Package cern.colt.matrix.linalg
Class Algebra
- java.lang.Object
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- cern.colt.PersistentObject
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- cern.colt.matrix.linalg.Algebra
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- All Implemented Interfaces:
java.io.Serializable
,java.lang.Cloneable
public class Algebra extends PersistentObject
Linear algebraic matrix operations operating onDoubleMatrix2D
; concentrates most functionality of this package.- Version:
- 1.0, 09/24/99
- See Also:
- Serialized Form
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Field Summary
Fields Modifier and Type Field Description static Algebra
DEFAULT
A default Algebra object; hasProperty.DEFAULT
attached for tolerance.protected Property
property
The property object attached to this instance.static Algebra
ZERO
A default Algebra object; hasProperty.ZERO
attached for tolerance.-
Fields inherited from class cern.colt.PersistentObject
serialVersionUID
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Method Summary
All Methods Static Methods Instance Methods Concrete Methods Modifier and Type Method Description private CholeskyDecomposition
chol(DoubleMatrix2D matrix)
Constructs and returns the cholesky-decomposition of the given matrix.java.lang.Object
clone()
Returns a copy of the receiver.double
cond(DoubleMatrix2D A)
Returns the condition of matrix A, which is the ratio of largest to smallest singular value.double
det(DoubleMatrix2D A)
Returns the determinant of matrix A.private EigenvalueDecomposition
eig(DoubleMatrix2D matrix)
Constructs and returns the Eigenvalue-decomposition of the given matrix.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.private LUDecomposition
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
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 toDoubleMatrix2D.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.private QRDecomposition
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(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'.private DoubleMatrix2D
subMatrix(DoubleMatrix2D A, int[] rowIndexes, int columnFrom, int columnTo)
Copies the columns of the indicated rows into a new sub matrix.private 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].private SingularValueDecomposition
svd(DoubleMatrix2D matrix)
Constructs and returns the SingularValue-decomposition of the given matrix.java.lang.String
toString(DoubleMatrix2D matrix)
Returns a String with (propertyName, propertyValue) pairs.java.lang.String
toVerboseString(DoubleMatrix2D matrix)
Returns the results of toString(A) and additionally the results of all sorts of decompositions applied to the given matrix.double
trace(DoubleMatrix2D A)
Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).DoubleMatrix2D
transpose(DoubleMatrix2D A)
Constructs and returns a new view which is the transposition of the given matrix A.protected DoubleMatrix2D
trapezoidalLower(DoubleMatrix2D A)
Modifies the matrix to be a lower trapezoidal matrix.private DoubleMatrix2D
xmultOuter(DoubleMatrix1D x, DoubleMatrix1D y)
Outer product of two vectors; Returns a matrix with A[i,j] = x[i] * y[j].private DoubleMatrix2D
xpowSlow(DoubleMatrix2D A, int k)
Linear algebraic matrix power; B = Ak <==> B = A*A*...*A.
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Field Detail
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DEFAULT
public static final Algebra DEFAULT
A default Algebra object; hasProperty.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.
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ZERO
public static final Algebra ZERO
A default Algebra object; hasProperty.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.
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property
protected Property property
The property object attached to this instance.
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Constructor Detail
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Algebra
public Algebra()
Constructs a new instance with an equality tolerance given by Property.DEFAULT.tolerance().
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Algebra
public Algebra(double tolerance)
Constructs a new instance with the given equality tolerance.- Parameters:
tolerance
- the tolerance to be used for equality operations.
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Method Detail
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chol
private CholeskyDecomposition chol(DoubleMatrix2D matrix)
Constructs and returns the cholesky-decomposition of the given matrix.
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clone
public java.lang.Object clone()
Returns a copy of the receiver. The attached property object is also copied. Hence, the property object of the copy is mutable.- Overrides:
clone
in classPersistentObject
- Returns:
- a copy of the receiver.
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cond
public double cond(DoubleMatrix2D A)
Returns the condition of matrix A, which is the ratio of largest to smallest singular value.
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det
public double det(DoubleMatrix2D A)
Returns the determinant of matrix A.- Returns:
- the determinant.
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eig
private EigenvalueDecomposition eig(DoubleMatrix2D matrix)
Constructs and returns the Eigenvalue-decomposition of the given matrix.
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hypot
protected static double hypot(double a, double b)
Returns sqrt(a^2 + b^2) without under/overflow.
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hypotFunction
protected static DoubleDoubleFunction hypotFunction()
Returns sqrt(a^2 + b^2) without under/overflow.
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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.
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lu
private LUDecomposition lu(DoubleMatrix2D matrix)
Constructs and returns the LU-decomposition of the given matrix.
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mult
public double mult(DoubleMatrix1D x, DoubleMatrix1D y)
Inner product of two vectors; Sum(x[i] * y[i]). Also known as dot product.
Equivalent to x.zDotProduct(y).- Parameters:
x
- the first source vector.y
- the second source matrix.- Returns:
- the inner product.
- Throws:
java.lang.IllegalArgumentException
- if x.size() != y.size().
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mult
public DoubleMatrix1D mult(DoubleMatrix2D A, DoubleMatrix1D y)
Linear algebraic matrix-vector multiplication; z = A * y. z[i] = Sum(A[i,j] * y[j]), i=0..A.rows()-1, j=0..y.size()-1.- Parameters:
A
- the source matrix.y
- the source vector.- Returns:
- z; a new vector with z.size()==A.rows().
- Throws:
java.lang.IllegalArgumentException
- if A.columns() != y.size().
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mult
public DoubleMatrix2D mult(DoubleMatrix2D A, DoubleMatrix2D B)
Linear algebraic matrix-matrix multiplication; C = A x B. C[i,j] = Sum(A[i,k] * B[k,j]), k=0..n-1.
Matrix shapes: A(m x n), B(n x p), C(m x p).- Parameters:
A
- the first source matrix.B
- the second source matrix.- Returns:
- C; a new matrix holding the results, with C.rows()=A.rows(), C.columns()==B.columns().
- Throws:
java.lang.IllegalArgumentException
- if B.rows() != A.columns().
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multOuter
public DoubleMatrix2D multOuter(DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix2D A)
Outer product of two vectors; Sets A[i,j] = x[i] * y[j].- Parameters:
x
- the first source vector.y
- the second source vector.A
- the matrix to hold the results. Set this parameter to null to indicate that a new result matrix shall be constructed.- Returns:
- A (for convenience only).
- Throws:
java.lang.IllegalArgumentException
- if A.rows() != x.size() || A.columns() != y.size().
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norm1
public double norm1(DoubleMatrix1D x)
Returns the one-norm of vector x, which is Sum(abs(x[i])).
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norm1
public double norm1(DoubleMatrix2D A)
Returns the one-norm of matrix A, which is the maximum absolute column sum.
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norm2
public double norm2(DoubleMatrix1D x)
Returns the two-norm (aka euclidean norm) of vector x; equivalent to mult(x,x).
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norm2
public double norm2(DoubleMatrix2D A)
Returns the two-norm of matrix A, which is the maximum singular value; obtained from SVD.
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normF
public double normF(DoubleMatrix2D A)
Returns the Frobenius norm of matrix A, which is Sqrt(Sum(A[i,j]2)).
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normInfinity
public double normInfinity(DoubleMatrix1D x)
Returns the infinity norm of vector x, which is Max(abs(x[i])).
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normInfinity
public double normInfinity(DoubleMatrix2D A)
Returns the infinity norm of matrix A, which is the maximum absolute row sum.
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permute
public DoubleMatrix1D permute(DoubleMatrix1D A, int[] indexes, double[] work)
Modifies the given vector A such that it is permuted as specified; Useful for pivoting. Cell A[i] will go into cell A[indexes[i]].Example:
Reordering [A,B,C,D,E] with indexes [0,4,2,3,1] yields [A,E,C,D,B] In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[2], A[3]<--A[3], A[4]<--A[1]. Reordering [A,B,C,D,E] with indexes [0,4,1,2,3] yields [A,E,B,C,D] In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[1], A[3]<--A[2], A[4]<--A[3].
- Parameters:
A
- the vector to permute.indexes
- the permutation indexes, must satisfy indexes.length==A.size() && indexes[i] >= 0 && indexes[i] < A.size();work
- the working storage, must satisfy work.length >= A.size(); set work==null if you don't care about performance.- Returns:
- the modified A (for convenience only).
- Throws:
java.lang.IndexOutOfBoundsException
- if indexes.length != A.size().
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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 toDoubleMatrix2D.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.
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permuteColumns
public DoubleMatrix2D permuteColumns(DoubleMatrix2D A, int[] indexes, int[] work)
Modifies the given matrix A such that it's columns are permuted as specified; Useful for pivoting. Column A[i] will go into column A[indexes[i]]. Equivalent to permuteRows(transpose(A), indexes, work).- Parameters:
A
- the matrix to permute.indexes
- the permutation indexes, must satisfy indexes.length==A.columns() && indexes[i] >= 0 && indexes[i] < A.columns();work
- the working storage, must satisfy work.length >= A.columns(); set work==null if you don't care about performance.- Returns:
- the modified A (for convenience only).
- Throws:
java.lang.IndexOutOfBoundsException
- if indexes.length != A.columns().
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permuteRows
public DoubleMatrix2D permuteRows(DoubleMatrix2D A, int[] indexes, int[] work)
Modifies the given matrix A such that it's rows are permuted as specified; Useful for pivoting. Row A[i] will go into row A[indexes[i]].Example:
Reordering [A,B,C,D,E] with indexes [0,4,2,3,1] yields [A,E,C,D,B] In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[2], A[3]<--A[3], A[4]<--A[1]. Reordering [A,B,C,D,E] with indexes [0,4,1,2,3] yields [A,E,B,C,D] In other words A[0]<--A[0], A[1]<--A[4], A[2]<--A[1], A[3]<--A[2], A[4]<--A[3].
- Parameters:
A
- the matrix to permute.indexes
- the permutation indexes, must satisfy indexes.length==A.rows() && indexes[i] >= 0 && indexes[i] < A.rows();work
- the working storage, must satisfy work.length >= A.rows(); set work==null if you don't care about performance.- Returns:
- the modified A (for convenience only).
- Throws:
java.lang.IndexOutOfBoundsException
- if indexes.length != A.rows().
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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).
- Parameters:
A
- the source matrix; must be square; stays unaffected by this operation.p
- the exponent, can be any number.- Returns:
- B, a newly constructed result matrix; storage-independent of A.
- Throws:
java.lang.IllegalArgumentException
- if !property().isSquare(A).
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property
public Property property()
Returns the property object attached to this Algebra, defining tolerance.- Returns:
- the Property object.
- See Also:
setProperty(Property)
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qr
private QRDecomposition qr(DoubleMatrix2D matrix)
Constructs and returns the QR-decomposition of the given matrix.
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rank
public int rank(DoubleMatrix2D A)
Returns the effective numerical rank of matrix A, obtained from Singular Value Decomposition.
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setProperty
public void setProperty(Property property)
Attaches the given property object to this Algebra, defining tolerance.- Parameters:
the
- Property object to be attached.- Throws:
java.lang.UnsupportedOperationException
- if this==DEFAULT && property!=this.property() - The DEFAULT Algebra object is immutable.java.lang.UnsupportedOperationException
- if this==ZERO && property!=this.property() - The ZERO Algebra object is immutable.- See Also:
property
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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.
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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.
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subMatrix
private 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:
java.lang.IndexOutOfBoundsException
- if columnFrom<0 || columnTo-columnFrom+1<0 || columnTo+1>matrix.columns() || for any row=rowIndexes[i]: row < 0 || row >= matrix.rows().
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subMatrix
private 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:
java.lang.IndexOutOfBoundsException
- if rowFrom<0 || rowTo-rowFrom+1<0 || rowTo+1>matrix.rows() || for any col=columnIndexes[i]: col < 0 || col >= matrix.columns().
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subMatrix
public DoubleMatrix2D subMatrix(DoubleMatrix2D A, int fromRow, int toRow, int fromColumn, int toColumn)
Constructs and returns a new sub-range view which is the sub matrix A[fromRow..toRow,fromColumn..toColumn]. The returned matrix is backed by this matrix, so changes in the returned matrix are reflected in this matrix, and vice-versa. Use idioms like result = subMatrix(...).copy() to generate an independent sub matrix.- Parameters:
A
- the source matrix.fromRow
- The index of the first row (inclusive).toRow
- The index of the last row (inclusive).fromColumn
- The index of the first column (inclusive).toColumn
- The index of the last column (inclusive).- Returns:
- a new sub-range view.
- Throws:
java.lang.IndexOutOfBoundsException
- if fromColumn<0 || toColumn-fromColumn+1<0 || toColumn>=A.columns() || fromRow<0 || toRow-fromRow+1<0 || toRow>=A.rows()
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svd
private SingularValueDecomposition svd(DoubleMatrix2D matrix)
Constructs and returns the SingularValue-decomposition of the given matrix.
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toString
public java.lang.String toString(DoubleMatrix2D matrix)
Returns a String with (propertyName, propertyValue) pairs. Useful for debugging or to quickly get the rough picture. For example,cond : 14.073264490042144 det : Illegal operation or error: Matrix must be square. norm1 : 0.9620244354009628 norm2 : 3.0 normF : 1.304841791648992 normInfinity : 1.5406551198102534 rank : 3 trace : 0
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toVerboseString
public java.lang.String toVerboseString(DoubleMatrix2D matrix)
Returns the results of toString(A) and additionally the results of all sorts of decompositions applied to the given matrix. Useful for debugging or to quickly get the rough picture. For example,A = 3 x 3 matrix 249 66 68 104 214 108 144 146 293 cond : 3.931600417472078 det : 9638870.0 norm1 : 497.0 norm2 : 473.34508217011404 normF : 516.873292016525 normInfinity : 583.0 rank : 3 trace : 756.0 density : 1.0 isDiagonal : false isDiagonallyDominantByColumn : true isDiagonallyDominantByRow : true isIdentity : false isLowerBidiagonal : false isLowerTriangular : false isNonNegative : true isOrthogonal : false isPositive : true isSingular : false isSkewSymmetric : false isSquare : true isStrictlyLowerTriangular : false isStrictlyTriangular : false isStrictlyUpperTriangular : false isSymmetric : false isTriangular : false isTridiagonal : false isUnitTriangular : false isUpperBidiagonal : false isUpperTriangular : false isZero : false lowerBandwidth : 2 semiBandwidth : 3 upperBandwidth : 2 ----------------------------------------------------------------------------- LUDecompositionQuick(A) --> isNonSingular(A), det(A), pivot, L, U, inverse(A) ----------------------------------------------------------------------------- isNonSingular = true det = 9638870.0 pivot = [0, 1, 2] L = 3 x 3 matrix 1 0 0 0.417671 1 0 0.578313 0.57839 1 U = 3 x 3 matrix 249 66 68 0 186.433735 79.598394 0 0 207.635819 inverse(A) = 3 x 3 matrix 0.004869 -0.000976 -0.00077 -0.001548 0.006553 -0.002056 -0.001622 -0.002786 0.004816 ----------------------------------------------------------------- QRDecomposition(A) --> hasFullRank(A), H, Q, R, pseudo inverse(A) ----------------------------------------------------------------- hasFullRank = true H = 3 x 3 matrix 1.814086 0 0 0.34002 1.903675 0 0.470797 0.428218 2 Q = 3 x 3 matrix -0.814086 0.508871 0.279845 -0.34002 -0.808296 0.48067 -0.470797 -0.296154 -0.831049 R = 3 x 3 matrix -305.864349 -195.230337 -230.023539 0 -182.628353 467.703164 0 0 -309.13388 pseudo inverse(A) = 3 x 3 matrix 0.006601 0.001998 -0.005912 -0.005105 0.000444 0.008506 -0.000905 -0.001555 0.002688 -------------------------------------------------------------------------- CholeskyDecomposition(A) --> isSymmetricPositiveDefinite(A), L, inverse(A) -------------------------------------------------------------------------- isSymmetricPositiveDefinite = false L = 3 x 3 matrix 15.779734 0 0 6.590732 13.059948 0 9.125629 6.573948 12.903724 inverse(A) = Illegal operation or error: Matrix is not symmetric positive definite. --------------------------------------------------------------------- EigenvalueDecomposition(A) --> D, V, realEigenvalues, imagEigenvalues --------------------------------------------------------------------- realEigenvalues = 1 x 3 matrix 462.796507 172.382058 120.821435 imagEigenvalues = 1 x 3 matrix 0 0 0 D = 3 x 3 matrix 462.796507 0 0 0 172.382058 0 0 0 120.821435 V = 3 x 3 matrix -0.398877 -0.778282 0.094294 -0.500327 0.217793 -0.806319 -0.768485 0.66553 0.604862 --------------------------------------------------------------------- SingularValueDecomposition(A) --> cond(A), rank(A), norm2(A), U, S, V --------------------------------------------------------------------- cond = 3.931600417472078 rank = 3 norm2 = 473.34508217011404 U = 3 x 3 matrix 0.46657 -0.877519 0.110777 0.50486 0.161382 -0.847982 0.726243 0.45157 0.51832 S = 3 x 3 matrix 473.345082 0 0 0 169.137441 0 0 0 120.395013 V = 3 x 3 matrix 0.577296 -0.808174 0.116546 0.517308 0.251562 -0.817991 0.631761 0.532513 0.563301
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trace
public double trace(DoubleMatrix2D A)
Returns the sum of the diagonal elements of matrix A; Sum(A[i,i]).
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transpose
public DoubleMatrix2D transpose(DoubleMatrix2D A)
Constructs and returns a new view which is the transposition of the given matrix A. Equivalent toA.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, 6transpose ==> 3 x 2 matrix:
1, 4
2, 5
3, 6transpose ==> 2 x 3 matrix:
1, 2, 3
4, 5, 6- Returns:
- a new transposed view.
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trapezoidalLower
protected DoubleMatrix2D trapezoidalLower(DoubleMatrix2D A)
Modifies the matrix to be a lower trapezoidal matrix.- Returns:
- A (for convenience only).
- See Also:
#triangulateLower(DoubleMatrix2D)
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xmultOuter
private 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.
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xpowSlow
private 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:
java.lang.IllegalArgumentException
- if !Testing.isSquare(A).
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