Interface Matrix
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- All Superinterfaces:
java.io.Serializable
- All Known Subinterfaces:
DenseMatrix
- All Known Implementing Classes:
JMatrix
public interface Matrix extends java.io.Serializable
An abstract interface of matrix. The most important method is the matrix vector multiplication, which is the only operation needed in many iterative matrix algorithms, e.g. biconjugate gradient method for solving linear equations and power iteration and Lanczos algorithm for eigen decomposition, which are usually very efficient for very large and sparse matrices.A matrix is a rectangular array of numbers. An item in a matrix is called an entry or an element. Entries are often denoted by a variable with two subscripts. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied. These operations have many of the properties of ordinary arithmetic, except that matrix multiplication is not commutative, that is, AB and BA are not equal in general.
Matrices are a key tool in linear algebra. One use of matrices is to represent linear transformations and matrix multiplication corresponds to composition of linear transformations. Matrices can also keep track of the coefficients in a system of linear equations. For a square matrix, the determinant and inverse matrix (when it exists) govern the behavior of solutions to the corresponding system of linear equations, and eigenvalues and eigenvectors provide insight into the geometry of the associated linear transformation.
There are several methods to render matrices into a more easily accessible form. They are generally referred to as matrix transformation or matrix decomposition techniques. The interest of all these decomposition techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.
The LU decomposition factors matrices as a product of lower (L) and an upper triangular matrices (U). Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The QR decomposition factors matrices as a product of an orthogonal (Q) and a right triangular matrix (R). QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm. Singular value decomposition expresses any matrix A as a product UDV', where U and V are unitary matrices and D is a diagonal matrix. The eigendecomposition or diagonalization expresses A as a product VDV-1, where D is a diagonal matrix and V is a suitable invertible matrix. If A can be written in this form, it is called diagonalizable.
- Author:
- Haifeng Li
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Method Summary
All Methods Static Methods Instance Methods Abstract Methods Default Methods Modifier and Type Method Description Matrix
aat()
Returns A * A'default double
apply(int i, int j)
Returns the entry value at row i and column j.Matrix
ata()
Returns A' * Adouble[]
atx(double[] x, double[] y)
y = A' * xdouble[]
atxpy(double[] x, double[] y)
y = A' * x + ydouble[]
atxpy(double[] x, double[] y, double b)
y = A' * x + b * ydouble[]
ax(double[] x, double[] y)
y = A * xdouble[]
axpy(double[] x, double[] y)
y = A * x + ydouble[]
axpy(double[] x, double[] y, double b)
y = A * x + b * ydefault double[]
diag()
Returns the diagonal elements.static DenseMatrix
diag(double[] A)
Returns a square diagonal matrix with the elements of vector diag on the main diagonal.default EVD
eigen(int k)
Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.default EVD
eigen(int k, double kappa, int maxIter)
Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.static DenseMatrix
eye(int n)
Returns an n-by-n identity matrix.static DenseMatrix
eye(int m, int n)
Returns an m-by-n identity matrix.double
get(int i, int j)
Returns the entry value at row i and column j.default boolean
isSymmetric()
Returns true if the matrix is symmetric.int
ncols()
Returns the number of columns.static DenseMatrix
newInstance(double[] A)
Returns a column vector/matrix initialized by given one-dimensional array.static DenseMatrix
newInstance(double[][] A)
Returns an matrix initialized by given two-dimensional array.static DenseMatrix
newInstance(int rows, int cols, double value)
Creates a matrix filled with given value.int
nrows()
Returns the number of rows.static DenseMatrix
ones(int rows, int cols)
Return an all-one matrix.static DenseMatrix
randn(int rows, int cols)
Returns a random matrix of standard normal distributed values with given mean and standard dev.static DenseMatrix
randn(int rows, int cols, double mu, double sigma)
Returns a random matrix of normal distributed values with given mean and standard dev.default void
setSymmetric(boolean symmetric)
Sets if the matrix is symmetric.default SVD
svd(int k)
Find k largest approximate singular triples of a matrix by the Lanczos algorithm.default SVD
svd(int k, double kappa, int maxIter)
Find k largest approximate singular triples of a matrix by the Lanczos algorithm.default java.lang.String
toString(boolean full)
Returns the string representation of matrix.default double
trace()
Returns the matrix trace.Matrix
transpose()
Returns the matrix transpose.static DenseMatrix
zeros(int rows, int cols)
Returns all-zero matrix.
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Method Detail
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newInstance
static DenseMatrix newInstance(double[][] A)
Returns an matrix initialized by given two-dimensional array.
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newInstance
static DenseMatrix newInstance(double[] A)
Returns a column vector/matrix initialized by given one-dimensional array.
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newInstance
static DenseMatrix newInstance(int rows, int cols, double value)
Creates a matrix filled with given value.
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zeros
static DenseMatrix zeros(int rows, int cols)
Returns all-zero matrix.
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ones
static DenseMatrix ones(int rows, int cols)
Return an all-one matrix.
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eye
static DenseMatrix eye(int n)
Returns an n-by-n identity matrix.
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eye
static DenseMatrix eye(int m, int n)
Returns an m-by-n identity matrix.
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diag
static DenseMatrix diag(double[] A)
Returns a square diagonal matrix with the elements of vector diag on the main diagonal.- Parameters:
A
- the array of diagonal elements.
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randn
static DenseMatrix randn(int rows, int cols)
Returns a random matrix of standard normal distributed values with given mean and standard dev.
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randn
static DenseMatrix randn(int rows, int cols, double mu, double sigma)
Returns a random matrix of normal distributed values with given mean and standard dev.
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toString
default java.lang.String toString(boolean full)
Returns the string representation of matrix.- Parameters:
full
- Print the full matrix if true. Otherwise only print top left 7 x 7 submatrix.
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isSymmetric
default boolean isSymmetric()
Returns true if the matrix is symmetric.
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setSymmetric
default void setSymmetric(boolean symmetric)
Sets if the matrix is symmetric. It is the caller's responability to make sure if the matrix symmetric. Also the matrix won't update this property if the matrix values are changed.
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nrows
int nrows()
Returns the number of rows.
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ncols
int ncols()
Returns the number of columns.
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transpose
Matrix transpose()
Returns the matrix transpose.
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get
double get(int i, int j)
Returns the entry value at row i and column j.
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apply
default double apply(int i, int j)
Returns the entry value at row i and column j. For Scala users.
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diag
default double[] diag()
Returns the diagonal elements.
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trace
default double trace()
Returns the matrix trace. The sum of the diagonal elements.
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ata
Matrix ata()
Returns A' * A
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aat
Matrix aat()
Returns A * A'
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ax
double[] ax(double[] x, double[] y)
y = A * x- Returns:
- y
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axpy
double[] axpy(double[] x, double[] y)
y = A * x + y- Returns:
- y
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axpy
double[] axpy(double[] x, double[] y, double b)
y = A * x + b * y- Returns:
- y
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atx
double[] atx(double[] x, double[] y)
y = A' * x- Returns:
- y
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atxpy
double[] atxpy(double[] x, double[] y)
y = A' * x + y- Returns:
- y
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atxpy
double[] atxpy(double[] x, double[] y, double b)
y = A' * x + b * y- Returns:
- y
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eigen
default EVD eigen(int k)
Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.- Parameters:
k
- the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.
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eigen
default EVD eigen(int k, double kappa, int maxIter)
Find k largest approximate eigen pairs of a symmetric matrix by the Lanczos algorithm.- Parameters:
k
- the number of eigenvalues we wish to compute for the input matrix. This number cannot exceed the size of A.kappa
- relative accuracy of ritz values acceptable as eigenvalues.maxIter
- Maximum number of iterations.
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svd
default SVD svd(int k)
Find k largest approximate singular triples of a matrix by the Lanczos algorithm.- Parameters:
k
- the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.
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svd
default SVD svd(int k, double kappa, int maxIter)
Find k largest approximate singular triples of a matrix by the Lanczos algorithm.- Parameters:
k
- the number of singular triples we wish to compute for the input matrix. This number cannot exceed the size of A.kappa
- relative accuracy of ritz values acceptable as singular values.maxIter
- Maximum number of iterations.
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