P -1, each component of which can be inexpensively computed (compared to the cost of obtaining the factors).Ī list of 3 Matrices, P, L, U, the PLU decomposition of a Matrix (these are the values returned from LUDecomposition when the output= option is given to that routine).
The first member is taken as the pivot Vector and the second member as the superimposed unit-lower and upper triangular LU factors (these are the default values returned from LUDecomposition when the method=NAG option is given to that routine). That is, the list items may be:Ī list of a Vector and Matrix, ipiv, LU, for an LU decomposition. These factors are in the form of returned values from LUDecomposition. If the first argument in the calling sequence is a list, then the elements of the list are taken as the Matrix factors of the Matrix A, due to some prefactorization. The subs method indicates that the input is already triangular, so only the appropriate forward or back substitution is done. The LU and Cholesky methods use the corresponding LUDecomposition method on the input Matrix (if not already prefactored) and then use forward and back substitution with a copy of the identity Matrix as a right-hand side. The polynom method uses an implementation of fraction-free Gaussian elimination. The univar method uses an evaluation method to reduce the Matrix to a Matrix of integers, then uses `Adjoint/integer`, and then uses genpoly to convert back to univariate polynomials. The integer method calls `Adjoint/integer` and divides it by the determinant. The complex and rational methods augment the input Matrix with a copy of the identity Matrix and then convert the system to reduced row echelon form. If m is included in the calling sequence, then the specified method is used to compute the inverse (except for 1 x 1, 2 x 2 and 3 x 3 Matrices where the calculation of the inverse is hard-coded for efficiency). If A is a non-square m x n Matrix, or if the option method = pseudo is specified, then the Moore-Penrose pseudo-inverse X is computed such that the following identities hold:
If A is recognized as a singular Matrix, an error message is returned. The MatrixInverse(A) function, where A is a nonsingular square Matrix, returns the Matrix inverse A -1. (optional) constructor options for the result object
(optional) equation of the form output=obj where obj is 'inverse' or 'proviso' or a list containing one or more of these names selects the result objects to compute (optional) equation of the form conjugate=true or false specifies whether to use the Hermitian transpose in the case of prefactored input from a Cholesky decomposition (optional) equation of the form methodoptions=list where the list contains options for specific methods (optional) equation of the form method = name where name is one of 'LU', 'Cholesky', 'subs', 'integer', 'univar', 'polynom', 'complex', 'rational', 'pseudo', or 'none' method used to factorize the inverse of A MatrixInverse( A, m, mopts, c, out, options ) Compute the inverse of a square Matrix or the Moore-Penrose pseudo-inverse of a Matrix
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