NAG Library Routine Document
F02FJF
1 Purpose
F02FJF finds eigenvalues and eigenvectors of a real sparse symmetric or generalized symmetric eigenvalue problem.
2 Specification
| SUBROUTINE F02FJF ( | N, M, K, NOITS, TOL, DOT, IMAGE, MONIT, NOVECS, X, LDX, D, WORK, LWORK, RUSER, LRUSER, IUSER, LIUSER, IFAIL) |
| INTEGER | N, M, K, NOITS, NOVECS, LDX, LWORK, LRUSER, IUSER(LIUSER), LIUSER, IFAIL |
| double precision | TOL, DOT, X(LDX,K), D(K), WORK(LWORK), RUSER(LRUSER) |
| EXTERNAL | DOT, IMAGE, MONIT |
3 Description
F02FJF finds the
m eigenvalues of largest absolute value and the corresponding eigenvectors for the real eigenvalue problem
where
C is an
n by
n matrix such that
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
the routine finds the
m eigenvalues of largest absolute magnitude for the standard symmetric eigenvalue problem
The routine is intended for the case where
A is sparse.
As a second example, when
where
the routine finds the
m eigenvalues of largest absolute magnitude for the generalized symmetric eigenvalue problem
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
For instance, in the above example, where
C=B-1A,
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-1B 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.
Eigenvalues
Required |
Problem |
| |
Ax=λx B=I |
Ax=λBx |
ABx=λx |
| Largest |
Compute w=Az |
Solve Bw=Az |
Compute w=ABz |
| Smallest (Find 1/λ) |
Solve Aw=z |
Solve Aw=Bz |
Solve Av=z, Bw=v |
Furthest from σ
(Find λ-σ) |
Compute
w=A-σIz |
Solve Bw=A-σBz |
Compute
w=AB-σIz |
Closest to σ
(Find 1/λ-σ) |
Solve A-σIw=z |
Solve A-σBw=Bz |
Solve AB-σIw=z |
Table 1The Requirement of
IMAGE for Various Problems.
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
wTBz.
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≤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 kth 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 by r (r≤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.
4 References
Nikolai P J (1979) Algorithm 538: Eigenvectors and eigenvalues of real generalized symmetric matrices by simultaneous iteration
ACM Trans. Math. Software 5 118–125
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
5 Parameters
- 1: N – INTEGERInput
On entry:
n, the order of the matrix C.
Constraint:
N≥1.
- 2: M – INTEGERInput/Output
On entry: m, the number of eigenvalues required.
Constraint:
M≥1.
On exit:
m′, the number of eigenvalues actually found. It is equal to
m if
IFAIL=0 on exit, and is less than
m if
IFAIL=2,
3 or
4. See
Sections 6 and
8 for further information.
- 3: 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:
K=M+4 will often be a reasonable choice in the absence of better information.
Constraint:
M<K≤N.
- 4: NOITS – INTEGERInput/Output
On entry: the maximum number of major iterations (eigenvalue sub-problems) to be performed. If
NOITS≤0, the value
100 is used in place of
NOITS.
On exit: the number of iterations actually performed.
- 5: TOL – double precisionInput
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.
di is accepted as an eigenvalue as soon as two successive approximations to
di differ by less than
d~i×TOL/10, where
d~i is the latest approximation to
di. Once an eigenvalue has been accepted, an eigenvector is accepted as soon as
difi/di-dk<TOL, where
fi is the normalized residual of the current approximation to the eigenvector (see
Section 8 for further information). The values of the
fi and
di can be printed from
MONIT. If
TOL is supplied outside the range (
ε,1.0), where
ε is the
machine precision, the value
ε is used in place of
TOL.
- 6: DOT – double precision FUNCTION, supplied by the user.External Procedure
DOT must return the value
wTBz for given vectors
w and
z. For the standard eigenvalue problem, where
B=I,
DOT must return the dot product
wTz.
DOT is called from F02FJF with the parameters
RUSER,
LRUSER,
IUSER and
LIUSER as supplied to F02FJF. You are free to use the arrays
RUSER and
IUSER to supply information to
DOT and to
IMAGE as an alternative to using COMMON global variables.
The specification of
DOT is:
| double precision FUNCTION DOT ( | IFLAG, N, Z, W, RUSER, LRUSER, IUSER, LIUSER) |
| INTEGER | IFLAG, N, LRUSER, IUSER(LIUSER), LIUSER |
| double precision | Z(N), W(N), RUSER(LRUSER) |
- 1: IFLAG – INTEGERInput/Output
On entry: is always non-negative.
On exit: may be used as a flag to indicate a failure in the computation of
wTBz. 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: N – INTEGERInput
On entry: the number of elements in the vectors z and w and the order of the matrix B.
- 3: Z(N) – double precision arrayInput
On entry: the vector z for which wTBz is required.
- 4: W(N) – double precision arrayInput
On entry: the vector w for which wTBz is required.
- 5: RUSER(LRUSER) – double precision arrayUser Workspace
- 6: LRUSER – INTEGERInput
-
On entry: the first dimension of the array
RUSER as declared in the (sub)program from which F02FJF is called.
- 7: IUSER(LIUSER) – INTEGER arrayUser Workspace
- 8: LIUSER – INTEGERInput
-
On entry: the first dimension of the array
IUSER as declared in the (sub)program from which F02FJF is called.
DOT must be declared as EXTERNAL in the (sub)program from which F02FJF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 7: IMAGE – SUBROUTINE, supplied by the user.External Procedure
IMAGE must return the vector
w=Cz for a given vector
z.
IMAGE is called from F02FJF with the parameters
RUSER,
LRUSER,
IUSER and
LIUSER as supplied to F02FJF. You are free to use the arrays
RUSER and
IUSER to supply information to
IMAGE and
DOT as an alternative to using COMMON global variables.
The specification of
IMAGE is:
| SUBROUTINE IMAGE ( | IFLAG, N, Z, W, RUSER, LRUSER, IUSER, LIUSER) |
| INTEGER | IFLAG, N, LRUSER, IUSER(LIUSER), LIUSER |
| double precision | Z(N), W(N), RUSER(LRUSER) |
- 1: IFLAG – INTEGERInput/Output
On entry: is always non-negative.
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: N – INTEGERInput
On entry: n, the number of elements in the vectors w and z, and the order of the matrix C.
- 3: Z(N) – double precision arrayInput
On entry: the vector z for which Cz is required.
- 4: W(N) – double precision arrayOutput
On exit: the vector w=Cz.
- 5: RUSER(LRUSER) – double precision arrayUser Workspace
- 6: LRUSER – INTEGERInput
-
On entry: the first dimension of the array
RUSER as declared in the (sub)program from which F02FJF is called.
- 7: IUSER(LIUSER) – INTEGER arrayUser Workspace
- 8: LIUSER – INTEGERInput
-
On entry: the first dimension of the array
IUSER as declared in the (sub)program from which F02FJF is called.
IMAGE must be declared as EXTERNAL in the (sub)program from which F02FJF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 8: 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 parameters
ISTATE and
NEXTIT allow selective printing by
MONIT.
The specification of
MONIT is:
| SUBROUTINE MONIT ( | ISTATE, NEXTIT, NEVALS, NEVECS, K, F, D) |
| INTEGER | ISTATE, NEXTIT, NEVALS, NEVECS, K |
| double precision | F(K), D(K) |
- 1: ISTATE – INTEGERInput
On entry: specifies the state of F02FJF.
- ISTATE=0
- No eigenvalue or eigenvector has just been accepted.
- ISTATE=1
- One or more eigenvalues have been accepted since the last call to MONIT.
- ISTATE=2
- One or more eigenvectors have been accepted since the last call to MONIT.
- ISTATE=3
- One or more eigenvalues and eigenvectors have been accepted since the last call to MONIT.
- ISTATE=4
- Return from F02FJF is about to occur.
- 2: NEXTIT – INTEGERInput
On entry: the number of the next iteration.
- 3: NEVALS – INTEGERInput
On entry: the number of eigenvalues accepted so far.
- 4: NEVECS – INTEGERInput
On entry: the number of eigenvectors accepted so far.
- 5: K – INTEGERInput
On entry: k, the number of simultaneous iteration vectors.
- 6: F(K) – double precision arrayInput
On entry: a vector of error quantities measuring the state of convergence of the simultaneous iteration vectors. See
TOL and
Section 8 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: D(K) – double precision arrayInput
On entry: Di contains the latest approximation to the absolute value of the ith eigenvalue of C.
MONIT must be declared as EXTERNAL in the (sub)program from which F02FJF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 9: NOVECS – INTEGERInput
On entry: the number of approximate vectors that are being supplied in
X. If
NOVECS is outside the range
0,K, the value
0 is used in place of
NOVECS.
- 10: X(LDX,K) – double precision arrayInput/Output
On entry: if
0<NOVECS≤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
IFAIL=0,
2,
3 or
4, the first
m′ columns contain the eigenvectors corresponding to the eigenvalues returned in the first
m′ elements of
D; and the next
k-m′-1 columns contain approximations to the eigenvectors corresponding to the approximate eigenvalues returned in the next
k-m′-1 elements of
D. Here
m′ is the value returned in
M, the number of eigenvalues actually found. The
kth column is used as workspace.
- 11: LDX – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which F02FJF is called.
Constraint:
LDX≥N.
- 12: D(K) – double precision arrayOutput
On exit: if
IFAIL=0,
2,
3 or
4, the first
m′ elements contain the first
m′ eigenvalues in decreasing order of magnitude; and the next
k-m′-1 elements contain approximations to the next
k-m′-1 eigenvalues. Here
m′ is the value returned in
M, the number of eigenvalues actually found.
Dk contains the value
e where
-e,e is the latest interval over which Chebyshev acceleration is performed.
- 13: WORK(LWORK) – double precision arrayWorkspace
- 14: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F02FJF is called.
Constraint:
LWORK≥3×K+maxK×K,2×N.
- 15: RUSER(LRUSER) – double precision arrayUser Workspace
RUSER is not used by F02FJF, but is passed directly to user-supplied subroutines
DOT and
IMAGE and may be used to supply information to these routines.
- 16: LRUSER – INTEGERInput
On entry: the dimension of the array
RUSER as declared in the (sub)program from which F02FJF is called.
Constraint:
LRUSER≥1.
- 17: IUSER(LIUSER) – INTEGER arrayUser Workspace
IUSER is not used by F02FJF, but is passed directly to
DOT and
IMAGE and may be used to supply information to these routines.
- 18: LIUSER – INTEGERInput
On entry: the dimension of the array
IUSER as declared in the (sub)program from which F02FJF is called.
Constraint:
LIUSER≥1.
- 19: IFAIL – INTEGERInput/Output
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if
IFAIL≠0 on exit, the recommended value is
-1.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
6 Error Indicators and Warnings
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
- IFAIL<0
-
A negative value of
IFAIL indicates an exit from F02FJF because you have set
IFLAG negative in
DOT or
IMAGE. The value of
IFAIL will be the same as your setting of
IFLAG.
- IFAIL=1
-
| On entry, | N<1, |
| or | M<1, |
| or | M≥K, |
| or | K>N, |
| or | LDX<N, |
| or | LWORK<3×K+maxK×K,2×N, |
| or | LRUSER<1, |
| or | LIUSER<1. |
- IFAIL=2
-
Not all the requested eigenvalues and vectors have been obtained. Approximations to the
rth eigenvalue are oscillating rapidly indicating that severe cancellation is occurring in the
rth eigenvector and so
M is returned as
r-1. A restart with a larger value of
K may permit convergence.
- IFAIL=3
-
Not all the 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.
- IFAIL=4
-
Not all the requested eigenvalues and vectors have been obtained.
NOITS iterations have been performed. A restart, possibly with a larger value of
K, may permit convergence.
- IFAIL=5
-
This error is very unlikely to occur, but indicates that convergence of the eigenvalue sub-problem has not taken place. Restarting with a different set of approximate vectors may allow convergence. If this error occurs you should check carefully that F02FJF is being called correctly.
7 Accuracy
Eigenvalues and eigenvectors will normally be computed to the accuracy requested by the parameter
TOL, but eigenvectors corresponding to small or to close eigenvalues may not always be computed to the accuracy requested by the parameter
TOL. Use of the
MONIT to monitor acceptance of eigenvalues and eigenvectors is recommended.
8 Further 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
nk2. 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=λ 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
d~h, to the
ith eigenvalue, is accepted as soon as
d~h and the previous approximation differ by less than
d~h×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
dr is the last eigenvalue in such a group and we define the residual
rj as
where
yr is the projection of
Cxj, with respect to
B, onto the space spanned by
x1,x2,…,
xr, and
xj is the current approximation to the
jth eigenvector, then the value
fi returned in
MONIT is given by
and each vector in the group is accepted as an eigenvector if
where
e is the current approximation to
d~k. The values of the
fi 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
Cxr, rather than forming
RTR.
9 Example
This example finds the four eigenvalues of smallest absolute value and corresponding eigenvectors for the generalized symmetric eigenvalue problem
Ax=λBx, where
A and
B are the
16 by
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. Details of the factorization of
A are passed from
F11JAF to
F11JCF by means of the COMMON block BLOCK1.
Output from
MONIT occurs each time
ISTATE is nonzero. Note that the required eigenvalues are the reciprocals of the eigenvalues returned by F02FJF.
9.1 Program Text
Program Text (f02fjfe.f)
9.2 Program Data
Program Data (f02fjfe.d)
9.3 Program Results
Program Results (f02fjfe.r)