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Once y is also eliminated from the third row, the result is a system of linear equations in triangular form, and so the first part of the algorithm is complete. From a computational point of view, it is faster to solve the variables in reverse order, a process known as back-substitution. One sees the solution is z = −1, y = 3, and x = 2. So ...
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations.
The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows: In the first equation, solve for one of the variables in terms of the others. Substitute this expression into the remaining equations. This yields a system of equations with one fewer equation and unknown.
The substitution that is needed to solve this Bernoulli equation is = Substituting = + directly into the Riccati equation yields the linear equation ′ + (+) = A set of solutions to the Riccati equation is then given by = + where z is the general solution to the aforementioned linear equation.
In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging iterative process.
Storing an LU decomposition of () and using forward and back substitution to solve the system of equations at each iteration is also of complexity O(n 3) + k O(n 2). Inverting the matrix will typically have a greater initial cost, but lower cost at each iteration.
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
Developed by R.H. Bartels and G.W. Stewart in 1971, [1] it was the first numerically stable method that could be systematically applied to solve such equations. The algorithm works by using the real Schur decompositions of and to transform = into a triangular system that can then be solved using forward or backward substitution.
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