7.10 Row-Based Algorithms
procedure
(matrix-gauss-elim M [ jordan? unitize-pivot? pivoting]) → (Values (Matrix Number) (Listof Index)) M : (Matrix Number) jordan? : Any = #f unitize-pivot? : Any = #f pivoting : (U 'first 'partial) = 'partial
If jordan? is true, row operations are done both above and below the pivot. If unitize-pivot? is true, the pivot’s row is scaled so that the pivot value is 1. When both are true, the algorithm is called Gauss-Jordan elimination, and the result matrix is in reduced row echelon form.
If pivoting is 'first, the first nonzero entry in the current column is used as the pivot. If pivoting is 'partial, the largest-magnitude nonzero entry is used, which improves numerical stability on average when M contains inexact entries.
The first return value is the result of Gaussian elimination.
The second return value is a list of indexes of columns that did not have a nonzero pivot.
See matrix-row-echelon for examples.
procedure
(matrix-row-echelon M [ jordan? unitize-pivot? pivoting]) → (Matrix Number) M : (Matrix Number) jordan? : Any = #f unitize-pivot? : Any = #f pivoting : (U 'first 'partial) = 'partial
> (define M (matrix [[2 1 -1] [-3 -1 2] [-2 1 2]])) > (matrix-row-echelon M) - : (Array Real)
(mutable-array #[#[-3 -1 2] #[0 5/3 2/3] #[0 0 1/5]])
> (matrix-row-echelon M #t) - : (Array Real)
(mutable-array #[#[-3 0 0] #[0 5/3 0] #[0 0 1/5]])
> (matrix-identity? (matrix-row-echelon M #t #t)) - : Boolean
#t
> (define B (col-matrix [8 -11 -3])) > (define MB (matrix-augment (list M B))) > (matrix-col (matrix-row-echelon MB #t #t) 3) - : (Array Real)
(array #[#[2] #[3] #[-1]])
> (matrix-solve M B) - : (Array Real)
(array #[#[2] #[3] #[-1]])
> (define MI (matrix-augment (list M (identity-matrix 3)))) > (submatrix (matrix-row-echelon MI #t #t) (::) (:: 3 #f))
- : #(struct:Array
(Indexes Index (Boxof Boolean) (-> Void) (-> Indexes Real))
#<syntax:.../array/typed-array-struct.rkt:56:13 prop:equal+hash>
#<syntax:.../array/typed-array-struct.rkt:55:13 prop:custom-write>
#<syntax:.../array/typed-array-struct.rkt:54:13 prop:custom-print-quotable>)
(array #[#[4 3 -1] #[-2 -2 1] #[5 4 -1]])
> (matrix-inverse M) - : (Array Real)
(array #[#[4 3 -1] #[-2 -2 1] #[5 4 -1]])
procedure
(matrix-lu M [fail])
→ (Values (U F (Matrix Number)) (Matrix Number)) M : (Matrix Number) fail : (-> F) = (λ () (error ...))
Because matrix-lu returns a unit lower-triangular matrix (i.e. a lower-triangular matrix with only ones on the diagonal), the decomposition is unique if it exists.
> (define-values (L U) (matrix-lu (matrix [[4 3] [6 3]]))) > (values L U) - : (values (Array Real) (Array Real))
(mutable-array #[#[1 0] #[3/2 1]])
(mutable-array #[#[4 3] #[0 -3/2]])
> (matrix* L U) - : (Array Real)
(array #[#[4 3] #[6 3]])