of vector x; equivalent to mult(x,x).


norm2

public double norm2(DoubleMatrix2D A)
Returns the two-norm of matrix A, which is the maximum singular value; obtained from SVD.


normF

public double normF(DoubleMatrix2D A)
Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]2)).


normInfinity

public double normInfinity(DoubleMatrix1D x)
Returns the infinity norm of vector x, which is Max(abs(x[i])).


normInfinity

public double normInfinity(DoubleMatrix2D A)
Returns the infinity norm of matrix A, which is the maximum absolute row sum.


permute

public DoubleMatrix1D permute(DoubleMatrix1D A,
                              int[] indexes,
                              double[] work)
Modifies the given vector A such that it is permuted as specified; Useful for pivoting. Cell A[i] will go into cell A[indexes[i]].

Example:

Reordering
[A,B,C,D,E] with indexes [0,4,2,3,1] yields 
[A,E,C,D,B]
In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[2], A[3]<--A[3], A[4]<--A[1].

Reordering
[A,B,C,D,E] with indexes [0,4,1,2,3] yields 
[A,E,B,C,D]
In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[1], A[3]<--A[2], A[4]<--A[3].

Parameters:
A - the vector to permute.
indexes - the permutation indexes, must satisfy indexes.length==A.size() && indexes[i] >= 0 && indexes[i] < A.size();
work - the working storage, must satisfy work.length >= A.size(); set work==null if you don't care about performance.
Returns:
the modified A (for convenience only).
Throws:
java.lang.IndexOutOfBoundsException - if indexes.length != A.size().

permute

public DoubleMatrix2D permute(DoubleMatrix2D A,
                              int[] rowIndexes,
                              int[] columnIndexes)
Constructs and returns a new row and column permuted selection view of matrix A; equivalent to DoubleMatrix2D.viewSelection(int[],int[]). The returned matrix is backed by this matrix, so changes in the returned matrix are reflected in this matrix, and vice-versa. Use idioms like result = permute(...).copy() to generate an independent sub matrix.

Returns:
the new permuted selection view.

permuteColumns

public DoubleMatrix2D permuteColumns(DoubleMatrix2D A,
                                     int[] indexes,
                                     int[] work)
Modifies the given matrix A such that it's columns are permuted as specified; Useful for pivoting. Column A[i] will go into column A[indexes[i]]. Equivalent to permuteRows(transpose(A), indexes, work).

Parameters:
A - the matrix to permute.
indexes - the permutation indexes, must satisfy indexes.length==A.columns() && indexes[i] >= 0 && indexes[i] < A.columns();
work - the working storage, must satisfy work.length >= A.columns(); set work==null if you don't care about performance.
Returns:
the modified A (for convenience only).
Throws:
java.lang.IndexOutOfBoundsException - if indexes.length != A.columns().

permuteRows

public DoubleMatrix2D permuteRows(DoubleMatrix2D A,
                                  int[] indexes,
                                  int[] work)
Modifies the given matrix A such that it's rows are permuted as specified; Useful for pivoting. Row A[i] will go into row A[indexes[i]].

Example:

Reordering
[A,B,C,D,E] with indexes [0,4,2,3,1] yields 
[A,E,C,D,B]
In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[2], A[3]<--A[3], A[4]<--A[1].

Reordering
[A,B,C,D,E] with indexes [0,4,1,2,3] yields 
[A,E,B,C,D]
In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[1], A[3]<--A[2], A[4]<--A[3].

Parameters:
A - the matrix to permute.
indexes - the permutation indexes, must satisfy indexes.length==A.rows() && indexes[i] >= 0 && indexes[i] < A.rows();
work - the working storage, must satisfy work.length >= A.rows(); set work==null if you don't care about performance.
Returns:
the modified A (for convenience only).
Throws:
java.lang.IndexOutOfBoundsException - if indexes.length != A.rows().

pow

public DoubleMatrix2D pow(DoubleMatrix2D A,
                          int p)
Linear algebraic matrix power; B = Ak <==> B = A*A*...*A. Implementation: Based on logarithms of 2, memory usage minimized.

Parameters:
A - the source matrix; must be square; stays unaffected by this operation.
p - the exponent, can be any number.
Returns:
B, a newly constructed result matrix; storage-independent of A.
Throws:
java.lang.IllegalArgumentException - if !property().isSquare(A).

property

public Property property()
Returns the property object attached to this Algebra, defining tolerance.

Returns:
the Property object.
See Also:
setProperty(Property)

rank

public int rank(DoubleMatrix2D A)
Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.


setProperty

public void setProperty(Property property)
Attaches the given property object to this Algebra, defining tolerance.

Parameters:
the - Property object to be attached.
Throws:
java.lang.UnsupportedOperationException - if this==DEFAULT && property!=this.property() - The DEFAULT Algebra object is immutable.
java.lang.UnsupportedOperationException - if this==ZERO && property!=this.property() - The ZERO Algebra object is immutable.
See Also:
property

solve

public DoubleMatrix2D solve(DoubleMatrix2D A,
                            DoubleMatrix2D B)
Solves A*X = B.

Returns:
X; a new independent matrix; solution if A is square, least squares solution otherwise.

solveTranspose

public DoubleMatrix2D solveTranspose(DoubleMatrix2D A,
                                     DoubleMatrix2D B)
Solves X*A = B, which is also A'*X' = B'.

Returns:
X; a new independent matrix; solution if A is square, least squares solution otherwise.

subMatrix

public DoubleMatrix2D subMatrix(DoubleMatrix2D A,
                                int fromRow,
                                int toRow,
                                int fromColumn,
                                int toColumn)
Constructs and returns a new sub-range view which is the sub matrix A[fromRow..toRow,fromColumn..toColumn]. The returned matrix is backed by this matrix, so changes in the returned matrix are reflected in this matrix, and vice-versa. Use idioms like result = subMatrix(...).copy() to generate an independent sub matrix.

Parameters:
A - the source matrix.
fromRow - The index of the first row (inclusive).
toRow - The index of the last row (inclusive).
fromColumn - The index of the first column (inclusive).
toColumn - The index of the last column (inclusive).
Returns:
a new sub-range view.
Throws:
java.lang.IndexOutOfBoundsException - if fromColumn<0 || toColumn-fromColumn+1<0 || toColumn>=A.columns() || fromRow<0 || toRow-fromRow+1<0 || toRow>=A.rows()

toString

public java.lang.String toString(DoubleMatrix2D matrix)
Returns a String with (propertyName, propertyValue) pairs. Useful for debugging or to quickly get the rough picture. For example,
cond          : 14.073264490042144
det           : Illegal operation or error: Matrix must be square.
norm1         : 0.9620244354009628
norm2         : 3.0
normF         : 1.304841791648992
normInfinity  : 1.5406551198102534
rank          : 3
trace         : 0


toVerboseString

public java.lang.String toVerboseString(DoubleMatrix2D matrix)
Returns the results of toString(A) and additionally the results of all sorts of decompositions applied to the given matrix. Useful for debugging or to quickly get the rough picture. For example,
A = 3 x 3 matrix
249  66  68
104 214 108
144 146 293

cond         : 3.931600417472078
det          : 9638870.0
norm1        : 497.0
norm2        : 473.34508217011404
normF        : 516.873292016525
normInfinity : 583.0
rank         : 3
trace        : 756.0

density                      : 1.0
isDiagonal                   : false
isDiagonallyDominantByColumn : true
isDiagonallyDominantByRow    : true
isIdentity                   : false
isLowerBidiagonal            : false
isLowerTriangular            : false
isNonNegative                : true
isOrthogonal                 : false
isPositive                   : true
isSingular                   : false
isSkewSymmetric              : false
isSquare                     : true
isStrictlyLowerTriangular    : false
isStrictlyTriangular         : false
isStrictlyUpperTriangular    : false
isSymmetric                  : false
isTriangular                 : false
isTridiagonal                : false
isUnitTriangular             : false
isUpperBidiagonal            : false
isUpperTriangular            : false
isZero                       : false
lowerBandwidth               : 2
semiBandwidth                : 3
upperBandwidth               : 2

-----------------------------------------------------------------------------
LUDecompositionQuick(A) --> isNonSingular(A), det(A), pivot, L, U, inverse(A)
-----------------------------------------------------------------------------
isNonSingular = true
det = 9638870.0
pivot = [0, 1, 2]

L = 3 x 3 matrix
1        0       0
0.417671 1       0
0.578313 0.57839 1

U = 3 x 3 matrix
249  66         68       
  0 186.433735  79.598394
  0   0        207.635819

inverse(A) = 3 x 3 matrix
 0.004869 -0.000976 -0.00077 
-0.001548  0.006553 -0.002056
-0.001622 -0.002786  0.004816

-----------------------------------------------------------------
QRDecomposition(A) --> hasFullRank(A), H, Q, R, pseudo inverse(A)
-----------------------------------------------------------------
hasFullRank = true

H = 3 x 3 matrix
1.814086 0        0
0.34002  1.903675 0
0.470797 0.428218 2

Q = 3 x 3 matrix
-0.814086  0.508871  0.279845
-0.34002  -0.808296  0.48067 
-0.470797 -0.296154 -0.831049

R = 3 x 3 matrix
-305.864349 -195.230337 -230.023539
   0        -182.628353  467.703164
   0           0        -309.13388 

pseudo inverse(A) = 3 x 3 matrix
 0.006601  0.001998 -0.005912
-0.005105  0.000444  0.008506
-0.000905 -0.001555  0.002688

--------------------------------------------------------------------------
CholeskyDecomposition(A) --> isSymmetricPositiveDefinite(A), L, inverse(A)
--------------------------------------------------------------------------
isSymmetricPositiveDefinite = false

L = 3 x 3 matrix
15.779734  0         0       
 6.590732 13.059948  0       
 9.125629  6.573948 12.903724

inverse(A) = Illegal operation or error: Matrix is not symmetric positive definite.

---------------------------------------------------------------------
EigenvalueDecomposition(A) --> D, V, realEigenvalues, imagEigenvalues
---------------------------------------------------------------------
realEigenvalues = 1 x 3 matrix
462.796507 172.382058 120.821435
imagEigenvalues = 1 x 3 matrix
0 0 0

D = 3 x 3 matrix
462.796507   0          0       
  0        172.382058   0       
  0          0        120.821435

V = 3 x 3 matrix
-0.398877 -0.778282  0.094294
-0.500327  0.217793 -0.806319
-0.768485  0.66553   0.604862

---------------------------------------------------------------------
SingularValueDecomposition(A) --> cond(A), rank(A), norm2(A), U, S, V
---------------------------------------------------------------------
cond = 3.931600417472078
rank = 3
norm2 = 473.34508217011404

U = 3 x 3 matrix
0.46657  -0.877519  0.110777
0.50486   0.161382 -0.847982
0.726243  0.45157   0.51832 

S = 3 x 3 matrix
473.345082   0          0       
  0        169.137441   0       
  0          0        120.395013

V = 3 x 3 matrix
0.577296 -0.808174  0.116546
0.517308  0.251562 -0.817991
0.631761  0.532513  0.563301


trace

public double trace(DoubleMatrix2D A)
Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).


transpose

public DoubleMatrix2D transpose(DoubleMatrix2D A)
Constructs and returns a new view which is the transposition of the given matrix A. Equivalent to A.viewDice(). This is a zero-copy transposition, taking O(1), i.e. constant time. The returned view is backed by this matrix, so changes in the returned view are reflected in this matrix, and vice-versa. Use idioms like result = transpose(A).copy() to generate an independent matrix.

Example:

2 x 3 matrix:
1, 2, 3
4, 5, 6
transpose ==> 3 x 2 matrix:
1, 4
2, 5
3, 6
transpose ==> 2 x 3 matrix:
1, 2, 3
4, 5, 6

Returns:
a new transposed view.

trapezoidalLower

protected DoubleMatrix2D trapezoidalLower(DoubleMatrix2D A)
Modifies the matrix to be a lower trapezoidal matrix.

Returns:
A (for convenience only).
See Also:
#triangulateLower(DoubleMatrix2D)