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The process is so called because for lower triangular matrices, one first computes , then substitutes that forward into the next equation to solve for , and repeats through to . In an upper triangular matrix, one works backwards, first computing x n {\displaystyle x_{n}} , then substituting that back into the previous equation to solve for x n ...
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 ...
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.
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 last equation involves only one unknown. Solving it in turn reduces the next last equation to one unknown, so that this backward substitution can be used to find all of the unknowns: = ′ = ′ ′ +; =,, …,
The backward algorithm complements the forward algorithm by taking into account the future history if one wanted to improve the estimate for past times. This is referred to as smoothing and the forward/backward algorithm computes (|:) for < <. Thus, the full forward/backward algorithm takes into account all evidence.
Once this decomposition is calculated, linear systems can be solved more efficiently, by a simple technique called forward and back substitution. Likewise, inverses of triangular matrices are algorithmically easier to calculate. The Gaussian elimination is a similar algorithm; it transforms any matrix to row echelon form. [51]
The Cholesky decomposition is commonly used in the Monte Carlo method for simulating systems with multiple correlated variables. The covariance matrix is decomposed to give the lower-triangular L . Applying this to a vector of uncorrelated observations in a sample u produces a sample vector Lu with the covariance properties of the system being ...