Crout Matrix Decomposition

In linear algebra, the Crout matrix decomposition is an LU decomposition which decomposes a matrix into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix (P).

The Crout matrix decomposition algorithm differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout method returns a lower triangular matrix and a unit upper triangular matrix.

So, if a matrix decomposition of a matrix A is such that:

A = LDU

being L a unit lower triangular matrix, D a diagonal matrix and U a unit upper triangular matrix, then Doolittle's method produces

A = L(DU)

and Crout's method produces

A = (LD)U.

being L a lower triangular matrix, D a diagonal matrix and U a normalised upper triangular matrix


C implementation:

void crout (double **A, double **L, double **U, int n){ int i,j,k; double sum=0; for (i=0; iOctave/Matlab impelentation

function =LUdecompCrout(A) = size(A); for i=1:R L(i,1)=A(i,1); U(i,i)=1; end for j=2:R U(1,j)=A(1,j)/L(1,1); end for i=2:R for j=2:i L(i,j)=A(i,j)-L(i,1:j-1)*U(1:j-1,j); end for j=i+1:R U(i,j)=(A(i,j)-L(i,1:i-1)*U(1:i-1,j))/L(i,i); end end

Famous quotes containing the word matrix:

    “As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
    —Margaret Atwood (b. 1939)