## Linear Algebra

#### CholeskyGetL

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

Collection CholeskyGetL(Collection A)

• 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)

• 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)

• X: The matrix to be analyzed.

#### count

Returns the number of cells in a tensor

Object count(Collection x, Collection cmp=nothing)

• x: is the tensor to count
• cmp: value to compare the cells

#### countRange

Returns the number of values within a given range.

Object countRange(Collection x, Collection min, Collection max)

• x: is the container for which you want to count the values.
• min: is the lower boundary of the range. The lower boundary is not included in the range.
• max: is the upper boundary of the range. The upper boundary is included in the range.

#### covarianceMatrix

Calculates the matrix product of the given matrices.

Collection covarianceMatrix(Collection m1, Collection m2)

• m1: The first matrix
• m2: The second matrix

#### determinant

Returns the determinant of the matrix.

float determinant(Collection X)

• X: Matrix for which the determinant is sought.

#### EigenproblemGetD

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

Collection EigenproblemGetD(Collection X)

• 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, boolean ahp=false)

• 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, boolean ahp=false)

• 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)

• X: The matrix for which the Inverse is sought.

#### isFullRank

Returns true if the matrix is of full rank

boolean isFullRank(Collection X)

• X: The Matrix to be tested.

#### isNonsingular

Returns true if the matrix is nonsingular

boolean isNonsingular(Collection X)

• X: The Matrix to be tested.

#### isSPD

Returns true if the matrix is symmetric positive definite.

boolean isSPD(Collection X)

• X: The matrix to be analyzed

#### linearMapping

Performs a mapping from a source vector to another vector space using either the weighted sum method or the proportional score method.

Collection linearMapping(Collection matrix, Collection x, integer sign=all, Object total=0.0, integer grouphandling=nothing, integer method=nothing, integer causeslevel=nothing, integer effectslevel=nothing, integer =nothing, integer =nothing)

• matrix: is the matrix describing the linear transformation.
• x: is the vector you want to map.
• sign: sets a filter for the matrix relations: all, pos(itive only), neg(ative only)
• total: total sum for normalization. Use 0 in order to skip normalization.
• grouphandling: defines how to handle parent items: shallow (leafs only), sums (accumulate hierarchically), levels (calculate system level and paramater level separately).
• method: wsm (weighted sum method) or prop (proportional score method)
• causeslevel: is the details level for the causes. Set to 0 in order to use the matrix default.
• effectslevel: is the details level for the effects. Set to 0 in order to use the matrix default.
• : is the details level for the causes. Set to 0 in order to use the matrix default.
• : is the details level for the effects. Set to 0 in order to use the matrix default.

#### LUGetL

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

Collection LUGetL(Collection X)

• X: The matrix.

#### LUGetPivot

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

Collection LUGetPivot(Collection X)

• X: TheMatrix.

#### LUGetU

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

Collection LUGetU(Collection X)

• X: The matrix.

#### LUSolve

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

Collection LUSolve(Collection A, Collection b)

• 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)

• m1: The first matrix
• m2: The second matrix

#### norm

Normalizes the numbers in a container to a given total.

Collection norm(Collection x, Numeric total, integer level=nothing, Object sumdim=nothing, Collection , Numeric , integer =nothing, Object =nothing)

• x: is the container which contains the values you want to normalize.
• total: is the sum you want to use for normalization, e.g. 100 if you want to express the values from container x in percent.
• level: is the level of details to operate on
• sumdim: specifies the dimension for the operation if the source is a multi-dimensional collection
• : is the container which contains the values you want to normalize.
• : is the sum you want to use for normalization, e.g. 100 if you want to express the values from container x in percent.
• : is the level of details to operate on
• : specifies the dimension for the operation if the source is a multi-dimensional collection

#### norm2

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

float norm2(Collection X)

• X: The matrix to be analyzed.

#### nsum

Adds all negative numbers in a container.

Object nsum(Collection x)

• x: is the container to sum up.

#### percentage

Normalizes the numbers in a container to a total of 1.0.

Collection percentage(Collection x, integer level=0, Object sumdim=nothing)

• x: is the container which contains the values you want to normalize.
• level: is the level of details to operate on
• sumdim: specifies the dimension for the operation if the source is a multi-dimensional collection

#### prod

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

Object prod(Collection x)

• x: is the container for which you want the product.

#### pseudoInverse

Returns the More-Penrose pseudoinverse of the matrix.

Collection pseudoInverse(Collection X)

• X: The matrix for which the Inverse is sought.

#### psum

Adds all positive numbers in a container.

Object psum(Collection x)

• x: is the container to sum up.

#### QRGetHouseholder

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

Collection QRGetHouseholder(Collection X)

• X: The matrix.

#### QRGetQ

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

Collection QRGetQ(Collection X)

• X: The matrix.

#### QRGetR

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

Collection QRGetR(Collection X)

• X: The matrix.

#### QRSolve

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

Collection QRSolve(Collection A, Collection b)

• 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)

• X: Returns the rank of the matrix.

#### sum

Adds all the numbers in a container.

Object sum(Collection x, Collection )

• x: is the container to sum up
• : is the container to sum up

#### SVDGetS

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

Collection SVDGetS(Collection X)

• X: The matrix.

#### SVDGetSV

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

Collection SVDGetSV(Collection X)

• X: The matrix.

#### SVDGetU

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

Collection SVDGetU(Collection X)

• X: The matrix.

#### SVDGetV

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

Collection SVDGetV(Collection X)

• X: The matrix.

#### SVDSolve

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

Collection SVDSolve(Collection A, Collection b)

• 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)

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

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