## x = A b; in Matlab

x = A b;

1. Is A square?
no => use QR to solve least squares problem.
2. Is A triangular or permuted triangular?
yes => sparse triangular solve
3. Is A symmetric with positive diagonal elements?
yes => attempt Cholesky after symmetric minimum degree.
4. Otherwise
=> use LU on A (:, colamd(A))

## Creating Sparse Finite-Element Matrices in MATLAB

I’m pleased to introduce Tim Davis as this week’s guest blogger. Tim is a professor at the University of Florida, and is the author or co-author of many of our sparse matrix functions (lu, chol, much of sparse backslash, ordering methods such as amd and colamd, and other functions such as etree and symbfact). He is also the author of a recent book, Direct Methods for Sparse Linear Systems, published by SIAM, where more details of MATLAB sparse matrices are discussed ( http://www.cise.ufl.edu/~davis ).

# Power iteration

In mathematics, the power iteration is an eigenvalue algorithm: given a matrix A, the algorithm will produce a number λ (theeigenvalue) and a nonzero vector v (the eigenvector), such that Av = λv.
The power iteration is a very simple algorithm. It does not compute a matrix decomposition, and hence it can be used when A is a very large sparse matrix. However, it will find only one eigenvalue (the one with the greatest absolute value) and it may converge only slowly.

# Krylov subspace

In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspacespanned by the images of b under the first r powers of A (starting from A0 = I), that is,
It is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper on this issue in 1931.[1]
Modern iterative methods for finding one (or a few) eigenvalues of large sparse matrices or solving large systems of linear equations avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors. Starting with a vector, b, one computes Ab, then one multiplies that vector by A to find A2b and so on. All algorithms that work this way are referred to as Krylov subspace methods; they are among the most successful methods currently available in numerical linear algebra.
Because the vectors tend very quickly to become almost linearly dependent, methods relying on Krylov subspace frequently involve some orthogonalization scheme, such as Lanczos iteration for Hermitian matrices or Arnoldi iteration for more general matrices.
The best known Krylov subspace methods are the ArnoldiLanczosConjugate gradientGMRES (generalized minimum residual),BiCGSTAB (biconjugate gradient stabilized), QMR (quasi minimal residual), TFQMR (transpose-free QMR), and MINRES (minimal residual) methods.

References

1. ^ Mike Botchev (2002). “A.N.Krylov, a short biography”.