class Fac_LU[MatT <: MatriD] extends Factorization with Error
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
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- new Fac_LU(a: MatT)
- a
the given m-by-n rectangular matrix
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- def bsolve(y: VectoD): VectorD
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.
- y
the constant vector
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- def det: Double
Compute the determinant of matrix 'a'.
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).
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- def factor(): Fac_LU[MatT]
Factor matrix 'a' into the product of 'l' and 'u'.
Factor matrix 'a' into the product of 'l' and 'u'.
- Definition Classes
- Fac_LU → Factorization
- def factor1(): MatriD
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.
- Definition Classes
- Factorization
- def factor12(): (MatriD, MatriD)
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.
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- Factorization
- def factor2(): MatriD
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.
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- Factorization
- val factored: Boolean
Flag indicating whether the matrix has been factored
Flag indicating whether the matrix has been factored
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- Factorization
- def factors: (MatriD, MatriD)
Return the 'l' and 'u' matrices.
Return the 'l' and 'u' matrices.
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- Fac_LU → Factorization
- final def flaw(method: String, message: String): Unit
Show the flaw by printing the error message.
Show the flaw by printing the error message.
- method
the method where the error occurred
- message
the error message
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- def inverse: MatriD
Compute the inverse of matrix 'a' from the LU Factorization.
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- def permute(d: VectoD): VectoD
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'.
- d
the vector to permute
- def permute(c: MatriD): Unit
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'.
- c
the matrix to permute
- def rank: Int
Return the rank (number of independent columns) of 'm-by-n' matrix 'a', by counting the number of non-zero diagonal elements in 'u'.
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.
- See also
en.wikipedia.org/wiki/Rank_%28linear_algebra%29
Fac_QR_RRorSVD
- def solve(b: VectoD): VectoD
Solve for 'x' in the equation 'l*u*x = b' via 'l*y = b' and 'u*x = y'.
Solve for 'x' in the equation 'l*u*x = b' via 'l*y = b' and 'u*x = y'. Return the solution vector 'x'.
- b
the constant vector
- Definition Classes
- Fac_LU → Factorization
- def split(): Unit
Split 'l' into lower and upper triangular matrices by placing the upper portion in 'u' and clearing, so 'l' is properly lower triangular.
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