The Complete Codes

DOOLITTLE Algorithm In the numerical method Doolittle Algorithm : LU Decomposition Algorithm (where LU stands for Lower and Upper and also called LU factorization Algorithm) factors a matrix as the product of a lower triangular matrix and an upper triangular matrix. Computers usually solve square systems of linear equations using the LU decomposition, and it is also a key step when inverting a matrix, or computing the determinant of a matrix. Let A be a square matrix. An LU factorization algorithm refers to the factorization of A, with proper row and/or column orderings or permutations, into two factors, a lower triangular matrix L and an upper triangular matrix U,  A=LU . Assume that  A  has a Crout factorization  A  =  LU . 

Improved Euler’s Method The Improved Euler’s method, also known as the Heun formula or the average slope method, gives a more accurate approximation than the Euler rule and gives an explicit formula for computing  y n+1 . The basic idea is to correct some errors of the original Euler's method. The syntax of the Improved Euler’s method is similar to that of the trapezoid rule, but the  y  value of the function in terms of  y n+1  consists of the sum of the  y  value and the product of  h  and the function in terms of  x n  and  y n .