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1 vote

Wanted: sequences of linear systems for recycling Krylov solver analysis

One problem I have worked on is dispersion relations, particularly for periodic materials. Consider a wave-like differential equation $$\mathcal{L} u(\mathbf{x}) = \omega^2 u(\mathbf{x}) \, ,\quad \...
nicoguaro's user avatar
  • 8,534
1 vote

Wanted: sequences of linear systems for recycling Krylov solver analysis

If you are willing to move away from PDEs, matrices arising from interior point methods could be a good candidate for your application. Interior point methods are used to solve convex optimization ...
filikat's user avatar
  • 31
2 votes

Getting singular matrices for lid driven cavity problem

Concerning why the GMRES finds a solution even if the matrix is singular (or nearly so), GMRES is able to find solutions of singular systems, under some conditions on the range and null space of the ...
filikat's user avatar
  • 31
3 votes

Solving linear system of equations with constraints on unknowns

I will use the notation $U_k$ and $U_{k,i}$ for rows and elements of matrix $U$, rather than small letter $u$, to avoid possible confusion with the $u_k$ notation in $y_k=au_k+bu_{k-1}+cu_{k−2}+pa^2u^...
jdgleeson's user avatar
  • 671
3 votes

Getting singular matrices for lid driven cavity problem

Determinants are a poor way to check singularity, since they are badly scaled. For instance, computing the determinant of $0.1I$ in Float64 (double precision) ...
Federico Poloni's user avatar
5 votes

On the calculation of the first m generalized eigenvectors

You can avoid inverting the entire matrix $B$ and instead deal only with triangular inversions/solves. Since $B$ is spd, it admits a Cholesky decomposition $B = LL^T$, where $L$ is lower triangular ...
whpowell96's user avatar
  • 2,656
3 votes
Accepted

Products of the Householder matrices during QR decomposition

You compute recursively as follows: $$ A_1 := (I-\beta_1 v_1 v_1^T)A = A - \beta_1 v_1 (v_1^TA); $$ $$ A_2 := (I-\beta_2 v_2 v_2^T)A_1 = A_1 - \beta_2 v_2 (v_2^TA_1); $$ $$ A_3 := (I-\beta_3 v_3 v_3^T)...
Federico Poloni's user avatar

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