This chapter provides routines for computing some eigenvalues and eigenvectors of large-scale (sparse) standard and generalized eigenvalue problems. It provides routines for:
solution of symmetric eigenvalue problems;
solution of nonsymmetric eigenvalue problems;
solution of generalized symmetric-definite eigenvalue problems;
solution of generalized nonsymmetric eigenvalue problems;
solution of polynomial eigenvalue problems;
partial singular value decomposition.
Routines are provided for both real and complex data.
The routines in this chapter whose short names begin with either f12a or f12f have all been derived from the
ARPACK software suite (see Lehoucq et al. (1998)),
a collection of Fortran 77 subroutines designed to solve large scale eigenvalue problems. The interfaces provided in this chapter have been chosen to combine ease of use with the flexibility of the original
software. The underlying iterative methods and algorithms remain essentially the same as those in ARPACK and are described fully in Lehoucq et al. (1998).
The algorithms used in the ARPACK routines are based upon an algorithmic variant of the Arnoldi process called the Implicitly Restarted Arnoldi Method. For symmetric matrices, this reduces to a variant of the Lanczos process called the Implicitly Restarted Lanczos Method. These variants may be viewed as a synthesis of the Arnoldi/Lanczos process with the Implicitly Shifted technique that is suitable for large scale problems. For many standard problems, a matrix factorization is not required. Only the action of the matrix on a vector is needed.
The ARPACK routines can be used to find the eigenvalues with the largest and/or smallest magnitudes, real part or imaginary part.
The routines in this chapter whose short names begin with ‘f12j’ have been derived from the FEAST software suite (see Polizzi (2009)). FEAST is a general purpose eigensolver for standard, generalized and polynomial eigenvalue problems. It is suitable for both sparse and dense matrices, and routines are available for real, complex, symmetric, Hermitian and non-Hermitian eigenvalue problems. The FEAST algorithm requires you to specify a particular region of interest in the complex plane within which eigenvalues are sought. The algorithm then performs a numerical quadrature computation, involving solving linear systems along a complex contour around the region of interest.
2Background to the Problems
This section is only a brief introduction to the solution of large-scale eigenvalue problems. For a more detailed discussion see, for example, Saad (1992) or Lehoucq (1995) in addition to Lehoucq et al. (1998). The basic factorization techniques and definitions of terms used for the different problem types are given in Section 2 in the F08 Chapter Introduction.
2.1Sparse Matrices and their Storage
A matrix may be described as sparse if the number of zero elements is so large that it is worthwhile using algorithms which avoid computations involving zero elements.
If is sparse, and the chosen algorithm requires the matrix coefficients to be stored, a significant saving in storage can often be made by storing only the nonzero elements. A number of different formats may be used to represent sparse matrices economically. These differ according to the amount of storage required, the amount of indirect addressing required for fundamental operations such as matrix-vector products, and their suitability for vector and/or parallel architectures. For a survey of some of these storage formats see Barrett et al. (1994).
of the routines in this chapter have been designed to be independent of the matrix storage format. This allows you to choose your own preferred format, or to avoid storing the matrix altogether.
Other routines are general purpose, which are easier to use, but are based on fixed storage formats. One such format is currently provided. This is the banded coordinate storage format as used in Chapters F07 and F08 (LAPACK) for storing general banded matrices.
2.2Symmetric Eigenvalue Problems
The symmetric eigenvalue problem is to find the eigenvalues, , and corresponding eigenvectors, , such that
For the Hermitian eigenvalue problem we have
For both problems the eigenvalues are real.
The basic task of the symmetric eigenproblem routines is to compute some of the values of and, optionally, corresponding vectors for a given matrix . For example, we may wish to obtain the first ten eigenvalues of largest magnitude, of a large sparse matrix .
This section is concerned with the solution of the generalized eigenvalue problems , , and , where and are real symmetric or complex Hermitian and is positive definite. Each of these problems can be reduced to a standard symmetric eigenvalue problem, using a Cholesky factorization of as either or ( or in the Hermitian case).
With , we have
Hence the eigenvalues of are those of , where is the symmetric matrix and . In the complex, case is Hermitian with and .
The basic task of the generalized symmetric eigenproblem routines is to compute some of the values of and, optionally, corresponding vectors for a given matrix . For example, we may wish to obtain the first ten eigenvalues of largest magnitude, of a large sparse matrix pair and .
2.4Nonsymmetric Eigenvalue Problems
The nonsymmetric eigenvalue problem is to find the eigenvalues, , and corresponding eigenvectors, , such that
More precisely, a vector as just defined is called a right eigenvector of , and a vector satisfying
is called a left eigenvector of .
A real matrix may have complex eigenvalues, occurring as complex conjugate pairs.
This problem can be solved via the Schur factorization of , defined in the real case as
where is an orthogonal matrix and is an upper quasi-triangular matrix with and diagonal blocks, the blocks corresponding to complex conjugate pairs of eigenvalues of . In the complex case, the Schur factorization is
where is unitary and is a complex upper triangular matrix.
The columns of are called the Schur vectors. For each (), the first columns of form an orthonormal basis for the invariant subspace corresponding to the first eigenvalues on the diagonal of . Because this basis is orthonormal, it is preferable in many applications to compute Schur vectors rather than eigenvectors. It is possible to order the Schur factorization so that any desired set of eigenvalues occupy the leading positions on the diagonal of .
The two basic tasks of the nonsymmetric eigenvalue routines are to compute, for a given matrix , some values of and, if desired, their associated right eigenvectors , and the Schur factorization.
2.5Generalized Nonsymmetric Eigenvalue Problem
The generalized nonsymmetric eigenvalue problem is to find the eigenvalues, , and corresponding eigenvectors, , such that
More precisely, a vector as just defined is called a right eigenvector of the matrix pair , and a vector satisfying
is called a left eigenvector of the matrix pair .
2.6The Polynomial Eigenvalue Problem
The polynomial eigenvalue problem is to find the eigenvalues, , and the corresponding eigenvectors, , such that
Here the are matrices, and is known as the degree of the problem.
More precisely, a vector as just defined is a right eigenvector of the problem and a vector satisfying
is a left eigenvector of the problem.
2.7The Singular Value Decomposition
The singular value decomposition (SVD) of an matrix is given by
where and are orthogonal (unitary) and is an diagonal matrix with real diagonal elements, , such that
The are the singular values of and the first columns of and are the left and right singular vectors of . The singular values and singular vectors satisfy
where and are the th columns of and respectively.
Thus selected singular values and the corresponding right singular vectors may be computed by finding eigenvalues and eigenvectors for the symmetric matrix (or the Hermitian matrix if is complex).
An alternative approach is to use the relationship
and thus compute selected singular values and vectors via the symmetric matrix
In many applications, one is interested in computing a few (say ) of the largest singular values and corresponding vectors. If , denote the leading columns of and respectively, and if denotes the leading principal submatrix of , then
is the best rank- approximation to in both the -norm and the Frobenius norm. Often a very small will suffice to approximate important features of the original or to approximately solve least squares problems involving .
Iterative methods for the solution of the standard eigenproblem
approach the solution through a sequence of approximations until some user-specified termination criterion is met or until some predefined maximum number of iterations has been reached. The number of iterations required for convergence is not generally known in advance, as it depends on the accuracy required, and on the matrix , its sparsity pattern, conditioning and eigenvalue spectrum.
3Choosing between ARPACK and FEAST Routines
Both the ARPACK and FEAST suites can handle standard, generalized, symmetric, Hermitian and non-Hermitian eigenvalue problems, with both left and right eigenvectors returned. However, the suites differ in the subset of eigenvalues that will be returned.
The ARPACK solvers can be instructed to find the eigenvalues with the largest and/or smallest magnitudes, real parts or imaginary parts.
The FEAST solvers allow you to specify a region in the complex plane (or an interval on the real line for Hermitian problems) within which eigenvalues will be found.
Note also that FEAST contains solvers for the polynomial eigenvalue problem.
4.1Recommendations on Choice and Use of Available Routines
4.1.1Types of Routine Available
The ARPACK routines available in this chapter divide essentially into three suites of basic reverse communication routines and some general purpose routines for banded systems.
Basic routines are grouped in suites of five, and
implement the underlying iterative method. Each suite comprises a setup routine, an options setting routine, a solver routine, a routine to return additional monitoring information and a post-processing routine. The solver routine is independent of the matrix storage format (indeed the matrix need not be stored at all) and the type of preconditioner. It uses reverse communication (see Section 7 in How to Use the NAG Library for further information), i.e., it returns repeatedly to the calling program with the argument irevcm set to specified values which require the calling program to carry out a specific task (either to compute a matrix-vector product or to solve the preconditioning equation), to signal the completion of the computation or to allow the calling program to monitor the solution. Reverse communication has the following advantages:
(i)Maximum flexibility in the representation and storage of sparse matrices. All matrix operations are performed outside the solver routine, thereby avoiding the need for a complicated interface with enough flexibility to cope with all types of storage schemes and sparsity patterns. This also applies to preconditioners.
(ii)Enhanced user interaction: you can closely monitor the solution and tidy or immediate termination can be requested. This is useful, for example, when alternative termination criteria are to be employed or in case of failure of the external routines used to perform matrix operations.
At present there are suites of basic routines for real symmetric and nonsymmetric systems, and for complex systems.
General purpose routines call basic routines in order to provide easy-to-use routines for particular sparse matrix storage formats. They are much less flexible than the basic routines, but do not use reverse communication, and may be suitable in many cases.
The structure of this part of the chapter has been designed to cater for as many types of application as possible. If a general purpose routine exists which is suitable for a given application you are recommended to use it. If you then decide you need some additional flexibility it is easy to achieve this by using basic and utility routines which reproduce the algorithm used in the general purpose routine, but allow more access to algorithmic control parameters and monitoring.
4.1.2Iterative Methods for Real Nonsymmetric and Complex Eigenvalue Problems
The suite of basic routines f12aaf,f12abf,f12acf,f12adfandf12aef implements the iterative solution of real nonsymmetric eigenvalue problems, finding estimates for a specified spectrum of eigenvalues. These eigenvalue estimates are often referred to as Ritz values and the error bounds obtained are referred to as the Ritz estimates. These routines allow a choice of termination criteria and many other options for specifying the problem type, allow monitoring of the solution process, and can return Ritz estimates of the calculated Ritz values of the problem .
For complex matrices there is an equivalent suite of routines. f12anf,f12apf,f12aqf,f12arfandf12asf are the basic routines which implement corresponding methods used for real nonsymmetric systems. Note that these routines are to be used for both Hermitian and non-Hermitian problems. Occasionally, when using these routines on a complex Hermitian problem, eigenvalues will be returned with small but nonzero imaginary part due to unavoidable round-off errors. These should be ignored unless they are significant with respect to the eigenvalues of largest magnitude that have been computed.
There are general purpose routines for the case where the matrices are known to be banded. In these cases an initialization routine is called first to set up default options, and the problem is solved by a single call to a solver routine. The matrices are supplied, in LAPACK banded-storage format, as arguments to the solver routine. For real general matrices these routines are f12aff and f12agf; and for complex matrices the pair is f12atf and f12auf. With each pair non-default options can be set, following a call to the initialization routine, using f12adf for real matrices and f12arf for complex matrices. For real matrices that can be supplied in the sparse matrix compressed column storage (CCS) format, the driver routine f02ekf is available. This routine uses routines from Chapter F12 in conjunction with direct solver routines from Chapter F11.
There is little computational penalty in using the non-Hermitian complex routines for a Hermitian problem. The only additional cost is to compute eigenvalues of a Hessenberg rather than a tridiagonal matrix. The difference in computational cost should be negligible compared to the overall cost.
4.1.3Iterative Methods for Real Symmetric Eigenvalue Problems
The suite of basic routines f12faf,f12fbf,f12fcf,f12fdfandf12fef implement a Lanczos method for the iterative solution of the real symmetric eigenproblem.
There is a general purpose routine pair for the case where the matrices are known to be banded. In this case an initialization routine, f12fff, is called first to set up default options, and the problem is solved by a single call to a solver routine, f12fgf. The matrices are supplied, in LAPACK banded-storage format, as arguments to f12fgf. Non-default options can be set, following a call to f12fff, using f12fdf.
4.1.4Iterative Methods for Singular Value Decomposition
The partial singular value decomposition, (as defined in Section 2.7), of an matrix can be computed efficiently using routines from this chapter. For real matrices, the suite of routines listed in Section 4.1.3 (for symmetric problems) can be used; for complex matrices, the corresponding suite of routines for complex problems can be used; however, there are no general purpose routines for complex problems.
The driver routine f02wgf is available for computing the partial SVD of real matrices. The matrix is not supplied to f02wgf; rather, a user-defined routine argument provides the results of performing Matrix-vector products.
For both real and complex matrices, you should use the default options (see, for example, the options listed in Section 11 in f12fdf) for problem type (Standard), computational mode (Regular) and spectrum (Largest Magnitude). The operation to be performed on request by the reverse communication routine (e.g., f12fbf) is, for real matrices, to multiply the returned vector by the symmetric matrix if , or by if . For complex matrices, the corresponding Hermitian matrices are and .
The right () or left () singular vectors are returned by the post-processing routine (e.g., f12fcf). The left (or right) singular vectors can be recovered from the returned singular vectors. Providing the largest singular vectors are not multiple or tightly clustered, there should be no problem in obtaining numerically orthogonal left singular vectors from the computed right singular vectors (or vice versa).
The second example in Section 10 in f12fbf illustrates how the partial singular value decomposition of a real matrix can be performed using the suite of routines for finding some eigenvalues of a real symmetric matrix. In this case , however, the program is easily amended to perform the same task in the case .
Similarly, routines in this part of the chapter may be used to estimate the -norm condition number,
This can be achieved by setting the option Both Ends to get the largest and smallest few singular values, then taking the ratio of largest to smallest computed singular values as your estimate.
Other routines for the solution of sparse linear eigenproblems can be found in Chapters F02 and F08. In particular, tridiagonal and band matrices are addressed in Chapter F08 whereas sparse matrices are addressed in Chapter F02.
4.2General Use of Routines
This section will describe the complete structure of the reverse communication interfaces. Numerous computational modes are available, including several shift-invert strategies designed to accelerate convergence. Two of the more sophisticated modes will be described in detail. The remaining ones are quite similar in principle, but require slightly different tasks to be performed with the reverse communication interface.
This section is structured as follows. The naming conventions used, and the data types available are described in Section 4.2.1, spectral transformations are discussed in Section 4.2.2. Spectral transformations are usually extremely effective but there are a number of problem dependent issues that determine which one to use. In Section 4.2.3 we describe the reverse communication interface needed to exercise the various shift-invert options. Each shift-invert option is specified as a computational mode and all of these are summarised in the remaining sections. There is a subsection for each problem type and hence these sections are quite similar and repetitive. Once the basic idea is understood, it is probably best to turn directly to the subsection that describes the problem setting that is most interesting to you.
Perhaps the easiest way to rapidly become acquainted with the modes in this part of the chapter is to run each of the example programs which use the various modes. These may be used as templates and adapted to solve specific problems.
Routines for solving nonsymmetric (real and complex) eigenvalue problems have as first letter after the chapter name, the letter ‘A’, e.g., f12abf; equivalent routines for symmetric eigenvalue problems will have this letter replaced by the letter ‘F’, e.g., f12fbf. For the letter following this, routines for real eigenvalue problems will have letters in the range ‘A to M’ while those for complex eigenvalue problems will have letters correspondingly shifted into the range ‘N to Z’; so, for example, the complex equivalent of f12adf is f12arf, while the real symmetric equivalent is f12fdf.
A suite of five routines are named consecutively, e.g., f12aaf,f12abf,f12acf,f12adfandf12aef.
Each general purpose routine has its own initialization routine, but uses the option setting routine from the suite relevant to the problem type. Thus each general purpose routine can be viewed as belonging to a suite of three routines, even though only two routines will be named consecutively. For example, f12adf,f12affandf12agf represent the suite of routines for solving a banded real symmetric eigenvalue problem.
4.2.2Shift and Invert Spectral Transformations
The most general problem that may be solved here is to compute a few selected eigenvalues and corresponding eigenvectors for
The shift and invert spectral transformation is used to enhance convergence to a desired portion of the spectrum. If is an eigen-pair for and then
This transformation is effective for finding eigenvalues near since the eigenvalues of that are largest in magnitude correspond to the eigenvalues of the original problem that are nearest to the shift in absolute value. These transformed eigenvalues of largest magnitude are precisely the eigenvalues that are easy to compute with a Krylov method. (See Barrett et al. (1994)). Once they are found, they may be transformed back to eigenvalues of the original problem. The direct relation is
and the eigenvector associated with in the transformed problem is also an eigenvector of the original problem corresponding to . Usually the Arnoldi process will rapidly obtain good approximations to the eigenvalues of of largest magnitude. However, to implement this transformation, you must provide the means to solve linear systems involving either with a matrix factorization or with an iterative method.
In general, will be non-Hermitian even if and are both Hermitian. However, this is easily remedied. The assumption that is Hermitian positive definite implies that the bilinear form
is an inner product. If is positive semidefinite and singular, then a semi-inner product results. This is a weighted -inner product and vectors , are called -orthogonal if . It is easy to show that if is Hermitian (self-adjoint) then is Hermitian self-adjoint with respect to this -inner product (meaning for all vectors , ). Therefore, symmetry will be preserved if we force the computed basis vectors to be orthogonal in this -inner product. Implementing this -orthogonality requires you to provide a matrix-vector product on request along with each application of . In the following sections we shall discuss some of the more familiar transformations to the standard eigenproblem. However, when
is positive (semi)definite, we recommend using the shift-invert spectral transformation with -inner products if at all possible. This is a far more robust transformation when is ill-conditioned or singular. With a little extra manipulation (provided automatically in the post-processing routines) the semi-inner product induced by prevents corruption of the computed basis vectors by roundoff-error associated with the presence of infinite eigenvalues. These very ill-conditioned eigenvalues are generally associated with a singular or highly ill-conditioned . A detailed discussion of this theory may be found in Chapter 4 of Lehoucq et al. (1998).
Shift-invert spectral transformations are very effective and should even be used on standard problems,
, whenever possible. This is particularly true when interior eigenvalues are sought or when the desired eigenvalues are clustered. Roughly speaking, a set of eigenvalues is clustered if the maximum distance between any two eigenvalues in that set is much smaller than the minimum distance between these eigenvalues and any other eigenvalues of .
If you have a generalized problem , then you must provide a way to solve linear systems with either , or a linear combination of the two matrices in order to use the reverse communication suites in this chapter. In this case, a sparse direct method should be used to factor the appropriate matrix whenever possible. The resulting factorization may be used repeatedly to solve the required linear systems once it has been obtained. If instead you decide to use an iterative method, the accuracy of the solutions must be commensurate with the convergence tolerance used for the Arnoldi iteration. A slightly more stringent tolerance is needed relative to the desired accuracy of the eigenvalue calculation.
The main drawback with using the shift-invert spectral transformation is that the coefficient matrix is typically indefinite in the Hermitian case and has zero-valued eigenvalues in the non-Hermitian case. These are often the most difficult situations for iterative methods and also for sparse direct methods.
The decision to use a spectral transformation on a standard eigenvalue problem or to use one of the simple modes is problem dependent. The simple modes have the advantage that you only need to supply a matrix vector product . However, this approach is usually only successful for problems where extremal non-clustered eigenvalues are sought. In non-Hermitian problems, extremal means eigenvalues near the boundary of the spectrum of . For Hermitian problems, extremal means eigenvalues at the left- or right-hand end points of the spectrum of . The notion of non-clustered (or well separated) is difficult to define without going into considerable detail. A simplistic notion of a well-separated eigenvalue for a Hermitian problem would be
for all with , where and are the smallest and largest algebraically. Unless a matrix vector product is quite difficult to code or extremely expensive computationally, it is probably worth trying to use the simple mode first if you are seeking extremal eigenvalues.
The remainder of this section discusses additional transformations that may be applied to convert a generalized eigenproblem to a standard eigenproblem. These are appropriate when is well-conditioned (Hermitian or non-Hermitian).
220.127.116.11 is Hermitian positive definite
If is Hermitian positive definite and well-conditioned
( is of modest size), then computing the Cholesky factorization and converting equation (2) to
provides a transformation to a standard eigenvalue problem. In this case, a request for a matrix vector product would be satisfied with the following three steps:
(ii)Matrix-vector multiply .
(iii)Solve for .
Upon convergence, a computed eigenvector for
is converted to an eigenvector of the original problem by solving the triangular system . This transformation is most appropriate when is Hermitian, is Hermitian positive definite and extremal eigenvalues are sought. This is because when is Hermitian, so is .
If is Hermitian positive definite and the smallest eigenvalues are sought, then it would be best to reverse the roles of and in the above description and ask for the largest algebraic eigenvalues or those of largest magnitude. Upon convergence, a computed eigenvalue
would then be converted to an eigenvalue of the original problem by the relation
18.104.22.168 is not Hermitian positive semidefinite
If neither nor is Hermitian positive semidefinite, then a direct transformation to standard form is required. One simple way to obtain a direct transformation of equation (2) to a standard eigenvalue problem is to multiply on the left by which results in . Of course, you should not perform this transformation explicitly since it will most likely convert a sparse problem into a dense one. If possible, you should obtain a direct factorization of and when a matrix-vector product involving is called for, it may be accomplished with the following two steps:
(i)Matrix-vector multiply .
(ii)Solve for .
Several problem-dependent issues may modify this strategy. If is singular or if you are interested in eigenvalues near a point then you may choose to work with but without using the -inner products discussed previously. In this case you will have to transform the converged eigenvalues of to eigenvalues of the original problem.
4.2.3Reverse Communication and Shift-invert Modes
The reverse communication interface routine for real nonsymmetric problems is f12abf; for complex problems is f12apf; and for real symmetric problems is f12fbf. First the reverse communication loop structure will be described and then the details and nuances of the problem setup will be discussed. We use the symbol for the operator that is applied to vectors in the Arnoldi/Lanczos process and will stand for the matrix to use in the weighted inner product described previously. For the shift-invert spectral transformation mode denotes .
The basic idea is to set up a loop that repeatedly call one of f12abf,f12apfandf12fbf. On each return, you must either apply or to a specified vector or exit the loop depending upon the value returned in the reverse communication argument irevcm.
22.214.171.124Shift and invert on a generalized eigenproblem
The example program in
Section 10 in f12aef
illustrates the reverse communication loop for f12abf in shift-invert mode for a generalized nonsymmetric eigenvalue problem. This loop structure will be identical for the symmetric problem calling f12fbf. The loop structure is also identical for the complex arithmetic subroutine f12apf.
In the example, the matrix is assumed to be symmetric and positive semidefinite. In the loop structure, you will have to supply a routine to obtain a matrix factorization of that may repeatedly be used to solve linear systems. Moreover, a routine needs to be provided to perform the matrix-vector product and a routine is required to solve linear systems of the form as needed using the previously computed factorization.
When convergence has taken place (indicated by and ), the reverse communication loop will be exited. Then, post-processing using the relevant routine from f12acf,f12aqfandf12fcf must be done to recover the eigenvalues and corresponding eigenvectors of the original problem. When operating in shift-invert mode, the eigenvalue selection option is normally set to Largest Magnitude. The post-processing routine is then used to convert the converged eigenvalues of to eigenvalues of the original problem (2). Also, when is singular or ill-conditioned, the post-processing routine takes steps to purify the eigenvectors and rid them of numerical corruption from eigenvectors corresponding to near-infinite eigenvalues. These procedures are performed automatically when operating in any one of the computational modes described above and later in this section.
You may wish to construct alternative computational modes using spectral transformations that are not addressed by any of the modes specified in this chapter. The reverse communication interface will easily accommodate these modifications. However, it will most likely be necessary to construct explicit transformations of the eigenvalues of to eigenvalues of the original problem in these situations.
126.96.36.199Using the computational modes
The problem set up is similar for all of the available computational modes. In the previous section, a detailed description of the reverse communication loop for a specific mode (Shift-invert for a Generalized Problem) was given. To use this or any of the other modes listed below, you are strongly urged to modify one of the example programs.
The first thing to decide is whether the problem will require a spectral transformation. If the problem is generalized, , then a spectral transformation will be required (see Section 4.2.2). Such a transformation will most likely be needed for a standard problem if the desired eigenvalues are in the interior of the spectrum or if they are clustered at the desired part of the spectrum. Once this decision has been made and has been specified, an efficient means to implement the action of the operator on a vector must be devised. The expense of applying to a vector will of course have direct impact on performance.
Shift-invert spectral transformations may be implemented with or without the use of a weighted -inner product. The relation between the eigenvalues of and the eigenvalues of the original problem must also be understood in order to make the appropriate eigenvalue selection option (e.g., Largest Magnitude) in order to recover eigenvalues of interest for the original problem. You must specify the number of eigenvalues to compute, which eigenvalues are of interest, the number of basis vectors to use, and whether or not the problem is standard or generalized. These items are controlled by setting options via the option setting routine.
Setting the number of eigenvalues nev and the number of basis vectors ncv (in the setup routine) for optimal performance is very much problem dependent. If possible, it is best to avoid setting nev in a way that will split clusters of eigenvalues. As a rule of thumb
is reasonable. There are trade-offs due to the cost of the user-supplied matrix-vector products and the cost of the implicit restart mechanism. If the user-supplied matrix-vector product is relatively cheap, then a smaller value of ncv may lead to more user matrix-vector products and implicit Arnoldi iterations but an overall decrease in computation time. Convergence behaviour can be quite different depending on which of the spectrum options (e.g., Largest Magnitude) is chosen. The Arnoldi process tends to converge most rapidly to extreme points of the spectrum. Implicit restarting can be effective in focusing on and isolating a selected set of eigenvalues near these extremes. In principle, implicit restarting could isolate eigenvalues in the interior, but in practice this is difficult and usually unsuccessful. If you are interested in eigenvalues near a point that is in the interior of the spectrum, a shift-invert strategy is usually required for reasonable convergence.
The integer argument irevcm is the reverse communication flag that will specify a requested action on return from one of the solver routines f12abf,f12apfandf12fbf. The options Standard and Generalized specify if this is a standard or generalized eigenvalue problem. The dimension of the problem is specified on the call to the initialization routine only; this value, together with the number of eigenvalues and the dimension of the basis vectors is passed through the communication array. There are a number of spectrum options which specify the eigenvalues to be computed; these options differ depending on whether a Hermitian or non-Hermitian eigenvalue problem is to be solved. For example, the Both Ends is specific to Hermitian (symmetric) problems while the Largest Imaginary is specific to non-Hermitian eigenvalue problems (see Section 11.1 in f12adf). The specification of problem type will be described separately but the reverse communication interface and loop structure is the same for each type of the basic modes Regular, Regular Inverse, Shifted Inverse (also Shifted Inverse Real and Shifted Inverse Imaginary for real nonsymmetric problems), and for the problem type: Standard or Generalized. There are some additional specialised modes for symmetric problems, Buckling and Cayley, and for real nonsymmetric problems with complex shifts applied in real arithmetic. You are encouraged to examine the documented example programs for these modes.
The Tolerance specifies the accuracy requested. If you wish to supply shifts for implicit restarting then the Supplied Shifts must be selected, otherwise the default Exact Shifts strategy will be used. The Supplied Shifts should only be used when you have a great deal of knowledge about the spectrum and about the implicit restarted Arnoldi method and its underlying theory. The Iteration Limit should be set to the maximum number of implicit restarts allowed. The cost of an implicit restart step (major iteration) is in the order of
floating-point operations for the dense matrix operations and
matrix-vector products with the matrix .
The choice of computational mode through the option setting routine is very important. The legitimate computational mode options available differ with each problem type and are listed below for each of them.
188.8.131.52Computational modes for real symmetric problems
The reverse communication interface subroutine for symmetric eigenvalue problems is f12fbf. The option for selecting the region of the spectrum of interest can be one of those listed in Table 1.
Eigenvalue spectrum options for symmetric eigenproblems
Table 4 lists the spectral transformation options for real nonsymmetric eigenvalue problems together with the specification of and for each mode and the problem type option setting. The equivalent listing for complex non-Hermitian eigenvalue problems is given in Table 5.
Problem types, computational modes and spectral transformations for real nonsymmetric eigenproblems
Note that there are two shifted inverse modes with complex shifts in Table 4. Since is complex, these both require the factorization of the matrix
in complex arithmetic even though, in the case of real nonsymmetric problems, both and are real. The only advantage of using this option for real nonsymmetric problems instead of using the equivalent suite for complex problems is that all of the internal operations in the Arnoldi process are executed in real arithmetic. This results in a factor of two saving in storage and a factor of four saving in computational cost. There is additional post-processing that is somewhat more complicated than the other modes in order to get the eigenvalues and eigenvectors of the original problem. These modes are only recommended if storage is extremely critical.
Problem types, computational modes and spectral transformations for complex non-Hermitian eigenproblems
On the final successful return from a reverse communication routine, the corresponding post-processing routine must be called to get eigenvalues of the original problem and the corresponding eigenvectors if desired. In the case of Shifted Inverse modes for Generalized problems, there are some subtleties to recovering eigenvectors when is ill-conditioned. This process is called eigenvector purification. It prevents eigenvectors from being corrupted with noise due to the presence of eigenvectors corresponding to near infinite eigenvalues. These operations are completely transparent to you. There is negligible additional cost to obtain eigenvectors. An orthonormal (Arnoldi/Lanczos) basis is always computed. The approximate eigenvalues of the original problem are returned in ascending algebraic order. The option relevant to this routine is Vectors which may be set to values that determine whether only eigenvalues are desired or whether corresponding eigenvectors and/or Schur vectors are required. The value of the shift used in spectral transformations must be passed to the post-processing routine through the appropriately named argument(s). The eigenvectors returned are normalized to have unit length with respect to the semi-inner product that was used. Thus, if then they will have unit length in the standard-norm. In general, a computed eigenvector will satisfy .
184.108.40.206Solution monitoring and printing
The option setting routine for each suite allows the setting of three options that control solution printing and the monitoring of the iterative and post-processing stages. These three options are: Advisory, Monitoring and Print Level. By default, no solution monitoring or printing is performed. The Advisory option controls where solution details are printed; the Monitoring option controls where monitoring details are to be printed and is mainly used for debugging purposes; the Print Level option controls the amount of detail to be printed, see individual option setting routine documents for specifications of each print level. The value passed to Advisory and Monitoring can be the same, but it is recommended that the two sets of information be kept separate. Note that the monitoring information can become very voluminous for the highest settings of Print Level.
The NAG FEAST suite of routines all have short names beginning with ‘f12j’. They are divided into the following types of routine:
Solving an eigenvalue problem using the FEAST algorithm involves the following routine calls.
1.Call f12jaf to initialize the handle to the internal data structure used by the routines and set options to their default values.
2.Optionally, call f12jbf to set any options if different from their defaults (for example, the number of quadrature nodes on the contour, or the location of the ellipse if such a contour is to be used). f12jbf should be called once for each option to be set.
3.Call one of the contour setting routines f12jef (for Hermitian and real symmetric problems), f12jff (for circular or elliptical contours) or f12jgf (for maximum flexibility in your choice of contour). These routines will generate a set of quadrature nodes and weights to be used by the solvers.
5.Call f12jzf to destroy the handle to the internal data structure.
The exact choice of which contour setting routine and which solver to use is problem-dependent and is detailed in Section 5.2.3.
5.1Contour Setting Routines
The contour setting routines create a set of nodes and weights describing the contour within which eigenvalues are required. There are three such routines.
f12jef is intended for use with Hermitian or real symmetric eigenvalue problems (the eigenvalues of such problems all lie on the real line). You need only specify the limits of the real interval within which eigenvalues will be sought. f12jef uses these to generate an elliptical contour, symmetric about the real axis. Prior to calling f12jef, you can set the eccentricity of the ellipse, and the number of contour integration points using the option setting routine f12jbf.
f12jff is intended for non-Hermitian eigenvalue problems. It generates nodes and weights for an elliptical contour in the complex plane. You need only specify the horizontal radius and the location of the centre of the ellipse. Prior to calling f12jff you can use f12jbf to rotate the ellipse, control its eccentricity and specify the number of integration points to use.
f12jgf gives you the maximum flexibility in creating your own contour. It is intended for non-Hermitian problems. Your contour can be made up of a combination of line segments and half ellipses. You must specify the start and end points of each segment of the contour, together with the number of integration points that should be assigned to each segment. f12jgf will use this information to generate the nodes and weights of a polygonal approximation to the contour. The contour must be convex (the behaviour of the solvers is undefined if a concave contour is used).
Note that f12jbf allows you to choose between three types of quadrature: Gauss-Legendre, Trapezoidal and (for Hermitian and real symmetric problems only) Zolotarev. The choice of quadrature will change the values of the nodes and weights computed by the contour setting routines. The type of quadrature, and the number of integration points used both influence the convergence rate of the algorithm. In general, increasing the number of integration points increases the convergence rate, at the expense of more expensive iterations, and using Zolotarev quadrature is recomended for Hermitian eigenvalue problems.
The solvers use reverse communication (see Section 7 in How to Use the NAG Library for further information). They return repeatedly to the calling program with the argument irevcm set to specified values which require the calling program to carry out a specific task (either to compute a matrix-vector product or to solve a linear system), or to signal the completion of the computation. Reverse communication offers maximum flexibility in the representation and storage of sparse matrices. All matrix operations are performed outside the solver routine, thereby avoiding the need for a complicated interface with enough flexibility to cope with all types of storage schemes and sparsity patterns.
When FEAST requires the calling program to solve a system of linear equations, this will occur in two stages.
(i)FEAST will first ask the calling program to compute a factorization of a matrix suitable for solving the linear system. For dense matrices this might be a Bunch-Kaufman factorization (f07nrf) or an decomposition (f07arf). For sparse matrices this could be an incomplete factorization (f11dnf) or even just a preconditioner. The factorization should be stored as it may be reused several times.
(ii)FEAST will ask the calling program to use the factorization computed in (i) to solve linear systems with different sets of righthand sides. When a new factorization is required (i.e., FEAST returns to step (i)), the factorization previously computed in step (i) can be overwritten.
Note that FEAST uses an inverse residual iteration algorithm which enables the linear systems to be solved with very low accuracy with no impact on the double precision convergence rate. Thus single precision solvers, and very high convergence tolerances are entirely acceptable when factorizing and solving the linear systems, provided the condition numbers of the linear systems are not so high as to prevent such low precision solvers from obtaining any degree of accuracy.
5.2.2Further Tips on the Use of the Solvers
The size of the search subspace m0 affects the convergence of the algorithm. Increasing m0 will improve convergence, but will require more memory and result in a more expensive computation. As a general rule of thumb, m0 should exceed the number of eigenvalues in the search contour by a factor of approximately (note that FEAST can be used to estimate the number of eigenvalues inside the contour prior to embarking on the full eigenvalue computation by setting the option in f12jbf).
In principal, the FEAST algorithm can be used to find many thousands of eigenpairs within a large search contour. However, in practice better performance will be achieved if the computation is split into multiple smaller contours (which could then be searched in parallel).
5.2.3Routine Choices for Different Problem Types
The following table shows which contour setting routine and which reverse communication solver should be used for the different problem types. Recall that for all problem types the initialization routine f12jaf should first be called, and the cleanup routine f12jzf should be called after the solver.
7Auxiliary Routines Associated with Library Routine Arguments
8 Withdrawn or Deprecated Routines
Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia
Lehoucq R B (1995) Analysis and implementation of an implicitly restarted iteration PhD Thesis Rice University, Houston, Texas
Lehoucq R B (2001) Implicitly restarted Arnoldi methods and subspace iteration SIAM Journal on Matrix Analysis and Applications23 551–562
Lehoucq R B and Scott J A (1996) An evaluation of software for computing eigenvalues of sparse nonsymmetric matrices Preprint MCS-P547-1195 Argonne National Laboratory
Lehoucq R B and Sorensen D C (1996) Deflation techniques for an implicitly restarted Arnoldi iteration SIAM Journal on Matrix Analysis and Applications17 789–821
Lehoucq R B, Sorensen D C and Yang C (1998) ARPACK Users' Guide: Solution of Large-scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods SIAM, Philadelphia
Polizzi E (2009) Density-Matrix-Based Algorithms for Solving Eigenvalue Problems Phys. Rev. B. 79 115112
Saad Y (1992) Numerical Methods for Large Eigenvalue Problems Manchester University Press, Manchester, UK