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)

Result: The linearMapping function maps a score for one dimension of a matrix to a score on the other dimension of a matrix using either the weighted sum method or the proportional score method.

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.

Calculation using weighted sum method:

1
for each row in the matrix multiply all values by the corresponding value in the input vector
2
calculate the sum of all columns in the matrix
3
optionally normalize the sum to a given total, e.g. 1.0: sum up all values in the output vector, then divide each value of the output vector by this sum.

Calculation using proportional score method:

1
calculate the sum of all columns in the matrix
2
for each row in the matrix multiply all values by the corresponding value in the input vector
3
divide all values in the matrix by the factor calculated in step 1.
4
calculate the sum of all columns in the matrix
5
optionally normalize the sum to a given total, e.g. 1.0: sum up all values in the output vector, then divide each value of the output vector by this sum.

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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Functions and Formulas

Linear Algebra

Linear Algebra

CholeskyGetL

CholeskySolve

condition

count

countRange

covarianceMatrix

determinant

EigenproblemGetD

Eigenvalues

Eigenvectors

inverse

isFullRank

isNonsingular

isSPD

linearMapping

Calculation using weighted sum method:

Calculation using proportional score method:

LUGetL

LUGetPivot

LUGetU

LUSolve

matrixProduct

norm

norm2

nsum

percentage

prod

pseudoInverse

psum

QRGetHouseholder

QRGetQ

QRGetR

QRSolve

rank

sum

SVDGetS

SVDGetSV

SVDGetU

SVDGetV

SVDSolve

transposed

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