Struct rulinalg::matrix::decomposition::FullPivLu

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pub struct FullPivLu<T> { /* private fields */ }

Expand description

LU decomposition with complete pivoting.

For any square matrix A, there exist two permutation matrices P and Q, a lower triangular matrix L and an upper triangular matrix U such that

PAQ = LU.

Unlike the LU decomposition computed with partial pivoting, this decomposition is stable for singular matrices. It is also a rank- revealing decomposition.

See PartialPivLu for applications of LU decompositions in general.

Implementations

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impl<T: 'static + Float> FullPivLu<T>

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pub fn decompose(matrix: Matrix<T>) -> Result<Self, Error>

Performs the decomposition.

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impl<T> FullPivLu<T>where
T: Any + Float,

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pub fn solve(&self, b: Vector<T>) -> Result<Vector<T>, Error>

Solves the linear system Ax = b.

Here, A is the decomposed matrix satisfying PAQ = LU. Note that this method is particularly well suited to solving multiple such linear systems involving the same A but different b.

Errors

If the matrix is very ill-conditioned, the function might fail to obtain the solution to the system.

Panics

The right-hand side vector b must have compatible size.

Examples

let x = Matrix::identity(4);
let lu = FullPivLu::decompose(x).unwrap();
let b = vector![3.0, 4.0, 2.0, 1.0];
let y = lu.solve(b)
          .expect("Matrix is invertible.");
assert_vector_eq!(y, vector![3.0, 4.0, 2.0, 1.0], comp = float);

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pub fn inverse(&self) -> Result<Matrix<T>, Error>

Computes the inverse of the matrix which this LUP decomposition represents.

Errors

The inversion might fail if the matrix is very ill-conditioned. The inversion fails if the matrix is not invertible.

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pub fn det(&self) -> T

Computes the determinant of the decomposed matrix.

Empty matrices are considered to have a determinant of 1.0.

Panics

If the underlying matrix is non-square.

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pub fn rank(&self) -> usize

Computes the rank of the decomposed matrix.

Examples

let x = matrix![1.0, 2.0, 3.0;
                4.0, 5.0, 6.0;
                5.0, 7.0, 9.0];
let lu = FullPivLu::decompose(x).unwrap();
assert_eq!(lu.rank(), 2);

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pub fn is_invertible(&self) -> bool

Returns whether the matrix is invertible.

Empty matrices are considered to be invertible for the sake of this function.

Examples

let x = Matrix::<f64>::identity(4);
let lu = FullPivLu::decompose(x).unwrap();
assert!(lu.is_invertible());

let y = matrix![1.0, 2.0, 3.0;
                4.0, 5.0, 6.0;
                5.0, 7.0, 9.0];
let lu = FullPivLu::decompose(y).unwrap();
assert!(!lu.is_invertible());