Parallel Colt 0.7.2

cern.colt.matrix.tdouble.algo
Class DoubleAlgebra

java.lang.Object
  extended by cern.colt.PersistentObject
      extended by cern.colt.matrix.tdouble.algo.DoubleAlgebra
All Implemented Interfaces:
Serializable, Cloneable

public class DoubleAlgebra
extends PersistentObject

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

Version:
1.0, 09/24/99, 1.1 10/20/2007
Author:
wolfgang.hoschek@cern.ch, Piotr Wendykier (piotr.wendykier@gmail.com)
See Also:
Serialized Form

Field Summary
static DoubleAlgebra DEFAULT
          A default Algebra object; has DoubleProperty.DEFAULT attached for tolerance.
static DoubleAlgebra ZERO
          A default Algebra object; has DoubleProperty.ZERO attached for tolerance.
 
Constructor Summary
DoubleAlgebra()
          Constructs a new instance with an equality tolerance given by Property.DEFAULT.tolerance().
DoubleAlgebra(double tolerance)
          Constructs a new instance with the given equality tolerance.
 
Method Summary
 DoubleMatrix1D backwardSolve(DoubleMatrix2D U, DoubleMatrix1D b)
           
 DoubleCholeskyDecomposition chol(DoubleMatrix2D matrix)
          Constructs and returns the cholesky-decomposition of the given matrix.
 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.
 DoubleEigenvalueDecomposition eig(DoubleMatrix2D matrix)
          Constructs and returns the Eigenvalue-decomposition of the given matrix.
 DoubleMatrix1D forwardSolve(DoubleMatrix2D L, DoubleMatrix1D b)
           
static double hypot(double a, double b)
          Returns sqrt(a^2 + b^2) without under/overflow.
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.
 DComplexMatrix1D kron(DComplexMatrix1D x, DComplexMatrix1D y)
          Computes the Kronecker product of two complex matrices.
 DoubleMatrix1D kron(double[] A, double[] B)
          Computes the Kronecker product of two arrays.
 DoubleMatrix1D kron(DoubleMatrix1D x, DoubleMatrix1D y)
          Computes the Kronecker product of two real matrices.
 DoubleLUDecomposition lu(DoubleMatrix2D matrix)
          Constructs and returns the LU-decomposition of the given matrix.
 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 norm(DoubleMatrix1D x, Norm type)
           
 double norm(DoubleMatrix2D A, Norm type)
           
 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 Sqrt(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(DComplexMatrix2D A)
          Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]2)).
 double normF(DoubleMatrix1D A)
          Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i]2)).
 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.
 DoubleProperty property()
          Returns the property object attached to this Algebra, defining tolerance.
 DoubleQRDecomposition qr(DoubleMatrix2D matrix)
          Constructs and returns the QR-decomposition of the given matrix.
 int rank(DoubleMatrix2D A)
          Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.
 void setProperty(DoubleProperty property)
          Attaches the given property object to this Algebra, defining tolerance.
 DoubleMatrix1D solve(DoubleMatrix2D A, DoubleMatrix1D b)
          Solves A*x = b.
 DoubleMatrix2D solve(DoubleMatrix2D A, DoubleMatrix2D B)
           
 DoubleMatrix2D solveTranspose(DoubleMatrix2D A, DoubleMatrix2D B)
          Solves X*A = B, which is also A'*X' = B'.
 DoubleMatrix2D subMatrix(DoubleMatrix2D A, int[] rowIndexes, int columnFrom, int columnTo)
          Copies the columns of the indicated rows into a new sub matrix.
 DoubleMatrix2D subMatrix(DoubleMatrix2D A, int rowFrom, int rowTo, int[] columnIndexes)
          Copies the rows of the indicated columns into a new sub matrix.
 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].
 DoubleSingularValueDecomposition svd(DoubleMatrix2D matrix)
          Constructs and returns the SingularValue-decomposition of the given matrix.
 DoubleSingularValueDecompositionDC svdDC(DoubleMatrix2D matrix)
          Constructs and returns the SingularValue-decomposition of the given matrix.
 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.
 DoubleMatrix2D trapezoidalLower(DoubleMatrix2D A)
          Modifies the matrix to be a lower trapezoidal matrix.
 double vectorNorm2(DoubleMatrix2D X)
          Returns the two-norm (aka euclidean norm) of vector X.vectorize();
 double vectorNorm2(DoubleMatrix3D X)
          Returns the two-norm (aka euclidean norm) of vector X.vectorize();
 DoubleMatrix2D xmultOuter(DoubleMatrix1D x, DoubleMatrix1D y)
          Outer product of two vectors; Returns a matrix with A[i,j] = x[i] * y[j].
 DoubleMatrix2D xpowSlow(DoubleMatrix2D A, int k)
          Linear algebraic matrix power; B = Ak <==> B = A*A*...*A.
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Field Detail

DEFAULT

public static final DoubleAlgebra DEFAULT
A default Algebra object; has DoubleProperty.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 DoubleAlgebra ZERO
A default Algebra object; has DoubleProperty.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

DoubleAlgebra

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


DoubleAlgebra

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

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

chol

public DoubleCholeskyDecomposition chol(DoubleMatrix2D 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 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.

eig

public DoubleEigenvalueDecomposition eig(DoubleMatrix2D matrix)
Constructs and returns the Eigenvalue-decomposition of the given matrix.


hypot

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


hypotFunction

public 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.

lu

public DoubleLUDecomposition lu(DoubleMatrix2D matrix)
Constructs and returns the LU-decomposition of the given matrix.


kron

public DComplexMatrix1D kron(DComplexMatrix1D x,
                             DComplexMatrix1D y)
Computes the Kronecker product of two complex matrices.

Parameters:
x -
y -
Returns:
the Kronecker product of two complex matrices

kron

public DoubleMatrix1D kron(double[] A,
                           double[] B)
Computes the Kronecker product of two arrays.

Parameters:
A -
B -
Returns:
the Kronecker product of two arrays

kron

public DoubleMatrix1D kron(DoubleMatrix1D x,
                           DoubleMatrix1D y)
Computes the Kronecker product of two real matrices.

Parameters:
x -
y -
Returns:
the Kronecker product of two real matrices

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 Sqrt(mult(x,x)).


vectorNorm2

public double vectorNorm2(DoubleMatrix2D X)
Returns the two-norm (aka euclidean norm) of vector X.vectorize();


vectorNorm2

public double vectorNorm2(DoubleMatrix3D X)
Returns the two-norm (aka euclidean norm) of vector X.vectorize();


norm

public double norm(DoubleMatrix2D A,
                   Norm type)

norm

public double norm(DoubleMatrix1D x,
                   Norm type)

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)).


normF

public double normF(DoubleMatrix1D A)
Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i]2)).


normF

public double normF(DComplexMatrix2D 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. 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 DoubleProperty property()
Returns the property object attached to this Algebra, defining tolerance.

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

qr

public DoubleQRDecomposition qr(DoubleMatrix2D matrix)
Constructs and returns the QR-decomposition of the given matrix.


rank

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


setProperty

public void setProperty(DoubleProperty 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 DoubleMatrix1D backwardSolve(DoubleMatrix2D U,
                                    DoubleMatrix1D b)

forwardSolve

public DoubleMatrix1D forwardSolve(DoubleMatrix2D L,
                                   DoubleMatrix1D b)

solve

public DoubleMatrix1D solve(DoubleMatrix2D A,
                            DoubleMatrix1D b)
Solves A*x = b.

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

solve

public DoubleMatrix2D solve(DoubleMatrix2D A,
                            DoubleMatrix2D 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[] 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 DoubleMatrix2D subMatrix(DoubleMatrix2D 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 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()

svd

public DoubleSingularValueDecomposition svd(DoubleMatrix2D matrix)
Constructs and returns the SingularValue-decomposition of the given matrix.


svdDC

public DoubleSingularValueDecompositionDC svdDC(DoubleMatrix2D matrix)
Constructs and returns the SingularValue-decomposition of the given matrix. This is a divide-and-conquer version.


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

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

Returns:
A (for convenience only).

xmultOuter

public DoubleMatrix2D xmultOuter(DoubleMatrix1D x,
                                 DoubleMatrix1D 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 DoubleMatrix2D xpowSlow(DoubleMatrix2D A,
                               int k)
Linear algebraic matrix power; B = Ak <==> 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).

Parallel Colt 0.7.2

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