Algebra (Blazegraph Database Platform 2.1.5 API)

java.lang.Object
- cern.colt.PersistentObject
- - cern.colt.matrix.linalg.Algebra

All Implemented Interfaces:

Serializable, Cloneable
```
public class Algebra
extends PersistentObject
```
Linear algebraic matrix operations operating on DoubleMatrix2D; concentrates most functionality of this package.

Version:

1.0, 09/24/99

Author:

wolfgang.hoschek@cern.ch

See Also:
Serialized Form

Field Summary

Fields
Modifier and Type	Field and Description
`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

Constructors
Constructor and Description
`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

Methods
Modifier and Type	Method and Description
`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 = A^k <==> B = AA...*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 XA = 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]`.
`String`	`toString(DoubleMatrix2D matrix)` Returns a String with (propertyName, propertyValue) pairs.
`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 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:

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:

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:

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:

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:

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:

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:

IndexOutOfBoundsException - if indexes.length != A.rows().

pow
```
public DoubleMatrix2D pow(DoubleMatrix2D A,
                 int p)
```
Linear algebraic matrix power; B = A^k <==> B = A*A*...*A.
- p >= 1: B = A*A*...*A.
- p == 0: B = identity matrix.
- p < 0: B = pow(inverse(A),-p).
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:

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:

UnsupportedOperationException - if this==DEFAULT && property!=this.property() - The DEFAULT Algebra object is immutable.

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:

IndexOutOfBoundsException - if fromColumn<0 || toColumn-fromColumn+1<0 || toColumn>=A.columns() || fromRow<0 || toRow-fromRow+1<0 || toRow>=A.rows()

toString

public 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 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)

Class Algebra

Field Summary

Fields inherited from class cern.colt.PersistentObject

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

DEFAULT

ZERO

property

Constructor Detail

Algebra

Algebra

Method Detail

clone

cond

det

hypot

hypotFunction

inverse

mult

mult

mult

multOuter

norm1

norm1

norm2

norm2

normF

normInfinity

normInfinity

permute

permute

permuteColumns

permuteRows

pow

property

rank

setProperty

solve

solveTranspose

subMatrix

toString

toVerboseString

trace

transpose

trapezoidalLower