 # Ublas :

From: Karl Meerbergen (Karl.Meerbergen_at_[hidden])
Date: 2005-01-13 10:35:51

Consulting a book about numerics, you would see that the determinant is
not a good measure of singularity. It is better to compute the condition
number. There are estimators in LAPACK, functions xTRCON(). But no
bindings are provided for these estimators, I think.

Karl

Gunter Winkler wrote:

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>On Thursday 13 January 2005 14:39, Rakesh K Sinha wrote:
>
>
>>I had posted this to the Boost Archives list sometime before. Since I did
>>not get any response there, I thought this list might be more appopriate
>>and hence am posting it here, again.
>>
>>For a particular application of mine, I ned to solve the equation Ax =
>>B,
>>i.e. x = inv(A) * B
>>
>>
>
>I would generally avoid the computation of any inverse. uBLAS currently does
>not provide and probably will not provide such a function.
>
>The way to solve linear equations is to first compute the LU factorization of
>the matrix A and then solve to triangular systems. The code looks like
>
>const int n=10;
>permutation_matrix<double> P(n);
>matrix<double> A(n,n);
>vector<double> x(n);
>vector<double> rhs(n);
>
>// fill matrix and rhs
>
>lu_factorize(A,P);
>x = rhs;
>lu_substitute(A,P,rhs);
>
>
>
>>This fits my need perfectly.
>>
>>
>
>in order to solve triangular systems you can use the solve() funtion. You can
>find examples in lu.hpp. If the documentation is missing or fuzzy, please
>send me a mail.
>
>
>
>>* Are there any other caveats with this method like it works only for
>>dense matrices. How does it scale to sparse matrices ?
>>
>>
>
>Yes. lu_factorize() works only for dense matrices reliably. The trianglular
>solves should work for any matrix type, but may fail for adapted matrices
>like trianglur_adaptor<compressed_matrix<...>>. We would be very pleased if
>you provide a short (regression-)test program if there are any problems.
>
>
>
>>* Also what does that lower_tag() essentially imply ? What is its
>>mathematical significance ?
>>
>>
>
>This should be explained in the documentation of matrix expressions

>
>There are lower (i<=j), stric_lower (i<j) and unit_lower (i<j + unit diagonal)
>
>
>
>>* Is there any function to calculate the singularity of a square matrix
>>
>>
>
>No. But you can do a LU decompositon, if it fails the determinant is zero,
>otherwise the product of the diagonal elements of the LU-Matrix give the
>absolute value of the determinant. The sign can be extracted from the
>characteristic of the permutation. More details should be in any good book
>about numerics.
>
>short: P * A = L * U; det(A) = prod(U_ii,i=1..n) * sign(P);
>
>(For solving sparse linear systems I prefer iterative methods.)
>
>mfg
>Gunter
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