The helical coordinate

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## Cholesky decomposition

Conceptually, the simplest computational method of spectral factorization might be Cholesky decomposition.'' For example, the matrix of (15) could have been found by Cholesky factorization of (14). The Cholesky algorithm takes a positive-definite matrix and factors it into a triangular matrix times its transpose, say .

It is easy to reinvent the Cholesky factorization algorithm. To do so, simply write all the components of a triangular matrix and then explicitly multiply these elements times the transpose matrix . You then find you have everything you need to recursively build the elements of from the elements of . Likewise, for a matrix, etc.

The case shows that the Cholesky algorithm requires square roots. Matrix elements are not always numbers. Sometimes, matrix elements are polynomials, such as -transforms. To avoid square roots, there is a variation of the Cholesky method. In this variation, we factor into , where is a diagonal matrix.

Once a matrix has been factored into upper and lower triangles, solving simultaneous equations is simply a matter of two back substitutions: (We looked at a special case of back substitution with Equation ().) For example, we often encounter simultaneous equations of the form . Suppose the positive-definite matrix has been factored into triangle form . To find , first backsolve for the vector . Then, we backsolve . When happens to be a band matrix, then the first back substitution is filtering down a helix, and the second is filtering back up it. Polynomial division is a special case of back substitution.

Poisson's equation requires boundary conditions, that we can honor when we filter starting from both ends. We cannot simply solve Poisson's equation as an initial-value problem. We could insert the Laplace operator into the polynomial division program, but the solution would diverge.

 The helical coordinate

Next: Toeplitz methods Up: INVERSE FILTERS AND OTHER Previous: Uniqueness and invertability

2015-03-25