A Large-Grained Parallel Algorithm for Nonlinear Eigenvalue Problems Using Complex Contour Integration Takeshi Amako, Yusaku Yamamoto and Shao-Liang Zhang Dept. of Computational Science & Engineering Nagoya University, Japan

Outline of the talk Introduction The nonlinear eigenvalue problem Existing algorithms Our objective The algorithm Formulation as a nonlinear equation Application of Kravanja et al s method Detecting and removing spurious eigenvalues Numerical results Accuracy of the computed eigenvalues Parallel performance Conclusion

Introduction The nonlinear eigenvalue problem Given A(z) C n × n, z: complex parameter Find z 1 C such that A(z 1 ) x = 0 has a nonzero solution x = x 1. z 1 and x 1 are called the eigenvalue and the corresponding eigenvector, respectively. Examples A(z) = A – zB + z 2 C : quadratic eigenvalue problem A(z) = A – zB + e z C : general nonlinear eigenvalue problem Applications Electronic structure calculation Nonlinear elasticity Theoretical fluid dynamics

Existing algorithms Multivariate Newtons method and its variants Locally quadratic convergence Requires good initial estimate both for z 1 and x 1. Nonlinear Arnoldi methods Nonlinear Jacobi-Davidson methods Efficient for large sparse matrices Not suitable for finding all eigenvalues within a specified region of the complex plane difficult to obtain

Our objective Let : closed Jordan curve on the complex plane, A(z) C n × n : analytical function of z in We propose an algorithm that can find all the eigenvalues within, and has large-grain parallelism. Im z Re z O Assumption: In the following, we mainly consider the case where is a circle centered at the origin and with radius r. Related work: Sakurai et al. propose an algorithm for linear generalized eigenvalue problems r

Our approach The basic idea Let f(z) = det(A(z)). Then f(z) is an analytical function of z in and the eigenvalues of A(z) are characterized as the zeros of f(z). Use Kravanjas method (Kravanja et al., 1999) to find the zeros of an analytic function.

Finding zeros of f(z) Let z 1, z 2,..., z m : zeros of f(z) in, and 1, 2,..., m : their multiplicity. Then f(z) can be written as Define the complex moments by Then f(z) = ×g(z) analytical and nonzero in analytical in unknowncomputable

Finding zeros of f(z) (cont'd) To extract information on {z k } from { p }, define the following matrices: Then it is easy to see that

Finding zeros of f(z) (cont'd) Noting that V m and D m are nonsingular, we have the following equivalence relation: That is, we can find the zeros of f(z) in by computing the complex moments 0, 1,..., 2m-1, constructing H m and H m <, and computing the eigenvalues of H m < – H m. is an eigenvalue of H m < – H m is an eigenvalue of m – k, = z k

Application to the nonlinear eigenvalue problem In our case, f(z) = det(A(z)) and By applying the trapezoidal rule with K points, we have where Im z Re z O

The algorithm The computationally intensive part. Large-grain parallelism

Detecting and removing spurious eigenvalues Usually, we do not know m, the number of eigenvalues of A(z) in, in advance and use some estimate M instead. When M > m, the eigenvalues of H m < – H m include spurious solutions that do not correspond to an eigenvalue of A(z). To detect them, we compute the corresponding eigenvector by inverse iteration and evaluate the relative residual defined by Of course, this quantity can also be used to check the accuracy of the computed eigenvalues. relative residual =

Numerical results Test problem A(z) = A – zI + B(z), where A(z) : real random nonsymmetric matrix B(z) : antidiagonal matrix with antidiagonal elements e z : parameter to specify the strength of nonlinearity Parameters n = 500, 1000, , 10 –4, 10 –3, 10 –2, 10 –1 Computational environment Fujitsu HPC2500 (SPARC 64IV), 1-16 processors Program written with C and MPI LAPACK routines were used to compute (A(z)) –1 and to compute the eigenvalues of H m < – H m.

Accuracy of the computed eigenvalues Parameters n = 500 and = 0.1 r = 0.85, K = 128 and M = 11. There are 7 eigenvalues in. Results Our algorithm succeeded in locating all the eigenvalues in. The relative residuals were all under 10 –10. Similar results for other cases. Im z Re z

Effect of K and M on the accuracy Effect of the number of sample points K Usually K=128 gives sufficient accuracy. Effect of the Hankel matrix size M It is better to take M a few more than the number of eigenvalues within (7 in this case). This is to mitigate the perturbation from eigenvalues outside. Residuals as a function of K. Residuals as a function of M. KM

Detecting and removing spurious eigenvalues Parameters n = 1000 and = 0.01 r = 0.7, K = 128 and M = 10. There are 9 eigenvalues in. Eigenvalues of H m < – H m 10 eigenvalues were found within. For 9 of the eigenvalues, the residual was less than 10 –11. For one eigenvalue, the residual was 10 –2. Im z Re z spurious eigenvalue

Parallel performance Performance on Fujitsu HPC2500 Matrix size: n = 500, 1000, 2000 Number of processors: P = 1, 2, 4, 8, 16 Execution time (sec) Number of processors Almost linear speedup was obtained in all cases due to large-grain parallelism.

Parallel performance (cont'd) Performance in a Grid environment Matrix size: n = 1000 Machine: Intel Xeon Cluster Master-worker type parallelization using OmniRPC (GridRPC) Execution time Number of processors Good scalability was obtained for up to 14 processors. 0:00:00 0:30:00 1:00:00 1:30:00 2:00: Speedup

Summary of this study We proposed a new algorithm for the nonlinear eigenvalue problem based on complex contour integration. Our algorithm can find all the eigenvalues within a closed curve on the complex plane. Moreover, it has large-grain parallelism and is expected to show excellent parallel performance. These advantages have been confirmed by numerical experiments.

Future work Performance evaluation on large-scale grid environments. Application to practical problems. Computation of scaling exponent in theoretical fluid dynamics Development of an efficient algorithm for computing