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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 ...
Forward and back substitution [ edit ] A matrix equation in the form L x = b {\displaystyle L\mathbf {x} =\mathbf {b} } or U x = b {\displaystyle U\mathbf {x} =\mathbf {b} } is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices.
In both cases we are dealing with triangular matrices (L and U), which can be solved directly by forward and backward substitution without using the Gaussian elimination process (however we do need this process or equivalent to compute the LU decomposition itself).
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. In 1979, G. Golub , C. Van Loan and S. Nash introduced an improved version of the algorithm, [ 2 ] known as the Hessenberg–Schur algorithm.
The first pass goes forward in time while the second goes backward in time; hence the name forward–backward algorithm. The term forward–backward algorithm is also used to refer to any algorithm belonging to the general class of algorithms that operate on sequence models in a forward–backward manner. In this sense, the descriptions in the ...
A classical algorithm for the numerical solution of the Sylvester equation is the Bartels–Stewart algorithm, which consists of transforming and into Schur form by a QR algorithm, and then solving the resulting triangular system via back-substitution. This algorithm, whose computational cost is () arithmetical operations, [citation needed] is ...
A first sweep eliminates the 's, and then an (abbreviated) backward substitution produces the solution. Examples of such matrices commonly arise from the discretization of 1D Poisson equation and natural cubic spline interpolation .
This algorithm uses a binary reduction tree to compute local householder QR decomposition at each node in the forward pass, and re-constitute the Q matrix in the backward pass. The binary tree structure aims at decreasing the amount of communication between processor to increase performance.