Difference between revisions of "Decompose"
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[[category:Matrix Functions]] | [[category:Matrix Functions]] | ||
| + | [[Category:Functions that operate on complex numbers]] | ||
[[Category:Doc Status C]] <!-- For Lumina use, do not change --> | [[Category:Doc Status C]] <!-- For Lumina use, do not change --> | ||
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such that L * L' = C, where L' is the transpose of L. | such that L * L' = C, where L' is the transpose of L. | ||
| − | = Testing for | + | = Testing for positive definiteness = |
To avoid an error, the following [[User-Defined Functions|UDF]] can be used to test for positive definiteness: | To avoid an error, the following [[User-Defined Functions|UDF]] can be used to test for positive definiteness: | ||
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Description: Returns true (1) if A is positive-definite, 0 (false) otherwise. | Description: Returns true (1) if A is positive-definite, 0 (false) otherwise. | ||
Definition: [[Min]]([[EigenDecomp]](A+[[Transpose]](A,I,J),I,J)[.Item='value'],J)>0 | Definition: [[Min]]([[EigenDecomp]](A+[[Transpose]](A,I,J),I,J)[.Item='value'],J)>0 | ||
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| + | = Complex numbers = | ||
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| + | ''new in [[Analytica 4.5]]''. | ||
| + | When «C» contains complex numbers, then the matrix must be [http://en.wikipedia.org/wiki/Hermitian_matrix Hermitian] and positive definite. The Hermitian condition is the same as the symmetric condition for real-valued matrices, but requires the transpose to be the complex conjugate of the matrix. | ||
= See Also = | = See Also = | ||
Revision as of 20:34, 15 April 2013
Decompose(C,I,J)
Returns the Cholesky decomposition (square root) matrix of matrix C along dimensions I and J. Matrix C must be symmetric and positive-definite. (Positive-definite means that v * C * v > 0, for all vectors v.)
Cholesky decomposition computes a lower diagonal matrix L such that L * L' = C, where L' is the transpose of L.
Testing for positive definiteness
To avoid an error, the following UDF can be used to test for positive definiteness:
Function IsPosDefinite( A : Number[I,J] ; I,J : Index ) Description: Returns true (1) if A is positive-definite, 0 (false) otherwise. Definition: Min(EigenDecomp(A+Transpose(A,I,J),I,J)[.Item='value'],J)>0
Complex numbers
new in Analytica 4.5. When «C» contains complex numbers, then the matrix must be Hermitian and positive definite. The Hermitian condition is the same as the symmetric condition for real-valued matrices, but requires the transpose to be the complex conjugate of the matrix.
See Also
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