Functions and Formulas

Linear Algebra

choleskyGetL

Returns the lower triangular factor L of a Cholesky decomposition, such that L*L'=A.

Collection choleskyGetL(Collection A)

Parameters

A :

Matrix to be decomposed, such that L*L'=A

choleskySolve

Returns the solution x of the problem Ax = b for x using Cholesky decomposition. x can be a matrix or vector.

Collection choleskySolve(Collection A, Collection b)

Parameters

A :	Matrix A of problem Ax=b
b :	Vector (or matrix) b of the problem Ax=b

condition

Returns the condition number of the matrix: max(S)/min(S).

float condition(Collection X)

Parameters

X :

The matrix to be analyzed.

cov

Calculates the matrix product of the given matrices.

Collection cov(Collection m1, Collection m2)

Parameters

m1 :	The first matrix
m2 :	The second matrix

determinant

Returns the determinant of the matrix.

float determinant(Collection X)

Parameters

X :

Matrix for which the determinant is sought.

eigenproblemGetD

Returns matrix D of A = V*D*V'.

Collection eigenproblemGetD(Collection X)

Parameters

X :

The matrix

eigenvalues

Returns a vector listing the complex eigenvalues of the matrix or a single eigenvalue.

Collection eigenvalues(Collection X, boolean complex=false, int index=0.0, boolean ahp=false)

Parameters

X :	The matrix to be tested.
complex :	true: return complex part of the eigenvalues (optional)
index :	Index (0 : all, 1 : greatest, 2...)
ahp :	true: calculate each group separately

eigenvectors

Returns a matrix listing the eigenvectors of the given matrix.as columns (in dimension 1) and the values in dimension 0.

Collection eigenvectors(Collection X, int index=0.0, boolean ahp=false)

Parameters

X :	The matrix
index :	Index (0 : all, 1 : greatest, 2...)
ahp :	true: calculate each group separately

inverse

Returns the inverse of the matrix. The matrix must be invertible, otherwise the result is undefined.

Collection inverse(Collection X)

Parameters

X :

The matrix for which the Inverse is sought.

isFullRank

Returns true if the matrix is of full rank

boolean isFullRank(Collection X)

Parameters

X :

The Matrix to be tested.

isNonsingular

Returns true if the matrix is nonsingular

boolean isNonsingular(Collection X)

Parameters

X :

The Matrix to be tested.

isSPD

Returns true if the matrix is symmetric positive definite.

boolean isSPD(Collection X)

Parameters

X :

The matrix to be analyzed

luGetL

Returns matrix L of the LU decomposition of the given matrix.

Collection luGetL(Collection X)

Parameters

X :

The matrix.

luGetPivot

Returns the Pivot vector of the LU decomposition of the given matrix.

Collection luGetPivot(Collection X)

Parameters

X :

TheMatrix.

luGetU

Returns matrix U of the LU decomposition of the given matrix.

Collection luGetU(Collection X)

Parameters

X :

The matrix.

luSolve

Returns solution x of the problem Ax = b for x using LU decomposition.

Collection luSolve(Collection A, Collection b)

Parameters

A :	Matrix A of the problem Ax = b. A can be an m-by-n matrix with m<>n.
b :	Vector (or matrix) b of the problem Ax = b.

matrixProduct

Calculates the matrix product of the given matrices.

Collection matrixProduct(Collection m1, Collection m2)

Parameters

m1 :	The first matrix
m2 :	The second matrix

norm2

Returns the 2 norm (max(S)) of the matrix

float norm2(Collection X)

Parameters

X :

The matrix to be analyzed.

nsum

Adds all negative numbers in a container.

Object nsum(Collection x)

Parameters

x :

is the container to sum up.

prod

Multiplies all the numbers in the given container and returns the product.

Object prod(Collection x)

Parameters

x :

is the container for which you want the product.

pseudoInverse

Returns the More-Penrose pseudoinverse of the matrix.

Collection pseudoInverse(Collection X)

Parameters

X :

The matrix for which the Inverse is sought.

psum

Adds all positive numbers in a container.

Object psum(Collection x)

Parameters

x :

is the container to sum up.

qrGetHouseholder

Returns the Householder vectors from the QR decomposition of the given matrix.

Collection qrGetHouseholder(Collection X)

Parameters

X :

The matrix.

qrGetQ

Returns matrix Q of the QR decomposition of the given matrix.

Collection qrGetQ(Collection X)

Parameters

X :

The matrix.

qrGetR

Returns matrix R of the QR decomposition of the given matrix.

Collection qrGetR(Collection X)

Parameters

X :

The matrix.

qrSolve

Returns solution x of the problem Ax = b for x using QR decomposition.

Collection qrSolve(Collection A, Collection b)

Parameters

A :	Matrix A of the problem Ax = b.
b :	Vector (or matrix) b of the problem Ax = b.

rank

Returns the rank of the matrix.

int rank(Collection X)

Parameters

X :

Returns the rank of the matrix.

sum

Adds all the numbers in a container.

Object sum(Collection x, Collection x-1)

Parameters

x :	is the container to sum up
x-1 :	is the container to sum up

svdGetS

Returns matrix S of the SVD decomposition of the given matrix.

Collection svdGetS(Collection X)

Parameters

X :

The matrix.

svdGetSV

Returns a vector of singular values. Values are ordered from large to small.

Collection svdGetSV(Collection X)

Parameters

X :

The matrix.

svdGetU

Returns matrix U of the SVD decomposition of the given matrix.

Collection svdGetU(Collection X)

Parameters

X :

The matrix.

svdGetV

Returns matrix V of the SVD decomposition of the given matrix.

Collection svdGetV(Collection X)

Parameters

X :

The matrix.

svdSolve

Solves the problem Ax = b for x using singular value decomposition

Collection svdSolve(Collection A, Collection b)

Parameters

A :	The matrix A of problem Ax=b.
b :	Vector (or matrix) b of the problem Ax = b.

transposed

Returns the transposed of the given auto matrix.

Collection transposed(Collection matrix)

Parameters

matrix :

The auto matrix for which you want to get the transposed matrix.

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