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On the Use of Sparse Direct Solver in a

Projection Method for Generalized Eigenvalue

Problems Using Numerical Integration

- Takamitsu Watanabe and Yusaku Yamamoto
- Dept. of Computational Science Engineering
- Nagoya University

Outline

- Background
- Objective of our study
- Projection method for generalized eigenvalue

problems using numerical integration - Application of the sparse direct solver
- Numerical results
- Conclusion

Background

- Generalized eigenvalue problems in quantum

chemistry and structural engineering

Given , find and

such that .

- Problem characteristics
- A and B are large and sparse.
- A is real symmetric and B is s.p.d.
- Eigenvalues are real.
- Eigenvalues in a specified interval are often

needed.

specified interval

real axis

eigenvalues

HOMO LUMO

Background (contd)

- A projection method using numerical integration
- Sakurai and Sugiura, A projection method for

generalized eigenvalue problems, - J. Comput. Appl. Math. (2003)
- Reduce the original problem to a small

generalized eigenvalue problem within a specified

region in the complex plane. - By solving the small problem, the eigenvalues

lying in the region can be obtained. - The main part of computation is to solve multiple

linear simultaneous equations. - Suited for parallel computation.

Small generalized eigenvalue problem within the

region

Original problem

region

Objective of our study

- Previous approach
- Solve the linear simultaneous equations by an

iterative method. - The number of iterations needed for convergence

differs from one simultaneous equations to

another. - This brings about load imbalance between

processors, decreasing parallel efficiency. - Our study
- Solve the linear simultaneous equations by a

sparse direct solver without pivoting. - Load balance will be improved since the

computational times are the same for all linear

simultaneous equations.

Projection method for generalized eigenvalue

problems using numerical integration

Suppose that has distinct

eigenvalues and that we need

that lie in a closed

curve .

Using two arbitrary complex vectors ,

define a complex function Then, f (z) can be

expanded as follows

?m1

.

?m2

C, g(z) polynomial in z.

,

Projection method for generalized eigenvalue

problems using numerical integration (contd)

Further define the moments by

and two Hankel matrices by

.

Th. are the m roots of

.

The original problem has been

reduced to a small problem

through contour integral.

Projection method for generalized eigenvalue

problems using numerical integration (contd)

Computation of the moments

- Set the path of integration G to a
- circle with center g and radius r .
- Approximate the integral using the
- trapezoidal rule.

The function values have to be computed for each

Path of integration

.

r

Solution of N independent linear simultaneous

equations is necessary (N 64 128).

Application of the sparse direct solver

- A and B sparse symmetric matrices, a

complex number

The coefficient matrix is a sparse complex

symmetric matrix.

- Application of the sparse direct solver
- For a sparse s.p.d. matrix, the sparse direct

solver provides an efficient way for solving the

linear simultaneous equations. - We adopt this approach by extending the sparse

direct solver to deal with complex symmetric

matrices.

The sparse direct solver

- Characteristics
- Reduce the computational work and memory

requirements of the Cholesky factorization by

exploiting the sparsity of the matrix. - Stability is guaranteed when the matrix is s.p.d.
- Efficient parallelization techniques are

available.

- Find a permutation of rows/columns that reduces
- computational work and memory requirements.

ordering

- Estimate the computational work and memory
- requirements.

symbolic factorization

- Prepare data structures to store the Cholesky
- factor.

Cholesky factorization

triangular solution

Extension of the sparse direct solver to complex

symmetric matrices

- Algorithm
- Extension is straightforward by using the

Cholesky factorization for complex symmetric

matrices. - Advantages such as reduced computational work,

reduced memory requirements and parallelizability

are carried over. - Accuracy and stability
- Theoretically, pivoting is necessary when

factorizing complex symmetric matrices. - Since our algorithm does not incorporate

pivoting, accuracy and stability is not

guaranteed. - We examine the accuracy and stability

experimentally by comparing the results with

those obtained using GEPP.

Numerical results

- Matrices used in the experiments

Harwell-Boeing Library

BCSSTK12 BCSSTK13

FMO

- For each matrix, we solve the equations with the

sparse direct solver - (with MD and ND ordering) and GEPP.
- We compare the computational time and accuracy

of the eigenvalues.

Computational time

Computational time (sec.) for one set of linear

simultaneous equations and speedup (PowerPC G5,

2.0GHz)

BCSSTK12 BCSSTK13

FMO

- The sparse direct solver is two to over one

hundred times faster than GEPP, depending on the

nonzero structure.

Accuracy of the eigenvalues (BCSSTK12)

Example of an interval containing 4 eigenvalues

Distribution of the eigenvalues and the specified

interval

eigenvalues specified interval

Relative errors in the eigenvalues for each

algorithm (N64)

- The errors were of the same order for all three

solvers. - Also, the growth factor for the sparse solver

was O(1).

Accuracy of the eigenvalues (BCSSTK13)

Example of an interval containing 3 eigenvalues

Distribution of the eigenvalues and the specified

interval

eigenvalues specified interval

Relative errors in the eigenvalues for each

algorithm (N64)

The errors were of the same order for all three

solvers.

Accuracy of the eigenvalues (FMO)

Example of an interval containing 4 eigenvalues

Distribution of the eigenvalues and the specified

interval

eigenvalues specified interval

Relative errors in the eigenvalues for each

algorithm (N64)

The errors were of the same order for all three

solvers.

Conclusion

- Summary of this study
- We applied a complex symmetric version of the

sparse direct solver to a projection method for

generalized eigenvalue problems using numerical

integration. - The sparse solver succeeded in solving the linear

simultaneous equations stably and accurately,

producing eigenvalues that are as accurate as

those obtained by GEPP. - Future work
- Apply our algorithm to larger matrices arising

from quantum chemistry applications. - Construct a hybrid method that uses an iterative

solver when the growth factor becomes too large. - Parallelize the sparse solver to enable more than

N processors to be used.

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