DenseFloatAlgebra (Parallel Colt 0.10.1

java.lang.Object
- cern.colt.PersistentObject
- - cern.colt.matrix.tfloat.algo.DenseFloatAlgebra

All Implemented Interfaces:

Serializable, Cloneable
```
public class DenseFloatAlgebra
extends PersistentObject
```
Linear algebraic matrix operations operating on dense matrices.

Version:

1.0, 09/24/99

Author:

wolfgang.hoschek@cern.ch, Piotr Wendykier (piotr.wendykier@gmail.com)

See Also:
Serialized Form

Field Summary

Fields
Modifier and Type	Field and Description
`static DenseFloatAlgebra`	`DEFAULT` A default Algebra object; has `FloatProperty.DEFAULT` attached for tolerance.
`static DenseFloatAlgebra`	`ZERO` A default Algebra object; has `FloatProperty.ZERO` attached for tolerance.

Constructor Summary

Constructors
Constructor and Description
`DenseFloatAlgebra()` Constructs a new instance with an equality tolerance given by `Property.DEFAULT.tolerance()`.
`DenseFloatAlgebra(float tolerance)` Constructs a new instance with the given equality tolerance.

Method Summary

Methods
Modifier and Type	Method and Description
`FloatMatrix1D`	`backwardSolve(FloatMatrix2D U, FloatMatrix1D b)` Solves the upper triangular system U*x=b;
`DenseFloatCholeskyDecomposition`	`chol(FloatMatrix2D matrix)` Constructs and returns the cholesky-decomposition of the given matrix.
`Object`	`clone()` Returns a copy of the receiver.
`float`	`cond(FloatMatrix2D A)` Returns the condition of matrix `A`, which is the ratio of largest to smallest singular value.
`float`	`det(FloatMatrix2D A)` Returns the determinant of matrix `A`.
`DenseFloatEigenvalueDecomposition`	`eig(FloatMatrix2D matrix)` Constructs and returns the Eigenvalue-decomposition of the given matrix.
`FloatMatrix1D`	`forwardSolve(FloatMatrix2D L, FloatMatrix1D b)` Solves the lower triangular system U*x=b;
`static float`	`hypot(float a, float b)` Returns sqrt(a^2 + b^2) without under/overflow.
`static FloatFloatFunction`	`hypotFunction()` Returns sqrt(a^2 + b^2) without under/overflow.
`FloatMatrix2D`	`inverse(FloatMatrix2D A)` Returns the inverse or pseudo-inverse of matrix `A`.
`FloatMatrix1D`	`kron(FloatMatrix1D x, FloatMatrix1D y)` Computes the Kronecker product of two real matrices.
`FloatMatrix2D`	`kron(FloatMatrix2D X, FloatMatrix2D Y)` Computes the Kronecker product of two real matrices.
`DenseFloatLUDecomposition`	`lu(FloatMatrix2D matrix)` Constructs and returns the LU-decomposition of the given matrix.
`float`	`mult(FloatMatrix1D x, FloatMatrix1D y)` Inner product of two vectors; `Sum(x[i] * y[i])`.
`FloatMatrix1D`	`mult(FloatMatrix2D A, FloatMatrix1D y)` Linear algebraic matrix-vector multiplication; `z = A * y`.
`FloatMatrix2D`	`mult(FloatMatrix2D A, FloatMatrix2D B)` Linear algebraic matrix-matrix multiplication; `C = A x B`.
`FloatMatrix2D`	`multOuter(FloatMatrix1D x, FloatMatrix1D y, FloatMatrix2D A)` Outer product of two vectors; Sets `A[i,j] = x[i] * y[j]`.
`float`	`norm(FloatMatrix1D x, Norm type)`
`float`	`norm(FloatMatrix2D A, Norm type)`
`float`	`norm1(FloatMatrix1D x)` Returns the one-norm of vector `x`, which is `Sum(abs(x[i]))`.
`float`	`norm1(FloatMatrix2D A)` Returns the one-norm of matrix `A`, which is the maximum absolute column sum.
`float`	`norm2(FloatMatrix1D x)` Returns the two-norm (aka euclidean norm) of vector `x`; equivalent to `Sqrt(mult(x,x))`.
`float`	`norm2(FloatMatrix2D A)` Returns the two-norm of matrix `A`, which is the maximum singular value; obtained from SVD.
`float`	`normF(FloatMatrix1D A)` Returns the Frobenius norm of matrix `A`, which is `Sqrt(Sum(A[i]²))`.
`float`	`normF(FloatMatrix2D A)` Returns the Frobenius norm of matrix `A`, which is `Sqrt(Sum(A[i,j]²))`.
`float`	`normInfinity(FloatMatrix1D x)` Returns the infinity norm of vector `x`, which is `Max(abs(x[i]))`.
`float`	`normInfinity(FloatMatrix2D A)` Returns the infinity norm of matrix `A`, which is the maximum absolute row sum.
`FloatMatrix1D`	`permute(FloatMatrix1D A, int[] indexes, float[] work)` Modifies the given vector `A` such that it is permuted as specified; Useful for pivoting.
`FloatMatrix2D`	`permute(FloatMatrix2D A, int[] rowIndexes, int[] columnIndexes)` Constructs and returns a new row and column permuted selection view of matrix `A`; equivalent to `FloatMatrix2D.viewSelection(int[],int[])`.
`FloatMatrix2D`	`permuteColumns(FloatMatrix2D A, int[] indexes, int[] work)` Modifies the given matrix `A` such that it's columns are permuted as specified; Useful for pivoting.
`FloatMatrix2D`	`permuteRows(FloatMatrix2D A, int[] indexes, int[] work)` Modifies the given matrix `A` such that it's rows are permuted as specified; Useful for pivoting.
`FloatMatrix2D`	`pow(FloatMatrix2D A, int p)` Linear algebraic matrix power; `B = A^k <==> B = AA...*A`.
`FloatProperty`	`property()` Returns the property object attached to this Algebra, defining tolerance.
`DenseFloatQRDecomposition`	`qr(FloatMatrix2D matrix)` Constructs and returns the QR-decomposition of the given matrix.
`int`	`rank(FloatMatrix2D A)` Returns the effective numerical rank of matrix `A`, obtained from Singular Value Decomposition.
`void`	`setProperty(FloatProperty property)` Attaches the given property object to this Algebra, defining tolerance.
`FloatMatrix1D`	`solve(FloatMatrix2D A, FloatMatrix1D b)` Solves A*x = b.
`FloatMatrix2D`	`solve(FloatMatrix2D A, FloatMatrix2D B)` Solves A*X = B.
`FloatMatrix2D`	`solveTranspose(FloatMatrix2D A, FloatMatrix2D B)` Solves XA = B, which is also A'X' = B'.
`FloatMatrix2D`	`subMatrix(FloatMatrix2D A, int[] rowIndexes, int columnFrom, int columnTo)` Copies the columns of the indicated rows into a new sub matrix.
`FloatMatrix2D`	`subMatrix(FloatMatrix2D A, int rowFrom, int rowTo, int[] columnIndexes)` Copies the rows of the indicated columns into a new sub matrix.
`FloatMatrix2D`	`subMatrix(FloatMatrix2D 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]`.
`DenseFloatSingularValueDecomposition`	`svd(FloatMatrix2D matrix)` Constructs and returns the SingularValue-decomposition of the given matrix.
`String`	`toString(FloatMatrix2D matrix)` Returns a String with (propertyName, propertyValue) pairs.
`String`	`toVerboseString(FloatMatrix2D matrix)` Returns the results of `toString(A)` and additionally the results of all sorts of decompositions applied to the given matrix.
`float`	`trace(FloatMatrix2D A)` Returns the sum of the diagonal elements of matrix `A`; `Sum(A[i,i])`.
`FloatMatrix2D`	`transpose(FloatMatrix2D A)` Constructs and returns a new view which is the transposition of the given matrix `A`.
`FloatMatrix2D`	`trapezoidalLower(FloatMatrix2D A)` Modifies the matrix to be a lower trapezoidal matrix.
`float`	`vectorNorm2(FloatMatrix2D X)` Returns the two-norm (aka euclidean norm) of vector `X.vectorize()`;
`float`	`vectorNorm2(FloatMatrix3D X)` Returns the two-norm (aka euclidean norm) of vector `X.vectorize()`;
`FloatMatrix2D`	`xmultOuter(FloatMatrix1D x, FloatMatrix1D y)` Outer product of two vectors; Returns a matrix with `A[i,j] = x[i] * y[j]`.
`FloatMatrix2D`	`xpowSlow(FloatMatrix2D A, int k)` Linear algebraic matrix power; `B = A^k <==> B = AA...*A`.

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Field Detail
  - DEFAULT
```
public static final DenseFloatAlgebra DEFAULT
```
    A default Algebra object; has FloatProperty.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 DenseFloatAlgebra ZERO
```
    A default Algebra object; has FloatProperty.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
  - DenseFloatAlgebra
```
public DenseFloatAlgebra()
```
    Constructs a new instance with an equality tolerance given by Property.DEFAULT.tolerance().
  - DenseFloatAlgebra
```
public DenseFloatAlgebra(float tolerance)
```
    Constructs a new instance with the given equality tolerance.
    
    Parameters:
    tolerance - the tolerance to be used for equality operations.
- Method Detail
  - chol
```
public DenseFloatCholeskyDecomposition chol(FloatMatrix2D matrix)
```
    Constructs and returns the cholesky-decomposition of the given matrix.
  - 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 float cond(FloatMatrix2D A)
```
    Returns the condition of matrix A, which is the ratio of largest to smallest singular value.
  - det
```
public float det(FloatMatrix2D A)
```
    Returns the determinant of matrix A.
    
    Returns:
    the determinant.
  - eig
```
public DenseFloatEigenvalueDecomposition eig(FloatMatrix2D matrix)
```
    Constructs and returns the Eigenvalue-decomposition of the given matrix.
  - hypot
```
public static float hypot(float a,
          float b)
```
    Returns sqrt(a^2 + b^2) without under/overflow.
  - hypotFunction
```
public static FloatFloatFunction hypotFunction()
```
    Returns sqrt(a^2 + b^2) without under/overflow.
  - inverse
```
public FloatMatrix2D inverse(FloatMatrix2D A)
```
    Returns the inverse or pseudo-inverse of matrix A.
    
    Returns:
    a new independent matrix; inverse(matrix) if the matrix is square, pseudoinverse otherwise.
  - lu
```
public DenseFloatLUDecomposition lu(FloatMatrix2D matrix)
```
    Constructs and returns the LU-decomposition of the given matrix.
  - kron
```
public FloatMatrix1D kron(FloatMatrix1D x,
                 FloatMatrix1D y)
```
    Computes the Kronecker product of two real matrices.
    
    Parameters:
    x -
    y -
    
    Returns:
    the Kronecker product of two real matrices
  - kron
```
public FloatMatrix2D kron(FloatMatrix2D X,
                 FloatMatrix2D Y)
```
    Computes the Kronecker product of two real matrices.
    
    Parameters:
    X -
    Y -
    
    Returns:
    the Kronecker product of two real matrices
  - mult
```
public float mult(FloatMatrix1D x,
         FloatMatrix1D 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 FloatMatrix1D mult(FloatMatrix2D A,
                 FloatMatrix1D 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 FloatMatrix2D mult(FloatMatrix2D A,
                 FloatMatrix2D 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 FloatMatrix2D multOuter(FloatMatrix1D x,
                      FloatMatrix1D y,
                      FloatMatrix2D 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 float norm1(FloatMatrix1D x)
```
    Returns the one-norm of vector x, which is Sum(abs(x[i])).
  - norm1
```
public float norm1(FloatMatrix2D A)
```
    Returns the one-norm of matrix A, which is the maximum absolute column sum.
  - norm2
```
public float norm2(FloatMatrix1D x)
```
    Returns the two-norm (aka euclidean norm) of vector x; equivalent to Sqrt(mult(x,x)).
  - vectorNorm2
```
public float vectorNorm2(FloatMatrix2D X)
```
    Returns the two-norm (aka euclidean norm) of vector X.vectorize();
  - vectorNorm2
```
public float vectorNorm2(FloatMatrix3D X)
```
    Returns the two-norm (aka euclidean norm) of vector X.vectorize();
  - norm
```
public float norm(FloatMatrix2D A,
         Norm type)
```
  - norm
```
public float norm(FloatMatrix1D x,
         Norm type)
```
  - norm2
```
public float norm2(FloatMatrix2D A)
```
    Returns the two-norm of matrix A, which is the maximum singular value; obtained from SVD.
  - normF
```
public float normF(FloatMatrix2D A)
```
    Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]²)).
  - normF
```
public float normF(FloatMatrix1D A)
```
    Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i]²)).
  - normInfinity
```
public float normInfinity(FloatMatrix1D x)
```
    Returns the infinity norm of vector x, which is Max(abs(x[i])).
  - normInfinity
```
public float normInfinity(FloatMatrix2D A)
```
    Returns the infinity norm of matrix A, which is the maximum absolute row sum.
  - permute
```
public FloatMatrix1D permute(FloatMatrix1D A,
                    int[] indexes,
                    float[] 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 FloatMatrix2D permute(FloatMatrix2D A,
                    int[] rowIndexes,
                    int[] columnIndexes)
```
    Constructs and returns a new row and column permuted selection view of matrix A; equivalent to FloatMatrix2D.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 FloatMatrix2D permuteColumns(FloatMatrix2D 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 FloatMatrix2D permuteRows(FloatMatrix2D 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 FloatMatrix2D pow(FloatMatrix2D 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 FloatProperty property()
```
    Returns the property object attached to this Algebra, defining tolerance.
    
    Returns:
    the Property object.
    See Also:
    setProperty(FloatProperty)
  - qr
```
public DenseFloatQRDecomposition qr(FloatMatrix2D matrix)
```
    Constructs and returns the QR-decomposition of the given matrix.
  - rank
```
public int rank(FloatMatrix2D A)
```
    Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.
  - setProperty
```
public void setProperty(FloatProperty property)
```
    Attaches the given property object to this Algebra, defining tolerance.
    
    Parameters:
    property - 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
  - backwardSolve
```
public FloatMatrix1D backwardSolve(FloatMatrix2D U,
                          FloatMatrix1D b)
```
    Solves the upper triangular system U*x=b;
    
    Parameters:
    U - upper triangular matrix
    b - right-hand side
    
    Returns:
    x, a new independent matrix;
  - forwardSolve
```
public FloatMatrix1D forwardSolve(FloatMatrix2D L,
                         FloatMatrix1D b)
```
    Solves the lower triangular system U*x=b;
    
    Parameters:
    L - lower triangular matrix
    b - right-hand side
    
    Returns:
    x, a new independent matrix;
  - solve
```
public FloatMatrix1D solve(FloatMatrix2D A,
                  FloatMatrix1D b)
```
    Solves A*x = b.
    
    Returns:
    x; a new independent matrix; solution if A is square, least squares solution otherwise.
  - solve
```
public FloatMatrix2D solve(FloatMatrix2D A,
                  FloatMatrix2D B)
```
    Solves A*X = B.
    
    Returns:
    X; a new independent matrix; solution if A is square, least squares solution otherwise.
  - solveTranspose
```
public FloatMatrix2D solveTranspose(FloatMatrix2D A,
                           FloatMatrix2D 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 FloatMatrix2D subMatrix(FloatMatrix2D A,
                      int[] rowIndexes,
                      int columnFrom,
                      int columnTo)
```
    Copies the columns of the indicated rows into a new sub matrix. sub[0..rowIndexes.length-1,0..columnTo-columnFrom] = A[rowIndexes(:),columnFrom..columnTo] ; The returned matrix is not backed by this matrix, so changes in the returned matrix are not reflected in this matrix, and vice-versa.
    
    Parameters:
    A - the source matrix to copy from.
    rowIndexes - the indexes of the rows to copy. May be unsorted.
    columnFrom - the index of the first column to copy (inclusive).
    columnTo - the index of the last column to copy (inclusive).
    
    Returns:
    a new sub matrix; with sub.rows()==rowIndexes.length; sub.columns()==columnTo-columnFrom+1 .
    
    Throws:
    
    IndexOutOfBoundsException - if columnFrom<0 || columnTo-columnFrom+1<0 || columnTo+1>matrix.columns() || for any row=rowIndexes[i]: row < 0 || row >= matrix.rows() .
  - subMatrix
```
public FloatMatrix2D subMatrix(FloatMatrix2D A,
                      int rowFrom,
                      int rowTo,
                      int[] columnIndexes)
```
    Copies the rows of the indicated columns into a new sub matrix. sub[0..rowTo-rowFrom,0..columnIndexes.length-1] = A[rowFrom..rowTo,columnIndexes(:)] ; The returned matrix is not backed by this matrix, so changes in the returned matrix are not reflected in this matrix, and vice-versa.
    
    Parameters:
    A - the source matrix to copy from.
    rowFrom - the index of the first row to copy (inclusive).
    rowTo - the index of the last row to copy (inclusive).
    columnIndexes - the indexes of the columns to copy. May be unsorted.
    
    Returns:
    a new sub matrix; with sub.rows()==rowTo-rowFrom+1; sub.columns()==columnIndexes.length .
    
    Throws:
    
    IndexOutOfBoundsException - if rowFrom<0 || rowTo-rowFrom+1<0 || rowTo+1>matrix.rows() || for any col=columnIndexes[i]: col < 0 || col >= matrix.columns() .
  - subMatrix
```
public FloatMatrix2D subMatrix(FloatMatrix2D 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()
  - svd
```
public DenseFloatSingularValueDecomposition svd(FloatMatrix2D matrix)
```
    Constructs and returns the SingularValue-decomposition of the given matrix.
  - toString
```
public String toString(FloatMatrix2D 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(FloatMatrix2D 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 float trace(FloatMatrix2D A)
```
    Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).
  - transpose
```
public FloatMatrix2D transpose(FloatMatrix2D 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
```
public FloatMatrix2D trapezoidalLower(FloatMatrix2D A)
```
    Modifies the matrix to be a lower trapezoidal matrix.
    
    Returns:
    A (for convenience only).
  - xmultOuter
```
public FloatMatrix2D xmultOuter(FloatMatrix1D x,
                       FloatMatrix1D y)
```
    Outer product of two vectors; Returns a matrix with A[i,j] = x[i] * y[j].
    
    Parameters:
    x - the first source vector.
    y - the second source vector.
    
    Returns:
    the outer product A.
  - xpowSlow
```
public FloatMatrix2D xpowSlow(FloatMatrix2D A,
                     int k)
```
    Linear algebraic matrix power; B = A^k <==> B = A*A*...*A.
    
    Parameters:
    A - the source matrix; must be square.
    k - the exponent, can be any number.
    
    Returns:
    a new result matrix.
    
    Throws:
    
    IllegalArgumentException - if !Testing.isSquare(A).

Class DenseFloatAlgebra

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

DEFAULT

ZERO

Constructor Detail

DenseFloatAlgebra

DenseFloatAlgebra

Method Detail

chol

clone

cond

det

eig

hypot

hypotFunction

inverse

lu

kron

kron

mult

mult

mult

multOuter

norm1

norm1

norm2

vectorNorm2

vectorNorm2

norm

norm

norm2

normF

normF

normInfinity

normInfinity

permute

permute

permuteColumns

permuteRows

pow

property

qr

rank

setProperty

backwardSolve

forwardSolve

solve

solve

solveTranspose

subMatrix

subMatrix

subMatrix

svd

toString

toVerboseString

trace

transpose

trapezoidalLower

xmultOuter

xpowSlow