The routine may be called by the names f02fjf or nagf_eigen_real_symm_sparse_eigsys.
3Description
f02fjf finds the $m$ eigenvalues of largest absolute value and the corresponding eigenvectors for the real eigenvalue problem
$$Cx=\lambda x$$
(1)
where $C$ is an $n\times n$ matrix such that
$$BC={C}^{\mathrm{T}}B$$
(2)
for a given positive definite matrix $B$. $C$ is said to be $B$-symmetric. Different specifications of $C$ allow for the solution of a variety of eigenvalue problems. For example, when
$$C=A\text{\hspace{1em} and \hspace{1em}}B=I\text{\hspace{1em} where \hspace{1em}}A={A}^{\mathrm{T}}$$
the routine finds the $m$ eigenvalues of largest absolute magnitude for the standard symmetric eigenvalue problem
$$Ax=\lambda x\text{.}$$
(3)
The routine is intended for the case where $A$ is sparse.
As a second example, when
$$C={B}^{-1}A$$
where
$$A={A}^{\mathrm{T}}$$
the routine finds the $m$ eigenvalues of largest absolute magnitude for the generalized symmetric eigenvalue problem
$$Ax=\lambda Bx\text{.}$$
(4)
The routine is intended for the case where $A$ and $B$ are sparse.
The routine does not require $C$ explicitly, but $C$ is specified via image which, given an $n$-element vector $z$, computes the image $w$ given by
$$w=Cz\text{.}$$
For instance, in the above example, where $C={B}^{-1}A$, image will need to solve the positive definite system of equations $Bw=Az$ for $w$.
To find the $m$ eigenvalues of smallest absolute magnitude of (3) we can choose $C={A}^{-1}$ and hence find the reciprocals of the required eigenvalues, so that image will need to solve $Aw=z$ for $w$, and correspondingly for (4) we can choose $C={A}^{-1}B$ and solve $Aw=Bz$ for $w$.
A table of examples of choice of image is given in Table 1. It should be remembered that the routine also returns the corresponding eigenvectors and that $B$ is positive definite. Throughout $A$ is assumed to be symmetric and, where necessary, nonsingularity is also assumed.
Table 1
The Requirement of image for Various Problems.
Eigenvalues Required
Problem
$Ax=\lambda x(B=I)$
$Ax=\lambda Bx$
$ABx=\lambda x$
Largest
Compute $w=Az$
Solve $Bw=Az$
Compute $w=ABz$
Smallest (Find $1/\lambda $)
Solve $Aw=z$
Solve $Aw=Bz$
Solve $Av=z$, $Bw=v$
Furthest from $\sigma $
(Find $\lambda -\sigma $)
Compute $w=(A-\sigma I)z$
Solve $Bw=(A-\sigma B)z$
Compute $w=(AB-\sigma I)z$
Closest to $\sigma $
(Find $1/(\lambda -\sigma )$)
Solve $(A-\sigma I)w=z$
Solve $(A-\sigma B)w=Bz$
Solve $(AB-\sigma I)w=z$
The matrix $B$ also need not be supplied explicitly, but is specified via dot which, given $n$-element vectors $z$ and $w$, computes the generalized dot product ${w}^{\mathrm{T}}Bz$.
f02fjf is based upon routine SIMITZ (see Nikolai (1979)), which is itself a derivative of the Algol procedure ritzit (see Rutishauser (1970)), and uses the method of simultaneous (subspace) iteration. (See Parlett (1998) for a description, analysis and advice on the use of the method.)
The routine performs simultaneous iteration on $k>m$ vectors. Initial estimates to $p\le k$ eigenvectors, corresponding to the $p$ eigenvalues of $C$ of largest absolute value, may be supplied to f02fjf. When possible $k$ should be chosen so that the $k$th eigenvalue is not too close to the $m$ required eigenvalues, but if $k$ is initially chosen too small then f02fjf may be re-entered, supplying approximations to the $k$ eigenvectors found so far and with $k$ then increased.
At each major iteration f02fjf solves an $r\times r$ ($r\le k$) eigenvalue sub-problem in order to obtain an approximation to the eigenvalues for which convergence has not yet occurred. This approximation is refined by Chebyshev acceleration.
4References
Nikolai P J (1979) Algorithm 538: Eigenvectors and eigenvalues of real generalized symmetric matrices by simultaneous iteration ACM Trans. Math. Software5 118–125
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia
Rutishauser H (1969) Computational aspects of F L Bauer's simultaneous iteration method Numer. Math.13 4–13
Rutishauser H (1970) Simultaneous iteration method for symmetric matrices Numer. Math.16 205–223
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $C$.
Constraint:
${\mathbf{n}}\ge 1$.
2: $\mathbf{m}$ – IntegerInput/Output
On entry: $m$, the number of eigenvalues required.
Constraint:
${\mathbf{m}}\ge 1$.
On exit: ${m}^{\prime}$, the number of eigenvalues actually found. It is equal to $m$ if ${\mathbf{ifail}}={\mathbf{0}}$ on exit, and is less than $m$ if ${\mathbf{ifail}}={\mathbf{2}}$, ${\mathbf{3}}$ or ${\mathbf{4}}$. See Sections 6 and 9 for further information.
3: $\mathbf{k}$ – IntegerInput
On entry: the number of simultaneous iteration vectors to be used. Too small a value of k may inhibit convergence, while a larger value of k incurs additional storage and additional work per iteration.
Suggested value:
${\mathbf{k}}={\mathbf{m}}+4$ will often be a reasonable choice in the absence of better information.
On entry: the maximum number of major iterations (eigenvalue sub-problems) to be performed. If ${\mathbf{noits}}\le 0$, the value $100$ is used in place of noits.
On exit: the number of iterations actually performed.
5: $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: a relative tolerance to be used in accepting eigenvalues and eigenvectors. If the eigenvalues are required to about $t$ significant figures, tol should be set to about ${10}^{-t}$. ${d}_{i}$ is accepted as an eigenvalue as soon as two successive approximations to ${d}_{i}$ differ by less than $(\left|{\stackrel{~}{d}}_{i}\right|\times {\mathbf{tol}})/10$, where ${\stackrel{~}{d}}_{i}$ is the latest approximation to ${d}_{i}$. Once an eigenvalue has been accepted, an eigenvector is accepted as soon as $\left({d}_{i}{f}_{i}\right)/({d}_{i}-{d}_{k})<{\mathbf{tol}}$, where ${f}_{i}$ is the normalized residual of the current approximation to the eigenvector (see Section 9 for further information). The values of the ${f}_{i}$ and ${d}_{i}$ can be printed from monit. If tol is supplied outside the range ($\epsilon ,1.0$), where $\epsilon $ is the machine precision, the value $\epsilon $ is used in place of tol.
6: $\mathbf{dot}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
dot must return the value ${w}^{\mathrm{T}}Bz$ for given vectors $w$ and $z$. For the standard eigenvalue problem, where $B=I$, dot must return the dot product ${w}^{\mathrm{T}}z$.
On exit: may be used as a flag to indicate a failure in the computation of ${w}^{\mathrm{T}}Bz$. If iflag is negative on exit from dot, f02fjf will exit immediately with ifail set to iflag. Note that in this case dot must still be assigned a value.
2: $\mathbf{n}$ – IntegerInput
On entry: the number of elements in the vectors $z$ and $w$ and the order of the matrix $B$.
3: $\mathbf{z}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector $z$ for which ${w}^{\mathrm{T}}Bz$ is required.
4: $\mathbf{w}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector $w$ for which ${w}^{\mathrm{T}}Bz$ is required.
5: $\mathbf{ruser}\left({\mathbf{lruser}}\right)$ – Real (Kind=nag_wp) arrayUser Workspace
dot is called with the argument ruser as supplied to f02fjf. You should use the array ruser to supply information to dot.
6: $\mathbf{lruser}$ – IntegerInput
On entry: the dimension of the array ruser as declared in the (sub)program from which f02fjf is called.
dot is called with the argument iuser as supplied to f02fjf. You should use the array iuser to supply information to dot.
8: $\mathbf{liuser}$ – IntegerInput
On entry: the dimension of the array iuser as declared in the (sub)program from which f02fjf is called.
dot must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f02fjf is called. Arguments denoted as Input must not be changed by this procedure.
Note:dot should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f02fjf. If your code inadvertently does return any NaNs or infinities, f02fjf is likely to produce unexpected results.
7: $\mathbf{image}$ – Subroutine, supplied by the user.External Procedure
image must return the vector $w=Cz$ for a given vector $z$.
On exit: may be used as a flag to indicate a failure in the computation of $w$. If iflag is negative on exit from image, f02fjf will exit immediately with ifail set to iflag.
2: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of elements in the vectors $w$ and $z$, and the order of the matrix $C$.
3: $\mathbf{z}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the vector $z$ for which $Cz$ is required.
4: $\mathbf{w}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the vector $w=Cz$.
5: $\mathbf{ruser}\left({\mathbf{lruser}}\right)$ – Real (Kind=nag_wp) arrayUser Workspace
image is called with the argument ruser as supplied to f02fjf. You should use the array ruser to supply information to image.
6: $\mathbf{lruser}$ – IntegerInput
On entry: the dimension of the array ruser as declared in the (sub)program from which f02fjf is called.
image is called with the argument iuser as supplied to f02fjf. You should use the array iuser to supply information to image.
8: $\mathbf{liuser}$ – IntegerInput
On entry: the dimension of the array iuser as declared in the (sub)program from which f02fjf is called.
image must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f02fjf is called. Arguments denoted as Input must not be changed by this procedure.
Note:image should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by f02fjf. If your code inadvertently does return any NaNs or infinities, f02fjf is likely to produce unexpected results.
8: $\mathbf{monit}$ – Subroutine, supplied by the NAG Library or the user.External Procedure
monit is used to monitor the progress of f02fjf. monit may be the dummy subroutine f02fjz if no monitoring is actually required. (f02fjz is included in the NAG Library.) monit is called after the solution of each eigenvalue sub-problem and also just prior to return from f02fjf. The arguments istate and nextit allow selective printing by monit.
No eigenvalue or eigenvector has just been accepted.
${\mathbf{istate}}=1$
One or more eigenvalues have been accepted since the last call to monit.
${\mathbf{istate}}=2$
One or more eigenvectors have been accepted since the last call to monit.
${\mathbf{istate}}=3$
One or more eigenvalues and eigenvectors have been accepted since the last call to monit.
${\mathbf{istate}}=4$
Return from f02fjf is about to occur.
2: $\mathbf{nextit}$ – IntegerInput
On entry: the number of the next iteration.
3: $\mathbf{nevals}$ – IntegerInput
On entry: the number of eigenvalues accepted so far.
4: $\mathbf{nevecs}$ – IntegerInput
On entry: the number of eigenvectors accepted so far.
5: $\mathbf{k}$ – IntegerInput
On entry: $k$, the number of simultaneous iteration vectors.
6: $\mathbf{f}\left({\mathbf{k}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: a vector of error quantities measuring the state of convergence of the simultaneous iteration vectors. See tol and Section 9 for further details. Each element of f is initially set to the value $4.0$ and an element remains at $4.0$ until the corresponding vector is tested.
7: $\mathbf{d}\left({\mathbf{k}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{d}}\left(i\right)$ contains the latest approximation to the absolute value of the $i$th eigenvalue of $C$.
monit must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f02fjf is called. Arguments denoted as Input must not be changed by this procedure.
9: $\mathbf{novecs}$ – IntegerInput
On entry: the number of approximate vectors that are being supplied in x. If novecs is outside the range $(0,{\mathbf{k}})$, the value $0$ is used in place of novecs.
10: $\mathbf{x}({\mathbf{ldx}},{\mathbf{k}})$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if $0<{\mathbf{novecs}}\le {\mathbf{k}}$, the first novecs columns of x must contain approximations to the eigenvectors corresponding to the novecs eigenvalues of largest absolute value of $C$. Supplying approximate eigenvectors can be useful when reasonable approximations are known, or when f02fjf is being restarted with a larger value of k. Otherwise it is not necessary to supply approximate vectors, as simultaneous iteration vectors will be generated randomly by f02fjf.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{2}}$, ${\mathbf{3}}$ or ${\mathbf{4}}$, the first ${m}^{\prime}$ columns contain the eigenvectors corresponding to the eigenvalues returned in the first ${m}^{\prime}$ elements of d; and the next $k-{m}^{\prime}-1$ columns contain approximations to the eigenvectors corresponding to the approximate eigenvalues returned in the next $k-{m}^{\prime}-1$ elements of d. Here ${m}^{\prime}$ is the value returned in m, the number of eigenvalues actually found. The $k$th column is used as workspace.
11: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f02fjf is called.
Constraint:
${\mathbf{ldx}}\ge {\mathbf{n}}$.
12: $\mathbf{d}\left({\mathbf{k}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{2}}$, ${\mathbf{3}}$ or ${\mathbf{4}}$, the first ${m}^{\prime}$ elements contain the first ${m}^{\prime}$ eigenvalues in decreasing order of magnitude; and the next $k-{m}^{\prime}-1$ elements contain approximations to the next $k-{m}^{\prime}-1$ eigenvalues. Here ${m}^{\prime}$ is the value returned in m, the number of eigenvalues actually found. ${\mathbf{d}}\left(k\right)$ contains the value $e$ where $(-e,e)$ is the latest interval over which Chebyshev acceleration is performed.
13: $\mathbf{work}\left({\mathbf{lwork}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
14: $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f02fjf is called.
iuser is not used by f02fjf, but is passed directly to dot and image and may be used to pass information to these routines.
18: $\mathbf{liuser}$ – IntegerInput
On entry: the dimension of the array iuser as declared in the (sub)program from which f02fjf is called.
19: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases f02fjf may return useful information.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{k}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{k}}\le {\mathbf{n}}$.
On entry, ${\mathbf{ldx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
On entry, lwork is too small. Minimum size required: $\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{k}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}<{\mathbf{k}}$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
Not all requested eigenvalues and vectors have been obtained.
Approximations to the $r$th eigenvalue are oscillating rapidly indicating that severe cancellation is occurring in the $r$th eigenvector and so m is returned as $(r-1)$. A restart with a larger value of k may permit convergence.
${\mathbf{ifail}}=3$
Not all requested eigenvalues and vectors have been obtained.
The rate of convergence of the remaining eigenvectors suggests that more than noits iterations would be required and so the input value of m has been reduced. A restart with a larger value of k may permit convergence.
${\mathbf{ifail}}=4$
Not all requested eigenvalues and vectors have been obtained.
noits iterations have been performed. A restart, possibly with a larger value of k, may permit convergence.
${\mathbf{ifail}}=5$
Convergence of eigenvalue sub-problem occurred.
Not all requested eigenvalues and vectors have been obtained.
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
Eigenvalues and eigenvectors will normally be computed to the accuracy requested by the argument tol, but eigenvectors corresponding to small or to close eigenvalues may not always be computed to the accuracy requested by the argument tol. Use of the monit to monitor acceptance of eigenvalues and eigenvectors is recommended.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f02fjf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f02fjf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The time taken by f02fjf will be principally determined by the time taken to solve the eigenvalue sub-problem and the time taken by dot and image. The time taken to solve an eigenvalue sub-problem is approximately proportional to $n{k}^{2}$. It is important to be aware that several calls to dot and image may occur on each major iteration.
As can be seen from Table 1, many applications of f02fjf will require the image to solve a system of linear equations. For example, to find the smallest eigenvalues of $Ax=\lambda Bx$, image needs to solve equations of the form $Aw=Bz$ for $w$ and routines from Chapters F01 and F04
will frequently be useful in this context. In particular, if $A$ is a positive definite variable band matrix, f04mcf may be used after $A$ has been factorized by f01mcf. Thus factorization need be performed only once prior to calling f02fjf. An illustration of this type of use is given in the example program.
An approximation ${\stackrel{~}{d}}_{h}$, to the $i$th eigenvalue, is accepted as soon as ${\stackrel{~}{d}}_{h}$ and the previous approximation differ by less than $\left|{\stackrel{~}{d}}_{h}\right|\times {\mathbf{tol}}/10$. Eigenvectors are accepted in groups corresponding to clusters of eigenvalues that are equal, or nearly equal, in absolute value and that have already been accepted. If ${d}_{r}$ is the last eigenvalue in such a group and we define the residual ${r}_{j}$ as
$${r}_{j}=C{x}_{j}-{y}_{r}$$
where ${y}_{r}$ is the projection of $C{x}_{j}$, with respect to $B$, onto the space spanned by ${x}_{1},{x}_{2},\dots ,{x}_{r}$, and ${x}_{j}$ is the current approximation to the $j$th eigenvector, then the value ${f}_{i}$ returned in
monit is given by
where $e$ is the current approximation to $\left|{\stackrel{~}{d}}_{k}\right|$. The values of the ${f}_{i}$ are systematically increased if the convergence criteria appear to be too strict. See Rutishauser (1970) for further details.
The algorithm implemented by f02fjf differs slightly from SIMITZ (see Nikolai (1979)) in that the eigenvalue sub-problem is solved using the singular value decomposition of the upper triangular matrix $R$ of the Gram–Schmidt factorization of $C{x}_{r}$, rather than forming ${R}^{\mathrm{T}}R$.
10Example
This example finds the four eigenvalues of smallest absolute value and corresponding eigenvectors for the generalized symmetric eigenvalue problem $Ax=\lambda Bx$, where $A$ and $B$ are the $16\times 16$ matrices
tol is taken as $0.0001$ and $6$ iteration vectors are used. f11jaf is used to factorize the matrix $A$, prior to calling f02fjf, and f11jcf is used within image to solve the equations $Aw=Bz$ for $w$.
Output from monit occurs each time istate is nonzero. Note that the required eigenvalues are the reciprocals of the eigenvalues returned by f02fjf.