Concepts inFORTRAN subroutines for general Toeplitz systems
Toeplitz matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: Any n×n matrix A of the form is a Toeplitz matrix. If the i,j element of A is denoted Ai,j, then we have
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Fortran
Fortran (previously FORTRAN) is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing.
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Subroutine
In computer science, a subroutine, also termed procedure, function, routine, method, or subprogram, is a part of source code within a larger computer program that performs a specific task and is relatively independent of the remaining code. As the name subprogram suggests, a subroutine behaves in much the same way as a computer program that is used as one step in a larger program or another subprogram.
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Levinson recursion
Levinson recursion or Levinson-Durbin recursion is a procedure in linear algebra to recursively calculate the solution to an equation involving a Toeplitz matrix. The algorithm runs in ¿(n) time, which is a strong improvement over Gauss-Jordan elimination, which runs in ¿(n). Newer algorithms, called asymptotically fast or sometimes superfast Toeplitz algorithms, can solve in ¿(n logn) for various p (e.g. p = 2, p = 3).
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Condition number
In the field of numerical analysis, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument. The "function" is the solution of a problem and the "arguments" are the data in the problem. A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.
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Numerical stability
In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is derived from the accuracy of the algorithm. An opposite phenomenon is instability.
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Positive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form (or a sesquilinear form in the complex case). The proper definition of positive-definite is unambiguous for Hermitian matrices, but there is no agreement in the literature on how this should be extended for non-Hermitian matrices, if at all. (See the section Non-Hermitian matrices below.)
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Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Let A be a symmetric matrix. Then: The entries of a symmetric matrix are symmetric with respect to the main diagonal (top left to bottom right). So if the entries are written as A = (aij), then for all indices i and j. The following 3×3 matrix is symmetric: Every diagonal matrix is symmetric, since all off-diagonal entries are zero.
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