Journal: Proceedings of the National Academy of Sciences of the United States of America

Article Title: Inflationary dynamics for matrix eigenvalue problems

doi: 10.1073/pnas.0801047105

Figure Lengend Snippet: Comparison of convergence of the present inflation method with the Lanczos and Power methods. The computational time m (to calculate both the lowest eigenvalue and corresponding eigenvector) is the number of matrix-vector multiplications. Note that the original Lanczos method requires two matrix-vector multiplications per step if the eigenstate as well as the eigenvalue are to obtained at the end of the calculation without storing all intermediate vectors. The implicitly restarted Arnoldi method (MATLAB/ARPACK) behaves very similarly to Lanczos, and is not shown here (but see Fig. 1). (A and B) Results for some test matrices taken from ref. 11. (C) Results for a random sparse matrix. The matrix used in D corresponds to a model of strongly correlated spin-polarized fermions on a triangular lattice.

Article Snippet: The implicitly restarted Arnoldi method (MATLAB/ARPACK) behaves very similarly to Lanczos, and is not shown here (but see ). ( A and B ) Results for some test matrices taken from ref. 11 . ( C ) Results for a random sparse matrix.

Techniques: Comparison, Plasmid Preparation

Journal: Proceedings of the National Academy of Sciences of the United States of America

Article Title: Inflationary dynamics for matrix eigenvalue problems

doi: 10.1073/pnas.0801047105

Figure Lengend Snippet: The convergence of the inflation method for the lowest four eigenpairs of a test matrix (11) is compared with the implicitly restarted Arnoldi method (as implemented in MATLAB/ARPACK). Exact eigenvalues are indicated by horizontal lines. In the inflation method, we diagonalize in a six-dimensional basis after every 6 dynamical steps. In the Arnoldi calculation, we use a basis of size 12. In each case, the computational time m represents the number of matrix-vector multiplications (i.e., we do not multiply Arnoldi iterations by 2, and we do count every matrix-vector multiplication on the horizontal axis; e.g., when inflating 6 eigenvalues, we count 6 matrix-vector multiplcations per iteration step).

Article Snippet: The implicitly restarted Arnoldi method (MATLAB/ARPACK) behaves very similarly to Lanczos, and is not shown here (but see ). ( A and B ) Results for some test matrices taken from ref. 11 . ( C ) Results for a random sparse matrix.

Techniques: Plasmid Preparation

Journal: Bulletin of mathematical biology

Article Title: Feedback Regulation in a Cancer Stem Cell Model can Cause an Allee Effect

doi: 10.1007/s11538-016-0161-5

Figure Lengend Snippet: The separatrix and its approximations for system (9). a, b pplane8 plots of (9) with parameters a Pr1 and b Pr2. The descending green curve in each graph is the pplane approximation of the stable manifold (separatrix M), and the ascending orange curve is the pplane approximation of the unstable manifold, U. The blue curves represent forward solutions of the system. The red curves represent the contours at which the self-renewal probability p = 0.5. Note that solutions to the right of the separatrix tend to nonzero (S, a), whereas solutions to the left tend to P1(0, 0). In each region, however, if the current values (S, a) are above the p = 0.5 contour, then S increases in time. Analogously, if (S, a) lies below the p = 0.5 contour, then S decreases. c, d The separatrices predicted by pplane8 (large dash, green) of c Pr1 and d Pr2 are plotted along with the SMT approximation of the separatrix, M* (solid black), the quadratic approximation, Mq∗ (small dash, blue), and the linear approximation, Ml∗ (cross, red) (Color figure online)

Article Snippet: Since it is cumbersome to further improve the approximation iteratively by the SMT, we check whether M * is a good approximation of M by comparing M * with the separatrix predicted for a given set of parameters by a numerical ODE analysis program (here we have used pplane8 in MATLAB; Arnold and Polking 1999 ).

Techniques: