crout¶

Purpose¶

Computes the Crout decomposition of a square matrix without row pivoting, such that: \(X = LU\).

Format¶

y = crout(x)¶

Parameters:	x (NxN matrix) – square nonsingular matrix.
Returns:	y (NxN matrix) – containing the lower (\(L\)) and upper (\(U\)) matrices of the Crout decomposition of x. The main diagonal of y is the main diagonal of the lower matrix L. The upper matrix has an implicit main diagonal of ones. Use `lowmat()` and `upmat1()` to extract the \(L\) and \(U\) matrices from y.

Examples¶

X = { 1 2 -1,
      2 3 -2,
      1 -2 1 };

// Perform crout decomposition
y = crout(x);

// Extract lower triangle of 'y' and assign it to 'L'
L = lowmat(y);

/*
** Extract upper triangle of 'y', fill the diagonal with
** ones and assign it to 'L'
*/
U = upmat1(y);

After the code above:

    1.0  2.0 -1.0       1.0  0.0  0.0       1.0  2.0 -1.0
y = 2.0 -1.0  0.0   L = 2.0 -1.0  0.0   U = 0.0  1.0  0.0
    1.0 -4.0  2.0       1.0 -4.0  2.0       0.0  0.0  1.0

Remarks¶

Since it does not do row pivoting, it is intended primarily for teaching purposes. See croutp() for a decomposition with pivoting.