CHOLESKI'S ALGORITHM
*
* To factor the positive definite n by n matrix A into LL**T,
* where L is lower triangular.
*
* INPUT: the dimension n; entries A(I,J), 1<=I, J<=n of A.
*
* OUTPUT: the entries L(I,J), 1<=J<=I, 1<=I<=n of L.
*
* the entries of U = L**T are U(I,J) = L(J,I), I<=J<=n, 1<=I<=n
CROUT
REDUCTION FOR TRIDIAGONAL LINEAR SYSTEMS ALGORITHM
*
* To solve the n x n linear system
*
* E1: A[1,1] X[1] + A[1,2] X[2] = A[1,n+1]
* E2: A[2,1] X[1] + A[2,2] X[2] + A[2,3] X[3] = A[2,n+1]
* :
* .
* E(n): A[n,n-1] X[n-1] + A[n,n] X[n] = A[n,n+1]
*
* INPUT: the dimension n; the entries of A.
*
* OUTPUT: the solution X(1), ..., X(N).
DIRECT
FACTORIZATION ALGORITHM
*
* To factor the n by n matrix A = (A(I,J)) into the product of the
* lower triangular matrix L = (L(I,J)) and U = (U(I,J)), that is
* A = LU, where the main diagonal of either L or U consists of all ones:
*
* INPUT: dimension n; the entries A(I,J), 1<=I, J<=n, of A;
* the diagonal L(1,1), ..., L(N,N) of L or the diagonal
* U(1,1), ..., U(N,N) of U.
*
* OUTPUT: the entries L(I,J), 1<=J<=I, 1<=I<=n of L and the entries
* U(I,J), I<=J<=n, 1<=I<=n of U.
GAUSSIAN
ELIMINATION WITH BACKWARD SUBSTITUTION ALGOTITHM
*
* To solve the n by n linear system
*
* E1: A[1,1] X[1] + A[1,2] X[2] +...+ A[1,n] X[n] = A[1,n+1]
* E2: A[2,1] X[1] + A[2,2] X[2] +...+ A[2,n] X[n] = A[2,n+1]
* :
* .
* EN: A[n,1] X[1] + A[n,2] X[2] +...+ A[n,n] X[n] = A[n,n+1]
*
* INPUT: number of unknowns and equations n; augmented
* matrix A = (A(I,J)) where 1<=I<=n and 1<=J<=n+1.
*
* OUTPUT: solution x(1), x(2),...,x(n) or a message that the
* linear system has no unique solution.
GAUSSIAN
ELIMINATION WITH PARTIAL PIVOTING ALGORITHM
*
* To solve the n by n linear system
*
* E1: A[1,1] X[1] + A[1,2] X[2] +...+ A[1,n] X[n] = A[1,n+1]
* E2: A[2,1] X[1] + A[2,2] X[2] +...+ A[2,n] X[n] = A[2,n+1]
* :
* .
* EN: A[n,1] X[1] + A[n,2] X[2] +...+ A[n,n] X[n] = A[n,n+1]
*
* INPUT: number of unknowns and equations n; augmented
* matrix A = (A(I,J)) where 1<=I<=n and 1<=J<=n+1.
*
* OUTPUT: solution x(1), x(2),...,x(n) or a message that the
* linear system has no unique solution.
GAUSSIAN
ELIMINATION WITH SCALED PARTIAL PIVOTING ALGORITHM
* To solve the n by n linear system
*
* E1: A[1,1] X[1] + A[1,2] X[2] +...+ A[1,n] X[n] = A[1,n+1]
* E2: A[2,1] X[1] + A[2,2] X[2] +...+ A[2,n] X[n] = A[2,n+1]
* :
* .
* EN: A[n,1] X[1] + A[n,2] X[2] +...+ A[n,n] X[n] = A[n,n+1]
*
* INPUT: number of unknowns and equations n; augmented
* matrix A = (A(I,J)) where 1<=I<=n and 1<=J<=n+1.
*
* OUTPUT: solution x(1), x(2),...,x(n) or a message that the
* linear system has no unique solution.
LDL-t ALGORITHM
*
* To factor the positive definite n by n matrix A into LDL**T,
* where L is a lower triangular matrix with ones along the diagonal
* and D is a diagonal matrix with positive entries on the
* diagonal.
*
* INPUT: the dimension n; entries A(I,J), 1<=I, J<=n of A.
*
* OUTPUT: the entries L(I,J), 1<=J<I, 1<=I<=N of L and D(I),
* 1<=I<=n of D.