The Fac_LU class provides methods to factor an m-by-n matrix into its lower and upper triangular products: A = LU when partial pivoting is not needed PA = LU where P is the permutation matrix A = QLU where Q = P.inverse where a is the given matrix, l is an m-by-n lower triangular matrix, and u is an n-by-n upper triangular matrix. The permutation matrix is represented by the piv vector. Once factored, can be used to solve a system of linear equations. Solve for x in Ax = b: Ax = QLUx = b => LUx = Pb using steps (1) and (2) (1) Solve Ly = Pb using forward substitution for y (2) Solve Ux = y using backward substitution for x
Value parameters
- a
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the given m-by-n rectangular matrix
Attributes
- Companion
- object
- Graph
-
- Supertypes
Members list
Value members
Concrete methods
Solve for x using back substitution in the equation u*x = y where matrix u is upper triangular.
Solve for x using back substitution in the equation u*x = y where matrix u is upper triangular.
Value parameters
- y
-
the constant vector FIX - replace with /~
Attributes
Compute the determinant of matrix a. The value of the determinant indicates, among other things, whether there is a unique solution to a system of linear equations (a nonzero determinant).
Compute the determinant of matrix a. The value of the determinant indicates, among other things, whether there is a unique solution to a system of linear equations (a nonzero determinant).
Attributes
Factor matrix a into the product of l and u.
Factor matrix a into the product of l and u.
Attributes
Return the lower l and upper u matrices.
Return the lower l and upper u matrices.
Attributes
Compute the inverse of matrix a from the LU Factorization.
Compute the inverse of matrix a from the LU Factorization.
Attributes
Permute matrix c, equivalent to qc, i.e., multiplying by the inverse of the permutation matrix p.
Permute matrix c, equivalent to qc, i.e., multiplying by the inverse of the permutation matrix p.
Value parameters
- c
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the matrix to permute
Attributes
Permute vector d, equivalent to pd, i.e., multiplying by the permutation matrix p.
Permute vector d, equivalent to pd, i.e., multiplying by the permutation matrix p.
Value parameters
- d
-
the vector to permute
Attributes
Return the rank (number of independent columns) of 'm-by-n' matrix 'a', by counting the number of non-zero diagonal elements in 'u'. If 'rank < n', then 'a' is singular.
Return the rank (number of independent columns) of 'm-by-n' matrix 'a', by counting the number of non-zero diagonal elements in 'u'. If 'rank < n', then 'a' is singular.
Attributes
- See also
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en.wikipedia.org/wiki/Rank_%28linear_algebra%29
Fac_QR_RRorSVD
Solve for x in the equation lux = b via ly = b and ux = y. Return the solution vector x.
Solve for x in the equation lux = b via ly = b and ux = y. Return the solution vector x.
Value parameters
- b
-
the constant vector
Attributes
Split l into lower and upper triangular matrices by placing the upper portion in u and clearing, so l is properly lower triangular.
Split l into lower and upper triangular matrices by placing the upper portion in u and clearing, so l is properly lower triangular.
Attributes
Inherited methods
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the first matrix.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the first matrix.
Attributes
- Inherited from:
- Factorization
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t' or a = q * r, returning both the first and second matrices.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t' or a = q * r, returning both the first and second matrices.
Attributes
- Inherited from:
- Factorization
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the second matrix.
Factor a matrix into the product of two matrices, e.g., 'a = l * l.t', returning only the second matrix.
Attributes
- Inherited from:
- Factorization
Return whether the matrix has been factored has aleady been factored.
Return whether the matrix has been factored has aleady been factored.
Attributes
- Inherited from:
- Factorization
Reset by setting factored to false.
Inherited fields
Flag indicating whether the matrix has been factored