LinearAlgebra UDF - LinearAlgebra für AutoIt mit BLAS/LAPACK

---- vector/matrix creation ----
_la_fromArray                 - converts a AutoIt array or array define string into a matrix map
_la_fromStruct                - creates a matrix/vector map from a DllStruct as used here in the UDF
_la_createVector              - creates new empty vector
_la_createMatrix              - creates new empty matrix
_la_createIdentity            - create identity matrix/vector
_la_duplicate                 - creates an independent copy of a matrix/vector map
_la_fromFile                  - reads a matrix or a vector from a file created by _la_toFile()

---- extraction/transforming ----
_la_join                      - combines 2 matrices
_la_transpose                 - transposes a matrix in-place or out-place and [optional] scaling
_la_ReDim                     - changes the shape of a matrix by by changing the number of columns (also matrix <-> vector conversion)
_la_getRow                    - extracts a row of a matrix as a vector
_la_getColumn                 - extracts a column of a matrix as a vector
_la_getDiag                   - extracts the diagonal of a matrix as a vector
_la_getTriangle               - extract upper or lower triangle part of a matrix
_la_VectorToDiag              - creates a diagonal matrix from a vector

---- data output ----
_la_display                   - displays a matrix/vector map, similar to _ArrayDisplay
_la_toArray                   - converts a matrix/vector map into an AutoIt array
_la_toFile                    - write a matrix/vector into a file

---- scalar operations ----
_la_rotate                    - applies a plane rotation to coordinate-pairs

---- matrix attributes ----
_la_isPositiveDefinite        - checks whether a matrix is positive definite
_la_isSymmetric               - checks whether a matrix is symmetrical
_la_rank                      - determines the rank of a matrix
_la_determinant               - calculate the determinant of a matrix
_la_conditionNumber           - determine the condition number of a matrix

---- unary operations ----
_la_inverse                   - calculates the inverse of a matrix
_la_sum                       - calculates the sum of the elements of a matrix, vector or parts thereof
_la_asum                      - calculate the sum of the absolute(!) values of a matrix/vector
_la_amin                      - finds the first element having the minimum absolute(!) value
_la_amax                      - finds the first element having the maximum absolute(!) value
_la_norm                      - calculate the euclidian norm of a vector
_la_mean                      - calculate the mean of a vector or parts of a matrix

---- element wise operations ----
_la_sqrtElements              - calculates the square root of each element of a matrix/vector
_la_squareElements            - calculates the square of each element of a matrix/vector
_la_invElements               - forms the reciprocal (1/x) for each element of the matrix/vector

---- addition subtraction ----
_la_sub                       - subtracts a matrix/vector B from matrix/vector A
_la_add                       - calculate the sum of a matrix/vector/scalar mA and a matrix/vector/scalar mB

---- multiplication ----
_la_mul                       - calculates a multiplication between a matrix/vector/scalar A and a matrix/vector/scalar B
_la_outerproduct              - calculates the outer product ("tensor product") of two vectors
_la_dot                       - calculate the "dot product"/"scalar product"/"inner product" of two vectors
_la_scale                     - multiplies the elements of a matrix/vector by a scalar value
_la_mulElementWise            - calculates the element-wise ("Hadarmard") product between two matrices/vectors
_la_cross                     - calculates the cross product between two 3-element vectors

---- factorization / decomposition ----
_la_LU                        - calculates the LU decomposition of a matrix
_la_QR                        - calculates the QR decomposition of a matrix
_la_SVD                       - calculates the singular value decomposition (SVD) of a matrix
_la_cholesky                  - calculate the cholesky decomposition of a symmetric, positive definite matrix ( A --> L * Lᵀ or A --> U * Uᵀ )

---- eigenvalues / eigenvectors ----
_la_eigen                     - computes for an N-by-N real matrix A, the eigenvalues and the left and/or right eigenvectors.

---- solve linear equation systems ----
_la_solve                     - computes the solution to a system of linear equations A * X = B

---- least squares solving ----
_la_lstsq                     - solves overdetermined or underdetermined [weighted] linear system

---- regression ----
_la_regression                - calculates an n-dimensional linear or non-linear regression

---- adjustment ----
_la_adjustment                - performs a least-squares adjustment calculation for a system of different [weighted] non-linear equations
_la_adjustment_l1             - performs a adjustment calculation to L1 norm for a system of different [weighted] non-linear equations
_la_adjustment_addObservation - adds an observation to the adjustment system

---- additional helper functions ----
_la_adj_showResult            - formats the results of _la_adjustment more clearly and display them in a window