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## cern.colt.matrix.linalg
Class Algebra

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

All Implemented Interfaces:
  Cloneable, Serializable

public class Algebra
extends PersistentObject

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

Version:
  1.0, 09/24/99
See Also:
  Serialized Form

Field Summary
`static Algebra` | `DEFAULT` A default Algebra object; has `Property.DEFAULT` attached for tolerance.
`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
`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`.
`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]`.
`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`.

Methods inherited from class java.lang.Object
`equals, 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.

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.

### 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 = Ak <==> 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.

  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.

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